A figurate number is a number that can be represented as a regular and discrete geometric pattern (e.g. dots). If the pattern is polytopic, the figurate is labeled a polytopic number, and may be a polygonal number or a polyhedral number. For other uses, see Geometry (disambiguation). ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... In mathematics, a polygonal number is a number that can be arranged as a regular polygon. ...

The first few triangular numbers can be built from rows of 1, 2, 3, 4, 5, and 6 items: A triangular number is the sum of the n natural numbers from 1 to n. ...

The n-th regular r-topic number is given by the formula: Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ... Image File history File links GrayDotX.svg Summary Light gray circle. ...

r! is the factorial of r, is a binomial coefficient, and n^(r) is the rising factorial. For factorial rings in mathematics, see unique factorisation domain. ... In mathematics, particularly in combinatorics, a binomial coefficient is a coefficient of any of the terms in the expansion of the binomial (x+1)n. ... In mathematics, the Pochhammer symbol is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial The empty product (x)0 is defined to be 1 in both cases. ...

Polytopic numbers for r = 2, 3, and 4 are:

• P₂(n) = 1/2 n(n + 1) (triangular numbers)
• P₃(n