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8,098,752,880,805,181,000 | Multiplying and Dividing Fractions DIGITAL TASK CARDS BUNDLE
Multiplying and Dividing Fractions DIGITAL TASK CARDS BUNDLE
Grade Levels
Common Core Standards
Product Rating
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5 Products in this Bundle
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1. Fractions as Division Digital Task Cards *Make sure to check out the preview for a look at all 20 task cards! ♻♻♻♻GO PAPERLESS!♻♻♻♻ This product includes 2 digital activities: *Digital Task Cards using Google Slides *Automatic Grading Tool using Google Forms Digital Task Cards 21 Task Cards Incl
2. Multiplying Fractions DIGITAL TASK CARDS Updated 10/18/17 to include Google Forms Grading Tool! Thanks for your support and make sure to re-download to gain access to this great new feature! This product includes 2 digital activities: *Digital Task Cards using Google Slides *Automatic Grading Tool
3. Fraction Multiplication as Scaling Digital Task Cards *Make sure to check out the preview for a look at all 20 task cards! ♻♻♻♻GO PAPERLESS!♻♻♻♻ Updated 6/25/18 to include Google Forms Grading Tool! Thanks for your support and make sure to re-download to gain access to this great new feature! Th
4. Multiplying Mixed Numbers DIGITAL TASK CARDS ♻♻♻♻GO PAPERLESS!♻♻♻♻ Updated 6/25/18 to include Google Forms Grading Tool! Thanks for your support and make sure to re-download to gain access to this great new feature! This product includes 2 digital activities: *Digital Task Cards using Google Slid
5. Dividing Unit Fractions DIGITAL TASK CARDS ♻♻♻♻GO PAPERLESS!♻♻♻♻ Updated 6/25/18 to include Google Forms Grading Tool! Thanks for your support and make sure to re-download to gain access to this great new feature! This product includes 2 digital activities: *Digital Task Cards using Google Slides
Bundle Description
Multiplying and Dividing Fractions (5.NF.B Cluster) Digital Task Cards BUNDLE for use with Google Classroom
SAVE 25% PERCENT BY PURCHASING THESE PRODUCTS IN A BUNDLE!
These 5 sets of Google Slides and Google Forms Activities include over 100 total problems covering the 5th Grade Common Core Math Standards in the NBT.B Cluster.
Apply and extend previous understandings of multiplication and division.
CCSS.MATH.CONTENT.5.NF.B.3
Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
CCSS.MATH.CONTENT.5.NF.B.4
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
CCSS.MATH.CONTENT.5.NF.B.5
Interpret multiplication as scaling (resizing), by:
CCSS.MATH.CONTENT.5.NF.B.5.A
Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
CCSS.MATH.CONTENT.5.NF.B.5.B
Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.
CCSS.MATH.CONTENT.5.NF.B.6
Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
CCSS.MATH.CONTENT.5.NF.B.7
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
Total Pages
100+
Answer Key
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N/A
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8,347,598,749,771,175,000 | Weighted Moving Average
Weighted Moving Average (WMA) is one of the configurations of simple moving average which accounts not only for price values but also their weight.
Calculated as per formula:
weighted_moving_average1
where: Pi — price value for the number of i-periods, (today i =1),
Wi — weight value for price for the number of i-periods.
In simpler words, elements with an account of their values are summed and divided for the sum of weights of those elements, thus, generally speaking, arithmetical average of those elements is calculated.
It is accepted that weight changes according to linear function where W1 takes the largest weight and then calculation uses simple arithmetical progression, for instance: 1, 2, 3, 4, 5, 6...; (or any other: 0,5, 0,75, 1, 1,25). Such representation is called Linear Weighted Moving Average, (LWMA). Let's take period equal to 5:
weighted_moving_average2 ,
where: P1 и P2 — are the prices for today and yesterday.
Some configurations may use more complicated formula with non-linear distribution, involving logarithmic, parabolic and other functions, for example, if following is accounted:
- the number of ticks in bar;
- the length of passed distance in candle (High - Low)
- weight average against the distance; - the size of candle body (|Close - Open|).
Price can also be different: Close, Open, High, Low, Median Price, Typical Price.
Application of WMA
Example of Weighted Moving Average
Weighted Moving Average is usually applied in the same cases in which simple moving average is applied for technical analysis purposes. Though under similar entrance and exit market alerts LWMA responds to price change faster because weight is accounted for the latest periods. That allows not to miss lucky moments for entering the market during important economic news, interventions and other significant moves.
For stock market analysis it is recommended to use parameters equal to 7 and 14, for currency market – 5 and 20. As you can see on the image, the larger period is, the smoother moving average is and the bigger fluctuation range is has.
Sine-Weighted Moving Average ( SWMA) uses sine function during its calculation as weight (W). Thanks to SWMA, it is possible to filter noises, determine bottom and top with a higher precision.
Pros and cons of WMA
Due to taking in account weight of elements, WMA is more sensitive towards price change in contrast to simple moving average, which allows getting entrance and exit alerts faster. However, as any other MA, weight also has a certain delay.
It is better to apply it in short- and mid-term strategies, because the latest price changes has the biggest weight. In other words, at high time-frame WMA looks smoother because of low market noise and it does not provide such clear alerts.
WMA is more sensitive towards change of prices
WMA is better to apply in short- and mid-term strategies
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100,637,948,586,101,920 | The 99 Club
The 99 Club is a mental-oral starter at Thirsk Community Primary School which aims to raise standards in maths through encouraging pupils to improve their mental calculations when attempting quick-fire multiplication and division problems.
The idea is that with repeated practice, the scheme should result in increased speed and confidence when tackling mental maths problems without relying on written workings and methods.
All pupils will begin at the 11 Club and work their way up, having three opportunities per week during to answer all calculations at their current level unaided and within the allotted time of five minutes.
If all of the calculations are answered correctly, the child moves up to the next level!
The initial 11 Club involves eleven problems which involve doubling numbers up to ten ie. 5+5, 8+8. The 22 Club then adds eleven further questions involving repeated addition for numbers from one to ten ie. 3+3+3+3, 5+5+5+5+5, while the 33 Club begins to introduce times tables.
Division facts are added by the time a pupil reaches the 77 Club, and in the 88 Club and 99 Club, pupils will be tackling a range of mixed multiplication and division problems.
The full breakdown of The 99 Club levels is as follows:
11 Club - 11 questions involving doubling numbers from one to ten
22 Club - 22 questions involving repeated addition of numbers from one to ten
33 Club - 33 questions introducing the 2x, 3x, 5x and 10x tables
44 Club - 44 questions adding the 1x, 4x and 6x tables
55 Club - 55 questions adding the 7x and 8x tables
66 Club - 66 questions adding the 9x, 11x and 12x tables
77 Club - 77 questions consisting of inverse division facts
88 Club - 88 questions of mixed multiplication and division facts
99 Club - 99 questions of mixed multiplication and division facts
The ultimate challenge is to complete all 99 questions of the 99 Club unaided, with no errors and within five minutes!
The 99 Club is not designed for the Early Years Foundation Stage, but if staff feel that children are ready to attempt it, they will be given the opportunity to enter the 11 Club!
Sheets are available to download below for pupils to practise their level at home. You can support your child by finding out which level they are working at and helping them to practise the relevant times tables.
Some especially skilled mathematicians at Thirsk Community Primary may even make it on to the newly-introduced BronzeSilver and even Gold Clubs which include over one hundred problems to complete in the five minute, including some very tricky maths involving squares, square roots and cubed numbers! Gosh!
Good luck, and keep practising! smile
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9,143,754,141,922,515,000 | viernes, 24 de febrero de 2012
Resumen capitulo 2
Transformaciones geométricas.
Habitualmente un paquete gráfico permite al usuario especificar que parte de una imagen definida se debe de visializar y donde esta parte se debe colocar en el dispositivo del visualización. Se compone por coodenadas.
Transformaciones bidimensionales.
Traslacion:En un objeto para cambiar su posicion a lo largo de la trayectoria de una linea recta de una direccion de coordenadas a otra. Asi que se convierte al agregar distancias de traslacion t y t con (x’ = x + tx’) o (y’ = y + ty), en ocasiones las ecucaciones de transformación matricianal se expresan en terminos de vectores en renglon de coordenadas en vez de vectores de columna.
Los poligonos se trasladas al sumar el vector de traslacion a la posición de coordenadas.
Rotación: Se aplica una rotación bidimensional en un objeto al cambiar su posición a lo largo de la trayectoria de una circunferencia en el plano de xy, para generar una rotación especificamos un angulo y la posición del punto de rotación en torno el cual se girara el objeto
Eslacion: Es una transformación de escalacion la cual altera el tamaño de un objeto. Se puede realizar esta operación para polígonos al multiplicar los valores de coordenadas de cada vértice por los factores de escalacion para producir coordenadas transformadas.
Coordenadas homogéneas y representación matricial
En las aplicaciones de diseño y de creación de imágenes, realizamos traslaciones, rotaciones y escalaciones para ajustar los componentes de la imagen en sus posiciones apropiadas. En este tema consideramos cómo se pueden volver a formular las representaciones de la matriz de modo que se pueden procesar de manera eficiente esas secuencias de transformación.
Composición de transformaciones bidimensionales.
Con las representaciones de matriz del tema anterior, podemos establecer una matriz para cualquier secuencia de transformaciones como una matriz de transformación compuesta al calcular el producto de la matriz de las transformaciones individuales.
P y P’ como vectores de columna de coordenadas homogéneas. Podemos verificar este resultado al calcular el producto de la matriz para las dos agrupaciones asociativas. Asimismo, la matriz de transformación compuesta para esta secuencia de transformaciones es:
Formula equivalente para lograr la rotación
Formula equivalente para lograr la escalacion.
Rotacion del punto pivote general
Se necesitan seguir 3 pequeños pasos a gran escala para desarrollarla.
1. Traslade el objeto de modo que se mueva la posición del punto pivote al origen de las coordenadas.
2. Gire el objeto con respecto del origen de las coordenadas.
3. Traslade el objeto de manera que se regrese el punto pivote a su posición original.
Escalacion del punto fijo general
Son una conjunto de secuencias las cuales se especifican en el siguiente dibujo.
1. Traslade el objeto de modo que el punto fijo coincida con el origen de las coordenadas.
2. Escale el objeto con respecto del origen de las coordenadas
3. Utilice la traslación inversa del paso 1 para regresar el objeto a su posición original.
Transformación ventana-área de vista.
Algunos paquetes gráficos permiten que el programador especifique coordenadas de primitivas de salida en un sistema de coordenadas de mundo de punto flotante, usando las unidades que sean relevantes para el programa de aplicación: angstroms, micras, metros, millas, años luz, etcétera. Se emplea el término de mundo porque el programa de aplicación representa un mundo que se crea o presenta interactivamente para el usuario.
Transformaciones de composición general y de eficiencia computacional.
Una transformación bidimensional general, que representa una combinación de traslaciones rotaciones y escalaciones se puede expresar como una matriz de 3x3.
Así como las transformaciones bidimensionales se pueden representar con matrices de 3x3 usando coordenadas homogéneas, las transformaciones tridimensionales se pueden representar con matrices de 4x4, siempre y cuando usemos representaciones de coordenadas homogéneas de los puntos en el espacio tridimensional.
Así, en lugar de representar un punto como (x,y,z), lo hacemos como (x, y, z, W), donde dos de estos cuádruplos representan el mismo punto si uno es un multiplicador distinto de cero del otro.
Las transformaciones geométricas son transformaciones afines. Esto es, pueden expresarse como una función lineal de posiciones de coordenadas. Traslación, rotación y escalación son transformaciones afines. Transforman líneas paralelas en líneas paralelas y posiciones de coordenadas finitas en posiciones finitas.
Representación matricial de transformación tridimensionales.
Así como las transformaciones bidimensionales se pueden representar con matrices de3 X 3 usando coordenadas homogéneas, las transformaciones tridimensionales se puedenrepresentar con matrices de 4 X 4, siempre y cuando usemos representaciones decoordenadas homogéneas de los puntos en el espacio tridimensional. Así, en lugar derepresentar un punto como (
x, y, z ), lo hacemos como (x, y, z, W ), donde dos de estoscuádruplos representan el mismo punto si uno es un multiplicador distinto de cero del otro:no se permite el cuádruplo (0, 0, 0, 0). Como sucede en el espacio bidimensional, larepresentación estándar de un punto (x, y, z, W ) con W ≠
0 se indica (x/W, y/W, z/W, 1).
La transformación de un punto a esta forma se denomina homogeneización, igual queantes. Además los puntos cuya coordenada W es cero se llaman puntos en el infinito.También existe una interpretación geométrica. Cada punto en el espacio tridimensional serepresenta con una línea que pasa por el origen en el espacio de cuatro dimensiones, y lasrepresentaciones homogeneizadas de estos puntos forman un subespacio tridimensional deun espacio de cuatro dimensiones definido por la ecuación
W = 1.
El sistema de coordenadas tridimensionales que se usará en los siguientes ejemplos esde mano derecha como se ilustra en la figura 2.16- Por convención las rotaciones positivasen el sistema de mano derecha son tales que, al ver hacia un eje positivo desde el origen,una rotación de 90˚ en sentido contrario al giro de las manecillas del reloj transformará uneje positivo en otro.
Composición de transformaciones tridimensionales
En este apartado se analizará la forma de componer matrices de transformación tridimensionales usando un ejemplo. El objetivo es transformar los segmentos de línea dirigida P1 P2 y P1 P3 de su posición inicial en la parte (a) a su posición finalen la parte (b). De esta manera, el punto P1 se trasladará al origen P1 P2 quedará en el ejepositivo y P1 P3 quedará en la mitad del eje positivo del plano (x, y). Las longitudes de laslíneas no se verán afectadas por la transformación.Se presentan dos formas de lograr la transformación deseada. El primer método escomponer las transformaciones primitivas T,Rx, R y y Rz. Este método, aunque es algotedioso, es fácil de ilustrar y su comprensión nos ayudará en nuestro conocimiento de las transformaciones. El segundo método, que utiliza las propiedades de las matrices ortogonales especiales que se analiza en la sección anterior, se explica de manera mas breve pero es más abstracto.
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} | 0c090d63199a0a01e3b08e4a255778a0 |
839,154,417,353,597,400 | [HOME] MAM2000 (Dimension) [Prev][Up][Next]
Dot Product in Three Dimensions
Geometrically, we know that two vectors are perpendicular if the Pythagorean Theorem holds, i.e. the square of the length of (a,b,c) plus the square of the length of (x,y,z) equals the square of the length of (a,b,c) - (x,y,z) = (a-x,b-y,c-z). This means that a2 + b2 + c2 + x2 + y2 + z2 = (a-x)2 + (b-y)2 + (c-z)2 = a2 - 2ax + x2 + b2 - 2by + y2 + c2 - 2cz + z2. From this it follows that 0 = -2ax - 2by - 2cz, so ax + by + cz = 0. Thus two vectors in R3 are perpendicular if and only if their dot product is zero.
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533,278,101,286,126,700 | Find answers, ask questions, and connect with our
community around the world.
Activity Discussion Math Maths
Tagged: ,
• Aashutosh
Member
May 30, 2021 at 10:46 pm
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::
A circle is a set of all those points that lie in a plane that is equidistant from a given point called “center”. It forms a closed two-dimensional figure.
The important basic properties of circles are listed below:
1. The outer line of a circle is equidistant from the center and is called the radius.
2. The diameter of the circle divides the circle into two equal parts.
3. Circles that have equal radii are said to be congruent to each other.
4. Circles that are different in size or having different radii are similar to each other.
5. Diameter of a circle is twice the radius of the circle.
6. The diameter of the circle is the largest chord of that circle.
7. Equal chords and equal circles have the equal circumference
8. The radius drawn perpendicular to the chord bisects the chord
9. A circle can circumscribe a square, trapezium, rectangle, triangle, and kite.
10. A circle can be inscribed in a square, triangle, and kite.
11. The chords that are equidistant from the center are equal in length.
12. The distance from the center of the circle to the longest chord (diameter) is zero
14. The tangents are parallel to each other if they are drawn at the end of the diameter.
15. An isosceles triangle is formed when the radii joining the ends of a chord to the center of a circle.
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2,506,685,838,132,140,500 | Unexpectedly Intriguing!
February 6, 2007
Although we don't go there often, we're not ones to shy away from personal topics here at Political Calculations. We are, after all, the only blog out there that gets into your paycheck, goes into your house to see if you should switch to compact fluorescents and helps you figure out how much diet soda your system can safely handle.
But now we're getting really personal with our latest tool, adapted from math posted by Geek Logik author Garth Sundem at his blog, asking the question "What are the chances your marriage will last?"
While the Geek Logik blog post contains three separate equations for helping decide various marital topics (the other two answer the questions "should we get married" and "how many kids should we have"), we were intrigued by the statistics that underlie the question of marital sustainability. Here's what Garth wrote about the data:
... the first is based on solid statistics -- an 11,000-person study by the CDC that expolored factors that help and hurt a marriage's chances of working (for example, they found that if a woman is married before age 24, her chances of staying married for 15 years decreased by 30%). These statistics were easy to write in math terms, and the equation does fairly accurately predict your chances of being married at time "T". Granted there are other factors that might help or hurt your specific marriage, but the CDC study found that, for most people, these are the biggest factors. Remember that the average for all marriages is only about 50% and if you get a low number, please accept my very best wishes in bucking the odds.
There's not much more than to go straight to the math, captured in our tool below:
Personal Data
Input Data Values
Her Age at Time of Marriage
Current Combined Years of Post-High School Education
Number of Kids from This Marriage
How Religious is the Couple?
(On a scale of 1-10 with 10 being "the Pope")
Combined Number of Divorces of Couple's Parents
Combined Previous Marriages
The Anniversary (Years of Marriage) for Which to Calculate the Probability
Will You Still Be Married?
Calculated Results Values
Probability at Given Year of Anniversary
Having coded the math, let's reassure you that the result isn't processed through any sort of normal probability distribution. It is, at best, an approximation. Just change the default "religiousness" value to 10 (aka "the pope") and you'll get a better than 100% probability level! Aside from these quirks of math however, you'll still be able get a somewhat realistic approximation of the odds that you'll be married for your "Xth" anniversary over a pretty wide range of the distribution curve.
Now that you've seen the generic probability that you'll still be married to your current spouse at the anniversary of your marriage that you entered, you may have more questions than answers. If the probability is really low, that might be a good place to begin a conversation with your spouse. If the probability is high, you may already have a strong foundation for a successful marriage. Just remember that it never hurts to make it stronger.
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4,605,250,094,556,951,600 | Plateforme N°1 de soutien en mathématique Post-bac Prépa
Résolution du problème suivant :
Bases mathématiques-Opérations-entre-ensembles-Partitions-MPSI
Ali Mkhida
Ali Mkhida
Dr. Agrégé en Mathématique & fondateur de Qoosmo.
Chapitre mentionné dans l'article :
Bases mathématiques-Opérations-entre-ensembles-Partitions-MPSI
Partitions
Définition
Bases mathématiques-Opérations-entre-ensembles-Partitions-MPSI
Une famille $\left(F_i\right)_{i \in I}$ de parties de $E$ est une partition de $E$ si
– Aucune partie n’est vide
$$
\forall i \in I, \quad F_i \neq \varnothing .
$$
– Deux parties distinctes sont disjointes
$$
\forall(i, j) \in I^2, \quad i \neq j \Rightarrow F_i \cap F_j=\varnothing .
$$
– La réunion est égale à $E$
$$
\cup_{i \in I} F_i=E .
$$
Exemple
La famille $\left(\left[n, n+1[)_{n \in \mathbb{Z}}\right.\right.$ est une partition de $\mathbb{R}$ : les parties sont non vides, disjointes et de réunion $\mathbb{R}$. Il s’agit de partitionner l’ensemble des réels selon leur partie entière.
Exemple
Les parties $2 \mathbb{Z}$ et $\{2 k+1, k \in \mathbb{Z}\}$ forment une partition de $\mathbb{Z}$ (selon la parité).
En fait, ces exemples relèvent d’un cadre plus général que nous détaillons maintenant.
Définition
Une relation d’équivalence sur un ensemble $E$ est une relation binaire $\sim$ qui vérifie les propriétés suivantes:
– est réflexive, c’est-à-dire, tout élément $x \in E$ est en relation avec lui-même pour :
$$
\forall x \in E, \quad x \sim x .
$$
– est symétrique, c’est-à-dire pour tous les éléments $x$ et $y \in E$ tels que $x$ est en relation avec $y$, on a aussi $y$ en relation avec $x$
$$
\forall x, y \in E, \quad x \sim y \quad \Leftrightarrow \quad y \sim x .
$$
– est transitive, c’est-à-dire pour tous les éléments $x, y$ et $z \in E$ tels que $x$ est en relation avec $y$ et $y$ est en relation avec $z$, on a aussi $x$ en relation avec $z$
$$
\forall x, y, z \in E, \quad x \sim y \text { et } y \sim z \quad \Rightarrow \quad x \sim z .
$$
Deux éléments en relation sont dits équivalents. La classe d’équivalence d’un élément $x$ pour la relation $\sim$ est l’ensemble des éléments de $E$ équivalents à $x$ pour $\sim$, à savoir
$$
\{y \in E, y \sim x\} .
$$
Exemple
Congruence sur les entiers
Soit $p \in \mathbb{N} \backslash\{0\}$. Deux entiers $m$ et $n$ sont congruents modulo $p$ si $p$ divise $m-n$. On note alors $m \equiv n[p]$ ou plus simplement $m=n[p]$ en gardant à l’esprit que ce signe d’égalité n’est pas une véritable égalité.
On vérifie immédiatement que la congruence modulo $p$ est une relation d’équivalence sur $\mathbb{Z}$ et que la classe d’équivalence de $a$ est
$$
\{a+b p, b \in \mathbb{Z}\}
$$
Généralisons l’exemple précédent aux réels (où, rappelons-le, la notion de divisibilité est peu pertinente puisque tout réel non nul divise tous les réels).
Exemple
Deux réels $x$ et $y$ sont congruents modulo $2 \pi$ si la différence $x-y$ est un multiple entier de $2 \pi$. On définit ainsi une relation d’équivalence sur $\mathbb{R}$ (le choix de $2 \pi$ correspond à une utilisation courante pour les fonctions trigonométriques, on peut bien évidemment choisir n’importe quel autre réel non nul).
Proposition
Soit une relation d’équivalence sur un ensemble $E$. Les classes d’équivalence pour $\sim$ forment une partition de $E$.
Démonstration
Il est clair qu’une classe d’équivalence est non vide et que $E$ est inclus dans la réunion des classes d’équivalence (car chaque élément est inclus dans sa propre classe d’équivalence). Montrons que deux classes d’équivalence distinctes sont disjointes. Pour cela, considérons $x$ et $y \in E$ tels que les classes d’équivalence de $x$ et $y$ ne soient pas disjointes et $z \in E$ à la fois équivalent à $x$ et $y$. Alors, par transitivité, la classe de $x$ est la classe de $z$; de même, la classe de $y$ est la classe de $z$. Ainsi, les classes de $x$ et de $y$ sont égales.
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6,445,490,058,772,371,000 | being a math tutor
Right now in my geometry class, I have an A+, but friend is failing that class. I offered to be her math tutor, to possibly meet in the library after school 3 days a week. Does anybody have any ideas on how to make our study sessions more productive (I haven't started yet -- maybe sometime next week)?
1. 👍
2. 👎
3. 👁
1. First, I suggest you talk with her geometry teacher who can probably offer you some good advice about how to help your friend. Both of you should go in together so that the teacher knows that s/he has your friend's consent to share that information with you.
Use as many visuals as possible. I think one of the problems that some students have with geometry is that they have trouble visualizing the shapes.
You could start each session by working together on problems she doesn't understand on that day's assignment. You could work through the first problem together. Then ask her to tackle the second problem while you stop her if she goes astray.
1. 👍
2. 👎
👤
Ms. Sue
2. Thank you very much for the advice! This will help me very much.
:-)
1. 👍
2. 👎
3. Good luck! I'm sure you'll be able to help your friend.
1. 👍
2. 👎
👤
Ms. Sue
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5,496,841,960,242,996,000 | Mandelbrot ... but with no main cardioid!
• 6 Replies
• 572 Views
0 Members and 1 Guest are viewing this topic.
Offline quadralienne
• *
• Fractal Fanatic
• ***
• Posts: 26
• infinite border, finite area
« on: June 20, 2019, 09:34:19 PM »
Once upon a time I stumbled across
https://commons.wikimedia.org/wiki/File:Parameter_plane_and_Mandelbrot_set_for_f(z)_%3D_z%5E4_%2B_m*z.png
and I worked through the C code (languages without native complex numbers make me sad) to convert it to a MathMap expression:
unit filter JustCircles ()
Mmax = 4096;
Escape = 3;
Escape2 = Escape * Escape;
if r < 1.0 || abs(ri:[x-4/3,y]) < 1/3 || abs(ri:[x+4/3*cos(pi/3),abs(y)-4/3*sin(pi/3)]) < 1/3 then
rgba:[0,0,0,1]
else
Mc = ri:[x,y];
ang = atan(-y,-x)/3.0;
rad = (0.0625 * (y*y+x*x)) ^ (1/6);
Mz = ri:[rad * cos(ang), rad * sin(ang)];
Mcount = 0;
while ((Mz[0]*Mz[0]+Mz[1]*Mz[1]) < Escape2) && (Mcount < Mmax) do
Mcount = Mcount + 1;
Mz = Mz ^ 4 + Mz * Mc;
end;
if Mcount < Mmax then
dist = log( Mcount + 1 - (log(log(abs(Mz)))/log(Escape)) ) / log(Mmax);
hue = pmod(arg(Mz)/(2*pi), 1);
sat = (1 - dist) ^ 2;
val = sqrt(dist);
toRGBA(hsva:[ hue, sat, val, 1 ])
else
rgba:[0,0,0,1]
end
end
end
filter JustCircles_p ()
JustCircles(xy:[0.25 + 2.5 * x, 2.5 * y])
end
Linkback: https://fractalforums.org/share-a-fractal/22/mandelbrot-but-with-no-main-cardioid/2890/
Offline Adam Majewski
• *
• Fractal Flamingo
• ****
• Posts: 330
« Reply #1 on: June 21, 2019, 04:19:23 PM »
You are right that c has also complex type and in new programs I use it.
Offline chronologicaldot
• *
• Fractal Friend
• **
• Posts: 10
• Unconventional Formulaic Object
• Personal Website
« Reply #2 on: July 29, 2019, 10:07:43 PM »
That's really cool! It's hard to tell whether it truly follows the Mandelbrot pattern on the tips or if it's now a bunch of successively shrinking circles.
There are no bad fractal parameters. There are simply those that haven't been tweaked enough.
Offline Adam Majewski
• *
• Fractal Flamingo
• ****
• Posts: 330
Offline 3DickUlus
• *
• 3f
• ******
• Posts: 1724
• Digilantism
« Reply #4 on: July 31, 2019, 05:16:00 AM »
https://upload.wikimedia.org/wikipedia/commons/0/03/Parameter_plane_and_Mandelbrot_set_for_f%28z%29_%3D_z%5E4_%2B_m%2Az.png
You can use BBC code for image URLs with weird characters and for wiki images use the image URL not the image page url ;)
Code: [Select]
[img]https://upload.wikimedia.org/wikipedia/commons/0/03/Parameter_plane_and_Mandelbrot_set_for_f%28z%29_%3D_z%5E4_%2B_m%2Az.png[/img]
[url=https://upload.wikimedia.org/wikipedia/commons/0/03/Parameter_plane_and_Mandelbrot_set_for_f%28z%29_%3D_z%5E4_%2B_m%2Az.png]https://upload.wikimedia.org/wikipedia/commons/0/03/Parameter_plane_and_Mandelbrot_set_for_f%28z%29_%3D_z%5E4_%2B_m%2Az.png[/url]
« Last Edit: July 31, 2019, 05:55:29 AM by 3DickUlus, Reason: typo »
Offline quadralienne
• *
• Fractal Fanatic
• ***
• Posts: 26
• infinite border, finite area
« Reply #5 on: August 02, 2019, 10:48:13 PM »
It's not just circles, it's Mandelbrotty all over!
Offline quadralienne
• *
• Fractal Fanatic
• ***
• Posts: 26
• infinite border, finite area
« Reply #6 on: August 02, 2019, 11:21:43 PM »
Here's an 8192x zoom at 1.98835 + 0.183i ... uh, fuzzy because single precision!
xx
Upload limits per day for Main and User Galleries?
Started by Anon on Discuss Fractalforums
13 Replies
799 Views
Last post September 22, 2017, 07:02:49 AM
by claude
question
Upload limits per day for Main and User Galleries (Sticky Topic/Post)
Started by Anon on Discuss Fractalforums
3 Replies
360 Views
Last post February 17, 2018, 08:13:59 PM
by 3DickUlus
xx
Mandelbrot Burning Ship Mandelbrot Mandelbrot hybrid 3
Started by claude on Fractal Image Gallery
0 Replies
377 Views
Last post January 17, 2018, 12:26:38 AM
by claude
xx
Mandelbrot Burning Ship Mandelbrot Mandelbrot hybrid
Started by claude on Fractal Image Gallery
0 Replies
358 Views
Last post January 16, 2018, 11:30:30 PM
by claude
xx
Mandelbrot Burning Ship Mandelbrot Mandelbrot hybrid 2
Started by claude on Fractal Image Gallery
0 Replies
314 Views
Last post January 17, 2018, 12:10:56 AM
by claude
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-6,792,524,150,457,260,000 | Resultado de 3(2x+1)-4=11
Solución simple y rápida para la ecuación 3(2x+1)-4=11. Nuestra respuesta es comprensible y explicada paso a paso.
Si no es lo que está buscando, escriba sus propios datos.
Resultado de 3(2x+1)-4=11:
3(2x+1)-4=11
Movemos todos los personajes a la izquierda:
3(2x+1)-4-(11)=0
Sumamos todos los números y todas las variables.
3(2x+1)-15=0
Multiplicar
6x+3-15=0
Sumamos todos los números y todas las variables.
6x-12=0
Movemos todos los términos que contienen x al lado izquierdo, todos los demás términos al lado derecho
6x=12
x=12/6
x=2
El resultado de la ecuación 3(2x+1)-4=11 para usar en su tarea doméstica.
Ver soluciones similares:
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-970,508,963,681,632,000 | mersenneforum.org
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Old 2009-06-13, 19:11 #1
ATH
Einyen
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Default Mersenne primes have highly composite p-1?
http://www.mersenneforum.org/showpos...6&postcount=85
Quote:
Originally Posted by akruppa View Post
Maybe there's a simple explanation why Mersenne primes M[I]p[/I] tend to have highly composite p-1. Trivially, 2p-1 and 2p-2 have no common factor, and the latter is 2(2p-1-1) which has a lot of algebraic factors if p-1 is highly composite, and so will have a lot of small prime factors, thus slightly reducing the probability that 2p-1 has small prime factors. The effect can't be very strong, though: divisors of 2p-1 are of form 2kp+1 and few factors of 2p-1-1 will be of this form. Still, it's the only thing I can think of how the number of divisors of p-1 might enter the picture.
If this effect is the reason for the smoother-than-expected p-1 in prime M[I]p[/I], then M[I]p[/I] with smooth p-1should simply have a slightly better chance of surviving trial division, but among the trial-divided candidates, the probability that M[I]p[/I] is prime should be independent of the smoothness of p-1 again.
Alex
http://www.mersenneforum.org/showpos...&postcount=345
Quote:
Originally Posted by akruppa View Post
I have no idea how to quantify this. An empirical test is the best I can think of. Only relatively small divisors should be affected, so one might check if those 2[I]p[/I] where p-1 has at least n divisors are more likely to survive trial division to, say, 240.
Alex
I took the list from 20M to 30M from the "Factoring limits" list, which is all those that has no known factor below 2^66 (20M) to 2^68 (30M).
There are 587,252 primes from 20M to 30M and of them 369,166 have known factors while 218,086 are in the factoring limit list and have no known factors. I made a small program to test number of factors in p-1 for all 587,252 primes and see if there was a difference:
Code:
p-1 factors A B
---------------------------------------------
2 factors 15617=4.23% 8278=3.80%
3 factors 52822=14.31% 29439=13.50%
4 factors 81916=22.19% 47299=21.69%
5 factors 80449=21.79% 48040=22.03%
6 factors 59649=16.16% 36367=16.68%
7 factors 37016=10.03% 22935=10.51%
8 factors 20575=5.57% 12596=5.78%
9 factors 10774=2,92% 6684=3.06%
10 factors 5394=1.46% 3265=1.50%
11 factors 2571=0.70% 1686=0.77%
12 factors 1280=0.35% 784=0.36%
13 factors 612=0.17% 369=0.17%
14 factors 265=0.07% 169=0.08%
15 factors 127 92
16 factors 57 39
17 factors 28 22
18 factors 7 8
19 factors 5 5
20 factors 0 6
21 factors 1 1
22 factors 1 1
23 factors 1 1
----------------------------------------------
Total 369166(100%) 218086(100%)
Column A are the exponents where 2^p-1 has factors below 2^66-2^68, and B where 2^p-1 have no factors below 2^66-2^68. Looking at the percentages there is no real difference in number of exponents with higher number of factors of p-1 in column B. Maybe we have to check at a lower factor level like 2^40 like Alex suggests.
Last fiddled with by ATH on 2009-06-13 at 19:19
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Old 2009-06-13, 20:04 #2
akruppa
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I rearranged a little to show the survival rate of exponents p depending on the number of factors of p-1:
Code:
p-1 factors #p #survivors #suvival rate
-----------------------------------------------------
2 factors 23895 8278 0.346
3 factors 82261 29439 0.358
4 factors 129215 47299 0.366
5 factors 128489 48040 0.374
6 factors 96016 36367 0.379
7 factors 59951 22935 0.383
8 factors 33171 12596 0.380
9 factors 17458 6684 0.383
10 factors 8659 3265 0.377
11 factors 4257 1686 0.396
12 factors 2064 784 0.380
13 factors 981 369 0.376
14 factors 434 169 0.389
So it looks like those p with few factors in p-1 do, in fact, have a lower chance of surviving trial division. ATH, I assume the number of factors is the number of prime factors with multiplicity in p-1? It might be interesting to make such a table for the number of proper divisors of p-1 as well.
My hypothesis isn't very convincing, though. By the same argument, 2[I]p[/I]-4 and 2[I]p[/I]-1 have at most the factor 3 in common, so the number of divisors in p-2 (and p-3 and p-4 etc.) should also affect the probability that Mp is prime.
Alex
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Old 2009-06-14, 23:04 #3
ATH
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Quote:
Originally Posted by akruppa View Post
So it looks like those p with few factors in p-1 do, in fact, have a lower chance of surviving trial division. ATH, I assume the number of factors is the number of prime factors with multiplicity in p-1? It might be interesting to make such a table for the number of proper divisors of p-1 as well.
If you mean distinct prime factors, here is the list:
Code:
p-1 factors Total(20M-30M) numbers without factors to 2^66-2^68
--------------------------------------------------------------------
2 factors 49855 17960 = 36.02%
3 factors 173824 63988 = 36.81%
4 factors 218645 81127 = 37.10%
5 factors 118143 44684 = 37.83%
6 factors 25228 9692 = 38.42%
7 factors 1548 629 = 40.63%
8 factors 9 6 (=66.67%)
--------------------------------------------------------------------
Total 587252 218086
There is a clear rising percentage of numbers "surviving" trialfactor to 2^66-2^68, the more distinct prime factors p-1 has.
If you mean all factors (not just prime factors) then the list is extensive, here is whole list (not counting 1 and p-1 as factors of p-1): mersennetest.txt
Here is the list abbriviated by combining the factor-categories with low number of members in them:
Code:
p-1 factors Total(20M-30M) numbers without factors to 2^66-2^68
--------------------------------------------------------------------
2 factors 23895 8278 = 34.64%
4 factors 12264 4577 = 37.32%
6 factors 76473 27329 = 35.74%
7-8 factors 3355 1230 = 36.66%
10 factors 47504 17911 = 37.70%
12-13 factors 985 376 = 38.17%
14 factors 98842 35686 = 36,10%
16 factors 5377 2038 = 37.90%
18 factors 10600 3991 = 37.65%
19-22 factors 68947 25980 = 37.68%
23-26 factors 2651 946 = 35.68%
28 factors 1955 753 = 38.52%
30 factors 66115 24718 = 37.39%
31-34 factors 13651 5150 = 37.73%
36-38 factors 12384 4728 = 38,18%
40-46 factors 49363 18813 = 38.11%
48-54 factors 3795 1435 = 37.81%
58 factors 4090 1563 = 38.22%
61-62 factors 24317 9256 = 38.06%
64-70 factors 12353 4747 = 38.43%
73-78 factors 6840 2622 = 38.33%
79-94 factors 18451 6969 = 37.77%
96-106 factors 1528 577 = 37.76%
108-110 factors 1234 465 = 37.68%
118 factors 2806 1091 = 38.88%
124-126 factors 4652 1819 = 39.10%
128-142 factors 4558 1761 = 38.64%
148-158 factors 1635 634 = 38.78%
160-178 factors 951 361 = 37.96%
180-190 factors 2707 1076 = 39.75%
194-214 factors 659 272 = 41.27%
218-254 factors 1294 508 = 39.26%
258-286 factors 546 230 = 42.12%
292-318 factors 147 58 = 39.46%
322-358 factors 141 57 = 40.43%
376-382 factors 108 43 = 39.81%
394-430 factors 48 28 = 58.33%
446-478 factors 20 7 = 35.00%
502-574 factors 11 3 = 27.27%
--------------------------------------------------------------------
Total 587252 218086 (=37.14%)
The trend is not so clear here, since there is so many categories with more or less members in. But 39+% happens only for >124 factors and 40+% only for >194factors.
Last fiddled with by ATH on 2009-06-14 at 23:11
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Old 2009-06-15, 13:11 #4
ATH
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Combined the categories on the last list even more (the one with all factors of p-1 except 1 and p-1):
Code:
p-1 factors Total(20M-30M) numbers without factors to 2^66-2^68
--------------------------------------------------------------------
2-6 factors 112632 40184 = 35.68%
7-13 factors 51844 19517 = 37.65%
14-18 factors 114819 41715 = 36.33%
19-22 factors 68947 25980 = 37.68%
23-30 factors 70721 26417 = 37.35%
31-46 factors 75398 28691 = 38.05%
48-62 factors 32202 12254 = 38.05%
64-94 factors 37644 14338 = 38.09%
96-142 factors 14778 5713 = 38.66%
148-254 factors 7246 2851 = 39.35%
258-382 factors 942 388 = 41.19%
394-574 factors 79 38 = 48.10%
--------------------------------------------------------------------
Total 587252 218086 (=37.14%)
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-998,444,846,774,835,700 | For the Students of Hindu Vedic Astrology by Dr. A. Shanker
Recent Posts
20130205
Encyclopedia of Vedic Astrology: Tajik Shastra and Annual Horoscopy: The Muntha, Chapter XI, Part - 1
Dr. Shanker Adawal
The Muntha
Part 1
1. The Muntha & its Progression: The Muntha is an important mathematical concept in Varshphal connecting the annual chart with the birth chart. At the time of birth, it is located in lagna/ Ascendant. Each year, the Muntha moves one rashi (sign) in direct motion. When the second year of life begins, the Muntha falls in the 2nd house from lagna. At the 3rd year of life, the Muntha will be in the 3rd house from lagna and so on. Because of its progression by one sign each year, the Muntha is also termed as the progressed ascendant. In the analysis of an annual chart, the Muntha is extremely important. The lord of Muntha is also very significant while analyzing an annual chart and is one of the five contenders for the post of lord of the year Since the Muntha progresses one sign or 30 degrees in a year, it progresses every month by 30/ 12 = 2.5 or 2 degree 30 min. Similarly, daily motion of the Muntha may also be calculated by dividing 2 degree 30 min by 30 (number of days in a month), giving a value of 5 min. These values are important if one intends to go into closer timing of events during a given year.
2. Calculation of the Muntha:
1. Note the number of sign wherein the lagna falls in the birth chart. Say if lagna is in Gemini, the number of sign is 3.
2. Add to it the number of completed years elapsed between the birth and the current year for which the Muntha is to be calculated.
3. Divide the total by 12 (total numbers of signs). The remainder so obtained would be the sign in which the Muntha is located in the annual chart.
4. If the remainder is zero, it should be treated as 12 sign or Pisces.
Based on the above, the Muntha in the example chart No. 21 of previous chapter would be in 3rd house of Aries and in chart No 2, in 2nd house of Sagittarius.
3. The Results of Muntha in different Houses
A well-placed and well-aspected Muntha strengthens the house it is in. The Muntha gives results according to its location in different houses, its association with different planets, and the disposition of the lord of the sign in which the Muntha is located. (This is some what similar to the position of dispositor in Parashari system). The general results are based on the “Tajika Neelkanthi”, the famous treatise by Acharya Neelkanth. The Muntha is considered very auspicious in houses 9, 10 & 11. In houses 1, 2, 3, & 5 it yields good results through personal efforts of the native. In the remaining houses 4, 6, 7, 8, & 12 in the annual chart, it is considered inauspicious. However experience shows that in 7th house, the Muntha does not give bad results except illness to the native. The general results of positioning of the Muntha in various houses are as under:
1. Lagna: It is considered to yield very good results and indicates dominance/ victory over opponents, dignity, favours from rulers, high status or a new job/ source of income, increase in power, comforts & money and good health through own efforts. It may also denote change in place, position or residence, transfer or the birth of a child.
Chart No. 29: PV Narsimha Rao: Born on Tuesday, 28 Jun 1921 at 1-02 PM at Warangal with Virgo Lagna; 70th annual chart for 1990-91
The Muntha is in lagna in own house with Mercury, Lagnesh of birth & annual chart and with benefics. The native became Prime Minister of India on 21 Jun 1991. The sudden & unexpected event was due to mutual aspect with 8-9th lord Saturn. However the Saturn’s aspect on Muntha and Muntha lord also gave illness in first part of the year.
Continue…
Shanker Adawal
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-4,610,528,528,350,969,300 | Slow Sum – Revisited
Good morning. I received a comment from Dmytro regarding the post Slow Sum suggesting the use of a priority queue instead of a stream. I appreciate the comment and suggestion.
Suppose we have a list of N numbers,
and repeat the following operation until we're left with only a single number:
Choose any two numbers and replace them with their sum.
Moreover, we associate a penalty with each operation equal to the value of the new number,
and call the penalty for the entire list as the sum of the penalties of each operation.
For example, given the list [1, 2, 3, 4, 5],
we could choose 2 and 3 for the first operation,
which would transform the list into [1, 5, 4, 5] and incur a penalty of 5.
The goal in this problem is to find the worst possible penalty for a given input.
Input:
An array arr containing N integers, denoting the numbers in the list.
Output format:
An int representing the worst possible total penalty.
Constraints:
o 1 ≤ N ≤ 10^6
o 1 ≤ Ai ≤ 10^7, where *Ai denotes the ith initial element of an array.
o The sum of values of N over all test cases will not exceed 5 * 10^6.
The description of the problem is the same as far as I can tell. I just copied it from the previous post.
1,2,3,4,5
main <<< arr: [1, 2, 3, 4, 5]
main <<< output: 50
4,2,1,3
main <<< arr: [4, 2, 1, 3]
main <<< output: 26
The test codes are the same. This time I did a screen capture using the implementation of the function of interest using a priority queue. The two test cases return the same values.
/**
* Test scaffolding
*
* @throws IOException
*/
public static void main(String[] args) throws IOException {
// **** open buffered reader ****
BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
// **** read input line and split values ****
String[] strs = br.readLine().trim().split(",");
// **** close buffered reader ****
br.close();
// **** create and populate array of integers ****
int[] arr = Arrays.stream(strs).mapToInt(Integer::parseInt).toArray();
// ???? ????
System.out.println("main <<< arr: " + Arrays.toString(arr));
// **** call function and display resuly ****
System.out.println("main <<< output: " + getTotalTime(arr));
}
Our test scaffold reads the single input line holding the values for the input array to the function of interest. The contents of the int[] array are displayed to make sure all is well so far.
The function of interest is called and the result is displayed.
/**
* Using a stream.
*
* Execution O(n log(n)) - Space: O(n)
*/
static int getTotalTime0(int[] arr) {
// **** sanity check(s) ****
if (arr.length == 1) return 0;
// ???? ????
System.out.println("<<< arr:: " + Arrays.toString(arr));
// **** sort array in descending order - O(n log(n)) ****
int[] rev = Arrays.stream(arr)
.boxed()
.sorted(Comparator.reverseOrder())
.mapToInt(Integer::intValue)
.toArray();
// ???? ????
System.out.println("<<< rev:: " + Arrays.toString(rev));
// **** initialization ****
int penalty = rev[0] + rev[1];
int penalties = penalty;
// **** loop counting penalties - O(n) ****
for (int i = 2; i < rev.length; i++) {
// **** generate penalty ****
penalty += rev[i];
// **** add penalty ****
penalties += penalty;
}
// **** return penalties ****
return penalties;
}
This is the previous implementation of the function of interest using Arrays.stream to sort the input array `arr` and generate a new int[] `rev` in which the values are sorted in monotonically descending order.
Please take a look at the comments section of the function in which the execution and space orders are listed.
/**
* Using a priority queue instead of a stream.
*
* Execution: O(n) - Space: O(n)
*/
static int getTotalTime(int[] arr) {
// **** sanity check(s) ****
if (arr.length == 1) return 0;
// **** initialization - O(n * log(n)) ****
PriorityQueue<Integer> rev = new PriorityQueue<>(arr.length, (a,b) -> b - a);
for (int i : arr) rev.add(i);
int penalty = rev.poll() + rev.poll();
int penalties = penalty;
// **** loop counting penalties - O(n) ****
while (!rev.isEmpty()) {
// **** generate penalty ****
penalty += rev.poll();
// **** add penalty ****
penalties += penalty;
}
// **** return penalties ****
return penalties;
}
This is the new implementation of the function of interest using a priority queue.
We start by performing a sanity check. There is no reason to proceed if the number of integers in the input int[] `arr` is 1.
We then initialize a priority queue with an initial capacity that matches the size of the input array and a comparator that will allow us to pull items in monotonically decreasing order. We then insert into the priority queue all the elements in the `arr`.
We then declare and initialize the `penalty` and `penalties` variables. We could have used a single variable but I used two to ease debugging by displaying variables during execution. I guess that at this point in the game I could have modified the code to use a single variable, but the code is in a different machine and I already pushed it to GitHub.
A loop is then entered. The loop will consume all the entries in the priority queue while updating the `penalties` value.
When all is said and done, our function returns the value in the `penalties` variable.
Hope you enjoyed solving this problem in a different way as suggested by Dmytro as much as I did. The entire code for this project can be found in my GitHub repository.
Please note that the code here presented might not be the best possible solution. In most cases, after solving the problem, you can take a look at the best accepted solution provided by the different websites (i.e., HackerRank, LeetCode). Since this problem came from a Facebook website, the number of test cases is limited. I believe in this case only two test cases were used to verify the solution.
If you have comments or questions regarding this, or any other post in this blog, please do not hesitate and leave me a note below. I will reply as soon as possible.
Keep on reading and experimenting. It is one of the best ways to learn, become proficient, refresh your knowledge and enhance your developer / engineering toolset.
Thanks for reading this post, feel free to connect with me John Canessa at LinkedIn.
Enjoy;
John
Leave a Reply
Your email address will not be published. Required fields are marked *
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Find the values of k so that lines are perpendicular using symetric equations
• Thread starter soulja101
• Start date
• #1
62
0
Homework Statement
Line 1: x-3/3k+1=Y+6/2=Z+3/2K
Line 2: x+7/3=y+8/-2k=z+9/-3
Homework Equations
Cross product and dot product
The Attempt at a Solution
vector equation for line 1: (x,y,z)=(4,-4,-3)+k(3,0,2)
vector equation for line 2: (x,y,z)=(-4,-8,-12)+k(0,-2,0)
The product of (3,0,2)and (0,-2,0) is zero so they are perpendicular but i don't know how to find k :confused:
Answers and Replies
• #2
HallsofIvy
Science Advisor
Homework Helper
41,795
925
Homework Statement
Line 1: x-3/3k+1=Y+6/2=Z+3/2K
a) Use parentheses: you mean, I think (x-3)/(3k+1)= (y+6)/2= (z+ 3)/2k
b) Don't use y and Y or k and K for the same thing. Those are different symbols and typically mean different values.
Setting each of those equal to t, (x-3)/(3k+1)= t so x- 3= (3k+1)t or x= 3+ (3k+1)t, (y+6)/2= t so y+ 6= 2t or y= -6+ 2t, and (z+3)/2k= t so z+ 3= 2kt or z= -3+ 2kt
Line 2: x+7/3=y+8/-2k=z+9/-3
(x+7)/3= t so x+ 7= 3t or x= -7+ 3t, (y+8)/(-2k)= t so y+8= -2kt or y= -8- 2kt, and (z+9)/(-3)= t so z+ 9= -3t or z= -9- 3t.
Homework Equations
Cross product and dot product
The Attempt at a Solution
vector equation for line 1: (x,y,z)=(4,-4,-3)+k(3,0,2)
No, that is not an equation for line 1. For one thing, k is given in the symmetric equation: one value of k corresponds to one line so without another parameter, this would be just a single point, not a line.
vector equation for line 2: (x,y,z)=(-4,-8,-12)+k(0,-2,0)
Same comment.
The product of (3,0,2)and (0,-2,0) is zero so they are perpendicular but i don't know how to find k :confused:
You don't have equations for the lines. As I said above your vector equation should be
(x, y, z)= (-7, -8, -3)+ t(3k+1, 2, 2k)t. The parameter t determines the point, k is fixed for a line.
For line 2, (x, y, z)= (-7, -8, -9)+ t(3, -2k, -3).
In order that the lines be perpendicular you must have (3k+1, 2, 2k).(3, -2k, -3)= 9k+ 3- 4k- 6k= -k+ 3= 0.
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-2,012,722,032,228,191,000 | Wednesday
May 4, 2016
Homework Help: smallest of 3 integers
Posted by Anonymous on Wednesday, June 27, 2012 at 1:15pm.
The sum of the reciprocals of three consecutive positive integers is equal to 47 divided by the product of the integers. What is the smallest of the three integers?
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-237,155,132,277,577,250 | login
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A121523 Number of up steps starting at an even level in all nondecreasing Dyck paths of semilength n. A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing. 2
1, 3, 10, 33, 103, 315, 941, 2770, 8051, 23171, 66138, 187486, 528365, 1481501, 4135756, 11500721, 31871625, 88054825, 242609585, 666783380, 1828452021, 5003697403, 13667302500, 37267071708, 101455834153, 275797332135 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) = Sum(k*A121522(n,k), k=1..n). a(n)+A121525(n)=n*fibonacci(2n-1).
LINKS
Table of n, a(n) for n=1..26.
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
FORMULA
G.f.: z(1-3z+z^2+5z^3-5z^4)/[(1+z)(1-3z+z^2)^2*(1-z-z^2)].
a(n) ~ (5-sqrt(5)) * (3+sqrt(5))^n * n / (5 * 2^(n+2)). - Vaclav Kotesovec, Mar 20 2014
EXAMPLE
a(3)=10 because we have (U)D(U)D(U)D, (U)D(U)UDD, (U)UDD(U)D, (U)UDUDD and (U)U(U)DDD, the up steps starting at even level being shown between parentheses (U=(1,1), D=(1,-1)).
MAPLE
G:=z*(1-3*z+z^2+5*z^3-5*z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): Gser:=series(G, z=0, 34): seq(coeff(Gser, z, n), n=1..30);
MATHEMATICA
Rest[CoefficientList[Series[x*(1-3*x+x^2+5*x^3-5*x^4)/(1+x)/(1-3*x+x^2)^2 /(1-x-x^2), {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)
CROSSREFS
Cf. A001519, A121522, A121525.
Sequence in context: A316411 A292549 A062454 * A115240 A027989 A096483
Adjacent sequences: A121520 A121521 A121522 * A121524 A121525 A121526
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 05 2006
STATUS
approved
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834,479,223,463,730,200 | 18.100B.PracticeFinal
18.100B.PracticeFinal - 18.100B/C Practice Final Exam...
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Unformatted text preview: 18.100B/C Practice Final Exam Monday, December 15, 2008, 1:30–4:30, in Johnson. Closed book, no calculators. YOUR NAME: This is a 180-minute exam. No notes, books, or calculators are permitted. Point values (out of 100) are indicated for each problem. There is a (hard) bonus question, Problem 9, at the end – do not attempt it until you have worked all other problems. (Note, you can achieve the full 100 points without attempting the bonus problem.) Do all the work on these pages. GRADING 1. /10 2. /10 3. /10 4. /15 5. /10 6. /10 7. /15 8. /20 9. /20 TOTAL BONUS /100 1 Problem 1. [10 points] Suppose that x ∈ R satisfies ≤ x ≤ for every > . Show that x = 0 , using only axioms of R as an ordered field. State the axioms you are using. (Note that the Archimedean and least upper bound properties are not ordered field axioms.) 2 Problem 2. [10 points: (a) /5 (b) /5] Let ( a n ) be a sequence of positive real numbers. (a) Suppose that the series ∞ X n =1 a n converges. Prove that ∞ X n =1 √ a n a n +1 also converges.also converges....
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This note was uploaded on 12/07/2011 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.
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18.100B.PracticeFinal - 18.100B/C Practice Final Exam...
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Unit Converter
Convert 146 Acres to Square Feet
To calculate 146 Acres to the corresponding value in Square Feet, multiply the quantity in Acres by 43560 (conversion factor). In this case we should multiply 146 Acres by 43560 to get the equivalent result in Square Feet:
146 Acres x 43560 = 6359760 Square Feet
146 Acres is equivalent to 6359760 Square Feet.
How to convert from Acres to Square Feet
The conversion factor from Acres to Square Feet is 43560. To find out how many Acres in Square Feet, multiply by the conversion factor or use the Area converter above. One hundred forty-six Acres is equivalent to six million three hundred fifty-nine thousand seven hundred sixty Square Feet.
Definition of Acre
The acre (symbol: ac) is a unit of land area used in the imperial and US customary systems. It is defined as the area of 1 chain by 1 furlong (66 by 660 feet), which is exactly equal to 1⁄640 of a square mile, 43,560 square feet, approximately 4,047 m2, or about 40% of a hectare. The most commonly used acre today is the international acre. In the United States both the international acre and the US survey acre are in use, but differ by only two parts per million, see below. The most common use of the acre is to measure tracts of land. One international acre is defined as exactly 4,046.8564224 square metres.
Definition of Square Foot
The square foot (plural square feet; abbreviated sq ft, sf, ft2) is an imperial unit and U.S. customary unit (non-SI, non-metric) of area, used mainly in the United States and partially in Bangladesh, Canada, Ghana, Hong Kong, India, Malaysia, Nepal, Pakistan, Singapore and the United Kingdom. It is defined as the area of a square with sides of 1 foot. 1 square foot is equivalent to 144 square inches (Sq In), 1/9 square yards (Sq Yd) or 0.09290304 square meters (symbol: m2). 1 acre is equivalent to 43,560 square feet.
Using the Acres to Square Feet converter you can get answers to questions like the following:
• How many Square Feet are in 146 Acres?
• 146 Acres is equal to how many Square Feet?
• How to convert 146 Acres to Square Feet?
• How many is 146 Acres in Square Feet?
• What is 146 Acres in Square Feet?
• How much is 146 Acres in Square Feet?
• How many ft2 are in 146 ac?
• 146 ac is equal to how many ft2?
• How to convert 146 ac to ft2?
• How many is 146 ac in ft2?
• What is 146 ac in ft2?
• How much is 146 ac in ft2? | {
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Acceleration as a function of displacement
1. Nov 13, 2014 #1
In one my classes my lecturer showed us the following derivation of acceleration as a function of displacement
dv/dt = v(dv/dx) = d(.5v^2)/dx
I understand how to get from the first to the second part. But I'm not sure how he got from the second part to the third. Its almost like he integrated v dv on the right hand side?
Any help would be appreciated.
2. jcsd
3. Nov 13, 2014 #2
ShayanJ
User Avatar
Gold Member
[itex]\frac{dv}{dt}=v\frac{dv}{dx}=\frac{1}{2}2v\frac{dv}{dx}=\frac{d}{dx}(\frac{1}{2} v^2)[/itex]
4. Nov 13, 2014 #3
Oh so you're really just reversing the chain rule
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"eai_open_web_math": 0.5778378844261169,
"eai_web_code": -0.000007990000085555948
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} | {
"free_decimal_correspondence": {
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"level_3": "Mechanics"
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"primary": {
"code": "0",
"label": "No missing content"
},
"secondary": {
"code": "4",
"label": "Missing Images or Figures"
}
},
"document_type_v2": {
"primary": {
"code": "18",
"label": "Q&A Forum"
},
"secondary": {
"code": "23",
"label": "Tutorial"
}
},
"reasoning_depth": {
"primary": {
"code": "2",
"label": "Basic Reasoning"
},
"secondary": {
"code": "3",
"label": "Intermediate Reasoning"
}
},
"technical_correctness": {
"primary": {
"code": "4",
"label": "Highly Correct"
},
"secondary": {
"code": "3",
"label": "Mostly Correct"
}
},
"education_level": {
"primary": {
"code": "3",
"label": "Undergraduate Level"
},
"secondary": {
"code": "2",
"label": "High School Level"
}
}
} | 0c090d63199a0a01e3b08e4a255778a0 |
-3,400,955,877,140,714,500 | Number 38500
[ thirty-eight thousand five hundred ]
Properties of number 38500
Cross Sum:
Factorization:
2 * 2 * 5 * 5 * 5 * 7 * 11
Divisors:
1, 2, 4, 5, 7, 10, 11, 14, 20, 22, 25, 28, 35, 44, 50, 55, 70, 77, 100, 110, 125, 140, 154, 175, 220, 250, 275, 308, 350, 385, 500, 550, 700, 770, 875, 1100, 1375, 1540, 1750, 1925, 2750, 3500, 3850, 5500, 7700, 9625, 19250, 38500
Count of divisors:
Sum of divisors:
Prime number?
No
Fibonacci number?
No
Bell Number?
No
Catalan Number?
No
Base 2 (Binary):
Base 3 (Ternary):
Base 4 (Quaternary):
Base 5 (Quintal):
Base 8 (Octal):
Base 16 (Hexadecimal):
Base 32:
15j4
sin(38500)
0.21624785196742
cos(38500)
-0.97633849996785
tan(38500)
-0.2214886045921
ln(38500)
10.558413520276
lg(38500)
4.5854607295085
sqrt(38500)
196.21416870349
Square(38500)
Number Look Up
Look Up
38500 which is pronounced (thirty-eight thousand five hundred) is a very unique figure. The cross sum of 38500 is 16. If you factorisate the figure 38500 you will get these result 2 * 2 * 5 * 5 * 5 * 7 * 11. 38500 has 48 divisors ( 1, 2, 4, 5, 7, 10, 11, 14, 20, 22, 25, 28, 35, 44, 50, 55, 70, 77, 100, 110, 125, 140, 154, 175, 220, 250, 275, 308, 350, 385, 500, 550, 700, 770, 875, 1100, 1375, 1540, 1750, 1925, 2750, 3500, 3850, 5500, 7700, 9625, 19250, 38500 ) whith a sum of 104832. The figure 38500 is not a prime number. The figure 38500 is not a fibonacci number. 38500 is not a Bell Number. The number 38500 is not a Catalan Number. The convertion of 38500 to base 2 (Binary) is 1001011001100100. The convertion of 38500 to base 3 (Ternary) is 1221210221. The convertion of 38500 to base 4 (Quaternary) is 21121210. The convertion of 38500 to base 5 (Quintal) is 2213000. The convertion of 38500 to base 8 (Octal) is 113144. The convertion of 38500 to base 16 (Hexadecimal) is 9664. The convertion of 38500 to base 32 is 15j4. The sine of 38500 is 0.21624785196742. The cosine of the number 38500 is -0.97633849996785. The tangent of the number 38500 is -0.2214886045921. The root of 38500 is 196.21416870349.
If you square 38500 you will get the following result 1482250000. The natural logarithm of 38500 is 10.558413520276 and the decimal logarithm is 4.5854607295085. that 38500 is special figure! | {
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} | 0c090d63199a0a01e3b08e4a255778a0 |
End of preview. Expand
in Data Studio
🧮 Taxonomy Math w/ FM
A high-quality mathematics dataset curated from web data using taxonomy-based filtering, containing 34 billion tokens of mathematical content.
🎯 Dataset Overview
This dataset is part of the Essential-Web project, which introduces a new paradigm for dataset curation using expressive metadata and simple semantic filters. Unlike traditional math datasets that require complex domain-specific pipelines, our approach leverages a 12-category taxonomy to efficiently identify and extract high-quality mathematical content.
🔬 Taxonomy Math w/ FM (34B tokens): Documents labeled as 51 - Mathematics in our taxonomy, with all 116M recalled documents then scored by the FineMath classifier and filtered to the top 34B tokens.
🏆 Performance
Our taxonomy-based approach achieves competitive results with significantly less curation effort:
| Dataset | GSM8K | MATH | MMLU-Math | Curation Complexity |
|---|---|---|---|---|
| FineMath 3+ | 26.4% ± 1.4 | 11.7% ± 0.4 | 32.3% ± 1.5 | Complex domain pipeline |
| OpenWebMath | 14.6% ± 1.1 | 9.3% ± 0.4 | 29.9% ± 1.5 | Complex domain pipeline |
| MegaMath Web (Top 10%) | 9.8% ± 0.9 | 7.9% ± 0.3 | 29.9% ± 1.5 | Complex domain pipeline |
| DCLM-baseline | 4.8% ± 0.7 | 4.4% ± 0.3 | 27.0% ± 1.4 | Standard baseline |
| EAI-Taxonomy Top Math | 21.3% ± 1.3 | 11.0% ± 0.4 | 30.5% ± 1.5 | Simple semantic filter |
| EAI-Taxonomy Math w/ FM | 22.4% ± 1.3 | 11.5% ± 0.4 | 30.9% ± 1.5 | + FineMath classifier |
Results show our EAI-Taxonomy datasets perform within 15% of SOTA on GSM8K while requiring minimal domain-specific tuning. On MATH and MMLU-Math, EAI-Taxonomy Math w/ FM performs within standard error of the leading FineMath 3+ dataset.
✨ Key Features
- 🎯 Direct Distribution Targeting: Leverage existing taxonomy labels to target math content from web-scale data without training custom high-recall classifiers
- 🚀 Rapid Curation: Skip the expensive classifier training phase and go straight to content selection
- 💰 Cost Effective: Avoid the need to train high-recall domain-specific classifiers for content discovery
- 🔍 Two-Stage Approach: Use taxonomy for recall, then apply existing quality classifiers for selection
- 🌐 Web-Scale: Access to math content identified across 23.6B web documents
🛠️ Curation Method
Our approach simplifies math dataset creation:
- Traditional Method: Train high-recall classifiers → Run on billions of documents
- Our Method: Query taxonomy metadata for
51 - Mathematics→ Apply FineMath classifier to all recalled documents → Select top-scoring content
Dataset Schema Documentation
Overview
This dataset contains web-crawled text data with comprehensive metadata, quality signals, and taxonomic classifications. Each record represents a document extracted from web archives with detailed provenance tracking and quality assessment metrics.
Core Fields
| Field | Type | Description | Path |
|---|---|---|---|
id |
Int64 |
Unique identifier based on document hash | id |
text |
String |
The main textual content of the document | text |
EAI Taxonomy Classification
Comprehensive hierarchical classification system with primary and secondary labels - the most important feature of this dataset. The taxonomy is designed to provide detailed subject categorization, document type identification, content quality assessment, and extraction quality indicators.
Free Decimal Correspondence (FDC)
A Dewey Decimal-inspired classification system with 3-level hierarchical labels. The FDC provides nested categories where each successive level refines its parent category. It's designed to be compatible with the Dewey Decimal System for library cataloging.
Level Structure:
- Level 1: Top-level categories (0-9) covering broad subject areas like General works, Philosophy, Religion, Social Sciences, etc.
- Level 2: Sub-divisions (00-99) that refine Level 1 categories
- Level 3: Specific categories (000-999) that further refine Level 2 categories
| Component | Description | Path |
|---|---|---|
| Primary Code | Main classification code | eai_taxonomy.free_decimal_correspondence.primary.code |
| Primary Level 1 | Top-level category (0=General works, 1=Philosophy, 2=Religion, 3=Social Sciences, 4=Language, 5=Science, 6=Technology, 7=Arts, 8=Literature, 9=History/Geography) | eai_taxonomy.free_decimal_correspondence.primary.labels.level_1 |
| Primary Level 2 | Mid-level category | eai_taxonomy.free_decimal_correspondence.primary.labels.level_2 |
| Primary Level 3 | Specific category | eai_taxonomy.free_decimal_correspondence.primary.labels.level_3 |
| Secondary Code | Alternative classification code | eai_taxonomy.free_decimal_correspondence.secondary.code |
| Secondary Level 1 | Alternative top-level category | eai_taxonomy.free_decimal_correspondence.secondary.labels.level_1 |
| Secondary Level 2 | Alternative mid-level category | eai_taxonomy.free_decimal_correspondence.secondary.labels.level_2 |
| Secondary Level 3 | Alternative specific category | eai_taxonomy.free_decimal_correspondence.secondary.labels.level_3 |
We recommend this viewer for easily navigating the FDC categories when curating filters: https://www.librarything.com/mds
Bloom's Taxonomy Integration
Based on Anderson and Krathwohl's 2001 revision of Bloom's Taxonomy of Educational Objectives, providing two complementary categorization dimensions for educational content analysis.
Knowledge Domain
Categorizes the type of knowledge demonstrated in the document:
| Component | Description | Path |
|---|---|---|
| Primary Code | Main knowledge domain code | eai_taxonomy.bloom_knowledge_domain.primary.code |
| Primary Label | Main knowledge domain label | eai_taxonomy.bloom_knowledge_domain.primary.label |
| Secondary Code | Alternative knowledge domain code | eai_taxonomy.bloom_knowledge_domain.secondary.code |
| Secondary Label | Alternative knowledge domain label | eai_taxonomy.bloom_knowledge_domain.secondary.label |
Possible Values:
| Code | Label | Description |
|---|---|---|
-1 |
Abstain | Unable to determine |
1 |
Factual | Basic elements to learn or solve problems |
2 |
Conceptual | Interrelationships between basic elements within larger context |
3 |
Procedural | Methods and techniques in the discipline |
4 |
Metacognitive | Awareness of how learning works in relation to oneself |
Cognitive Processing Level
Assesses the learning and thinking skill levels demonstrated by the document author:
| Component | Description | Path |
|---|---|---|
| Primary Code | Main cognitive process code | eai_taxonomy.bloom_cognitive_process.primary.code |
| Primary Label | Main cognitive process label | eai_taxonomy.bloom_cognitive_process.primary.label |
| Secondary Code | Alternative cognitive process code | eai_taxonomy.bloom_cognitive_process.secondary.code |
| Secondary Label | Alternative cognitive process label | eai_taxonomy.bloom_cognitive_process.secondary.label |
Possible Values:
| Code | Label | Description |
|---|---|---|
-1 |
Abstain | Unable to determine |
1 |
Remember | Retrieve relevant knowledge from memory |
2 |
Understand | Determine meaning of instructional messages |
3 |
Apply | Use a procedure in a given situation |
4 |
Analyze | Break materials into components and determine relationships |
5 |
Evaluate | Make judgments based on criteria and standards |
6 |
Create | Create new or original work |
Document Characteristics
Document Type v1
In-house classification of common web document types and formats:
| Component | Description | Path |
|---|---|---|
| Primary Code | Main document type code | eai_taxonomy.document_type_v1.primary.code |
| Primary Label | Main document type label | eai_taxonomy.document_type_v1.primary.label |
| Secondary Code | Alternative document type code | eai_taxonomy.document_type_v1.secondary.code |
| Secondary Label | Alternative document type label | eai_taxonomy.document_type_v1.secondary.label |
Possible Values:
| Code | Label | Examples |
|---|---|---|
-1 |
Abstain | Unable to classify |
1 |
News/Editorial | CNN articles, opinion columns |
2 |
Academic/Research | ArXiv papers, research articles |
3 |
Reference/Encyclopedic/Educational | FAQs, Wikipedia entries |
4 |
Code/Software | GitHub repos, code examples |
5 |
Social/Forum | Conversation threads, Q&A boards |
6 |
Promotional/Advertisement | Product pages, calls to action |
7 |
Search/Directory/Bibliography | Link pages, search results |
8 |
Adult/Pornographic | Adult content |
9 |
Personal/Misc | Blogs, user profiles |
10 |
Machine-Generated | Lorem ipsum, garbled text |
11 |
Legal/Regulatory | Contracts, terms of service |
12 |
Government/Political | Legislation, press releases |
13 |
Literary/Creative | Poems, short stories |
14 |
Reviews/Critiques | Film critiques, product reviews |
15 |
E-Commerce/Marketplace | eBay listings, Amazon pages |
16 |
Images/Videos/Audio | YouTube videos, Imgur pages |
17 |
Other/Unclassified | Documents that resist classification |
Document Type v2
Updated classification based on WebOrganizer taxonomy with refined categories for improved document classification accuracy:
| Component | Description | Path |
|---|---|---|
| Primary Code | Main document type code (v2) | eai_taxonomy.document_type_v2.primary.code |
| Primary Label | Main document type label (v2) | eai_taxonomy.document_type_v2.primary.label |
| Secondary Code | Alternative document type code (v2) | eai_taxonomy.document_type_v2.secondary.code |
| Secondary Label | Alternative document type label (v2) | eai_taxonomy.document_type_v2.secondary.label |
Complete Value Mapping:
| Code | Label | Examples |
|---|---|---|
-1 |
Abstain | Documents requiring human review |
1 |
About (Org.) | Company about pages, mission statements |
2 |
About (Personal) | Personal bios, LinkedIn profiles |
3 |
Academic Writing | Research papers, abstracts, dissertations |
4 |
Audio Transcript | Interview transcripts, court records, captions |
5 |
Comment Section | Reddit threads, blog comments |
6 |
Content Listing | Site maps, product catalogs, directory listings |
7 |
Creative Writing | Song lyrics, novel excerpts, poetry |
8 |
Documentation | API docs, README files, user manuals |
9 |
FAQ | FAQ pages, Q&A lists |
10 |
Knowledge Article | Wikipedia articles, Britannica entries |
11 |
Legal Notices | Privacy policies, license agreements, terms of service |
12 |
Listicle | Buzzfeed-style articles, "Top 10" lists |
13 |
News (Org.) | Government blog posts, corporate announcements |
14 |
News Article | Newspaper articles, CNN content, breaking news |
15 |
Nonfiction Writing | Editorials, obituaries, memoirs, opinion pieces |
16 |
Personal Blog | Personal journals, diary entries, lifestyle blogs |
17 |
Product Page | Product descriptions, course offerings, sales pages |
18 |
Q&A Forum | Quora posts, Stack Exchange discussions |
19 |
Spam / Ads | SEO keyword stuffing, promotional spam |
20 |
Structured Data | Datasheets, glossaries, JSON files, databases |
21 |
Customer Support | Help articles, troubleshooting guides |
22 |
Truncated | Paywalled sites, image galleries, partial content |
23 |
Tutorial | Cooking recipes, WikiHow pages, step-by-step guides |
24 |
User Review | Yelp reviews, TripAdvisor feedback, product reviews |
25 |
Other/Unclassified | Miscellaneous documents not fitting other categories |
Extraction Artifacts
Assessment of technical extraction quality, identifying issues from HTML-to-text conversion:
| Component | Description | Path |
|---|---|---|
| Primary Code | Main extraction artifact code | eai_taxonomy.extraction_artifacts.primary.code |
| Primary Label | Main extraction artifact label | eai_taxonomy.extraction_artifacts.primary.label |
| Secondary Code | Alternative extraction artifact code | eai_taxonomy.extraction_artifacts.secondary.code |
| Secondary Label | Alternative extraction artifact label | eai_taxonomy.extraction_artifacts.secondary.label |
Possible Values:
| Code | Label | Description |
|---|---|---|
-1 |
Abstain | Unable to determine |
0 |
No Artifacts | Clean text with no leftover HTML or irrelevant elements |
1 |
Leftover HTML | HTML/code artifacts remaining after extraction |
2 |
Text Extraction Errors | Broken math expressions, encoding errors, improperly parsed tables |
3 |
Irrelevant Content | Headers, footers, nav menus extracted by mistake |
4 |
Indeterminate | Insufficient content to judge |
Missing Content
Assessment of content completeness and extraction success:
| Component | Description | Path |
|---|---|---|
| Primary Code | Main missing content code | eai_taxonomy.missing_content.primary.code |
| Primary Label | Main missing content label | eai_taxonomy.missing_content.primary.label |
| Secondary Code | Alternative missing content code | eai_taxonomy.missing_content.secondary.code |
| Secondary Label | Alternative missing content label | eai_taxonomy.missing_content.secondary.label |
Possible Values:
| Code | Label | Description |
|---|---|---|
-1 |
Abstain | Unable to determine |
0 |
No Missing Content | Complete and coherent text |
1 |
Truncated Snippets | Obvious "...", incomplete paragraphs, cut-off text |
2 |
Click Here References | "Download here", "Click here" without linked content |
3 |
Incoherent Flow | Unreadable or illogical flow due to missing context |
4 |
Missing Images or Figures | Placeholders or references to missing visual content |
5 |
Missing Referenced Data | References to absent tables/datasets (e.g., "See Table 3") |
6 |
Indeterminate | Insufficient content to judge |
Text Structure Information
| Field | Type | Description | Path |
|---|---|---|---|
| Line Start Indices | List[Int32] |
Starting indices of each line | line_start_n_end_idx.line_start_idx |
| Line End Indices | List[Int32] |
Ending indices of each line | line_start_n_end_idx.line_end_idx |
Content Quality Dimensions
Quality assessment inspired by NaturalReasoning and FineWeb efforts to categorize web data by information sophistication.
Reasoning Depth
Assesses the complexity and sophistication of logical reasoning in the document:
| Component | Description | Path |
|---|---|---|
| Primary Code | Main reasoning depth code | eai_taxonomy.reasoning_depth.primary.code |
| Primary Label | Main reasoning depth label | eai_taxonomy.reasoning_depth.primary.label |
| Secondary Code | Alternative reasoning depth code | eai_taxonomy.reasoning_depth.secondary.code |
| Secondary Label | Alternative reasoning depth label | eai_taxonomy.reasoning_depth.secondary.label |
Possible Values:
| Code | Label | Description |
|---|---|---|
-1 |
Abstain | Unable to determine |
1 |
No Reasoning | Facts present but no evidence of reasoning |
2 |
Basic Reasoning | Basic analysis with minimal explanation and summarization |
3 |
Intermediate Reasoning | Some logical steps connecting ideas and structured thinking |
4 |
Advanced Reasoning | Multi-step reasoning and thorough analysis with well-developed explanations |
5 |
Exceptional Reasoning | Novel abstractions, theoretical frameworks, long chain-of-thought, original insights, or proofs |
6 |
Indeterminate | Insufficient context to judge |
Technical Correctness
Evaluates the accuracy and precision of technical information:
| Component | Description | Path |
|---|---|---|
| Primary Code | Main technical correctness code | eai_taxonomy.technical_correctness.primary.code |
| Primary Label | Main technical correctness label | eai_taxonomy.technical_correctness.primary.label |
| Secondary Code | Alternative technical correctness code | eai_taxonomy.technical_correctness.secondary.code |
| Secondary Label | Alternative technical correctness label | eai_taxonomy.technical_correctness.secondary.label |
Possible Values:
| Code | Label | Description |
|---|---|---|
-1 |
Abstain | Unable to determine |
1 |
Technically Flawed | Significant errors undermining content validity |
2 |
Partially Correct | Some correctness but contains flaws, omissions, or errors |
3 |
Mostly Correct | Technical correctness with minor flaws or incomplete explanations |
4 |
Highly Correct | High technical correctness with precise definitions and clear explanations |
5 |
Exceptionally Correct | Exceptional technical correctness with formal proofs and flawless content |
6 |
Not Applicable/Indeterminate | No technical content or insufficient context |
Education Level
Assesses the appropriate educational background required to comprehend the content:
| Component | Description | Path |
|---|---|---|
| Primary Code | Main education level code | eai_taxonomy.education_level.primary.code |
| Primary Label | Main education level label | eai_taxonomy.education_level.primary.label |
| Secondary Code | Alternative education level code | eai_taxonomy.education_level.secondary.code |
| Secondary Label | Alternative education level label | eai_taxonomy.education_level.secondary.label |
Possible Values:
| Code | Label | Description |
|---|---|---|
-1 |
Abstain | Unable to determine |
1 |
General Audience | Accessible to anyone with basic literacy; simple terms |
2 |
High School Level | Requires high school education; specialized terminology explained for non-experts |
3 |
Undergraduate Level | Requires college education; uses specialized terminology and assumes background knowledge |
4 |
Graduate/Expert Level | Requires graduate education or domain expertise; assumes deep background knowledge |
5 |
Indeterminate | Insufficient content to judge educational level |
Metadata
Metadata Structure
The metadata field contains a nested structure with web archive information:
| Field | Type | Description | Path |
|---|---|---|---|
| URL Information | |||
| URL | String |
Original URL of the document | metadata.url |
| Source Domain | String |
Domain name of the source | metadata.source_domain |
| Snapshot ID | String |
Identifier for the web archive snapshot | metadata.snapshot_id |
| WARC Metadata | WARC (Web ARChive) format metadata | ||
| Content Length | String |
Size of the content | metadata.warc_metadata.Content-Length |
| Content Type | String |
MIME type of the content | metadata.warc_metadata.Content-Type |
| Block Digest | String |
Checksum of the WARC block | metadata.warc_metadata.WARC-Block-Digest |
| Concurrent To | String |
Related WARC records | metadata.warc_metadata.WARC-Concurrent-To |
| Date | String |
Timestamp of the crawl | metadata.warc_metadata.WARC-Date |
| IP Address | String |
Source server IP address | metadata.warc_metadata.WARC-IP-Address |
| Payload Type | String |
Identified content type | metadata.warc_metadata.WARC-Identified-Payload-Type |
| Payload Digest | String |
Checksum of the payload | metadata.warc_metadata.WARC-Payload-Digest |
| Record ID | String |
Unique WARC record identifier | metadata.warc_metadata.WARC-Record-ID |
| Target URI | String |
Original target URL | metadata.warc_metadata.WARC-Target-URI |
| Truncated | String |
Truncation status | metadata.warc_metadata.WARC-Truncated |
| Type | String |
WARC record type | metadata.warc_metadata.WARC-Type |
| Warcinfo ID | String |
Associated warcinfo record | metadata.warc_metadata.WARC-Warcinfo-ID |
| Additional Info | |||
| WARC Info | String |
Additional WARC information | metadata.warc_info |
Quality Signals
The dataset includes two comprehensive quality assessment frameworks:
Red Pajama v2 Quality Metrics
Text quality indicators derived from the Red Pajama v2 filtering pipeline:
Content Structure Metrics
| Metric | Description | Path |
|---|---|---|
| Original Length | Original document length | quality_signals.red_pajama_v2.ccnet_original_length |
| Original Lines | Number of lines in original document | quality_signals.red_pajama_v2.ccnet_original_nlines |
| Sentence Count | Total sentence count | quality_signals.red_pajama_v2.rps_doc_num_sentences |
| Word Count | Total word count | quality_signals.red_pajama_v2.rps_doc_word_count |
| Mean Word Length | Average word length | quality_signals.red_pajama_v2.rps_doc_mean_word_length |
Language Quality Metrics
| Metric | Description | Path |
|---|---|---|
| Stop Word Fraction | Proportion of stop words | quality_signals.red_pajama_v2.rps_doc_stop_word_fraction |
| Unique Words Fraction | Fraction of unique words | quality_signals.red_pajama_v2.rps_doc_frac_unique_words |
| All Caps Words | Fraction of words in all capitals | quality_signals.red_pajama_v2.rps_doc_frac_all_caps_words |
| Non-Alphabetic Words | Fraction of non-alphabetic words | quality_signals.red_pajama_v2.rps_doc_frac_no_alph_words |
| Unigram Entropy | Entropy measure of word distribution | quality_signals.red_pajama_v2.rps_doc_unigram_entropy |
Content Pattern Analysis
| Metric | Description | Path |
|---|---|---|
| Curly Bracket Density | Curly bracket density (code indicator) | quality_signals.red_pajama_v2.rps_doc_curly_bracket |
| Symbol-to-Word Ratio | Symbol-to-word ratio | quality_signals.red_pajama_v2.rps_doc_symbol_to_word_ratio |
| Ellipsis Line Endings | Lines ending with ellipsis | quality_signals.red_pajama_v2.rps_doc_frac_lines_end_with_ellipsis |
| Lorem Ipsum Detection | Lorem ipsum text detection | quality_signals.red_pajama_v2.rps_doc_lorem_ipsum |
| Offensive Content | Potentially offensive content detection | quality_signals.red_pajama_v2.rps_doc_ldnoobw_words |
| UT1 Blacklist | UT1 blacklist filtering score | quality_signals.red_pajama_v2.rps_doc_ut1_blacklist |
Duplication Detection
| Metric | Description | Path |
|---|---|---|
| 5-gram Duplication | Character-level duplication for 5-grams | quality_signals.red_pajama_v2.rps_doc_frac_chars_dupe_5grams |
| 6-gram Duplication | Character-level duplication for 6-grams | quality_signals.red_pajama_v2.rps_doc_frac_chars_dupe_6grams |
| 7-gram Duplication | Character-level duplication for 7-grams | quality_signals.red_pajama_v2.rps_doc_frac_chars_dupe_7grams |
| 8-gram Duplication | Character-level duplication for 8-grams | quality_signals.red_pajama_v2.rps_doc_frac_chars_dupe_8grams |
| 9-gram Duplication | Character-level duplication for 9-grams | quality_signals.red_pajama_v2.rps_doc_frac_chars_dupe_9grams |
| 10-gram Duplication | Character-level duplication for 10-grams | quality_signals.red_pajama_v2.rps_doc_frac_chars_dupe_10grams |
| Top 2-gram Coverage | Most frequent 2-gram coverage | quality_signals.red_pajama_v2.rps_doc_frac_chars_top_2gram |
| Top 3-gram Coverage | Most frequent 3-gram coverage | quality_signals.red_pajama_v2.rps_doc_frac_chars_top_3gram |
| Top 4-gram Coverage | Most frequent 4-gram coverage | quality_signals.red_pajama_v2.rps_doc_frac_chars_top_4gram |
Domain Importance Scores
| Metric | Description | Path |
|---|---|---|
| Books Importance | Similarity to book content | quality_signals.red_pajama_v2.rps_doc_books_importance |
| Books Importance (Length Corrected) | Length-corrected books similarity | quality_signals.red_pajama_v2.rps_doc_books_importance_length_correction |
| OpenWebText Importance | Similarity to OpenWebText | quality_signals.red_pajama_v2.rps_doc_openwebtext_importance |
| OpenWebText Importance (Length Corrected) | Length-corrected OpenWebText similarity | quality_signals.red_pajama_v2.rps_doc_openwebtext_importance_length_correction |
| Wikipedia Importance | Similarity to Wikipedia | quality_signals.red_pajama_v2.rps_doc_wikipedia_importance |
| Wikipedia Importance (Length Corrected) | Length-corrected Wikipedia similarity | quality_signals.red_pajama_v2.rps_doc_wikipedia_importance_length_correction |
FastText Classification Scores
Domain and content type classification probabilities:
| Metric | Description | Path |
|---|---|---|
| DCLM Score | DataComp-LM classifier score | quality_signals.fasttext.dclm |
| English Confidence | English language confidence | quality_signals.fasttext.english |
| Educational Content | Educational content approximation | quality_signals.fasttext.fineweb_edu_approx |
| General Math | General mathematics content | quality_signals.fasttext.eai_general_math |
| Web Math | OWM Web-based mathematics content | quality_signals.fasttext.eai_open_web_math |
| Code Content | Code content detection | quality_signals.fasttext.eai_web_code |
How to Load the Dataset
This section provides examples of how to load the EssentialAI/eai-taxonomy-math-w-fm dataset using different Python libraries and frameworks.
Using Hugging Face Datasets (Standard Method)
The simplest way to load the dataset is using the Hugging Face datasets library:
from datasets import load_dataset
# Load the entire dataset
dataset = load_dataset("EssentialAI/eai-taxonomy-math-w-fm")
# View dataset structure
print(dataset)
print(f"Number of examples: {len(dataset['train'])}")
You can also load the dataset in streaming mode to avoid downloading the entire dataset at once:
from datasets import load_dataset
# Load in streaming mode
dataset = load_dataset("EssentialAI/eai-taxonomy-math-w-fm", streaming=True)
data_stream = dataset["train"]
# Iterate through examples
for example in data_stream.take(5):
print(example)
Using PySpark
For large-scale distributed processing, you can load the dataset using PySpark with the pyspark_huggingface library:
# First install the required library:
# pip install pyspark_huggingface
import pyspark_huggingface
from pyspark.sql import SparkSession
# Initialize Spark session
spark = SparkSession.builder.appName("EAI-Taxonomy-Math").getOrCreate()
# Load the dataset using the "huggingface" data source
df = spark.read.format("huggingface").load("EssentialAI/eai-taxonomy-math-w-fm")
# Basic dataset exploration
print(f"Dataset shape: {df.count()} rows, {len(df.columns)} columns")
df.show(10)
df.printSchema()
# Load only specific columns for efficiency
df_subset = (
spark.read.format("huggingface")
.option("columns", '["column1", "column2"]') # Replace with actual column names
.load("EssentialAI/eai-taxonomy-math-w-fm")
)
# Run SQL queries on the dataset
df.createOrReplaceTempView("eai_math_dataset")
result = spark.sql("""
SELECT COUNT(*) as total_examples
FROM eai_math_dataset
""")
result.show()
Using Daft
Daft provides a modern DataFrame library optimized for machine learning workloads. You can load the dataset directly from Hugging Face:
import daft
# Load the entire dataset
df = daft.read_parquet("hf://datasets/EssentialAI/eai-taxonomy-math-w-fm")
# Basic exploration
print("Dataset schema:")
df.schema()
print("First 5 rows:")
df.show(5)
If you need to access private datasets or use authentication:
import daft
from daft.io import IOConfig, HTTPConfig
io_config = IOConfig(http=HTTPConfig(bearer_token="your_token"))
df = daft.read_parquet("hf://datasets/EssentialAI/eai-taxonomy-math-w-fm", io_config=io_config)
Installation Requirements
Make sure you have the required libraries installed:
# For Hugging Face datasets
pip install datasets
# For PySpark with Hugging Face integration
pip install pyspark_huggingface
# For Daft
pip install daft
🎓 Citation
If you use this dataset, please cite our EssentialWeb paper:
@misc{ai2025essentialwebv1024ttokens,
title={Essential-Web v1.0: 24T tokens of organized web data},
author={Essential AI and : and Andrew Hojel and Michael Pust and Tim Romanski and Yash Vanjani and Ritvik Kapila and Mohit Parmar and Adarsh Chaluvaraju and Alok Tripathy and Anil Thomas and Ashish Tanwer and Darsh J Shah and Ishaan Shah and Karl Stratos and Khoi Nguyen and Kurt Smith and Michael Callahan and Peter Rushton and Philip Monk and Platon Mazarakis and Saad Jamal and Saurabh Srivastava and Somanshu Singla and Ashish Vaswani},
year={2025},
eprint={2506.14111},
archivePrefix={arXiv},
primaryClass={cs.CL},
url={https://arxiv.org/abs/2506.14111},
}
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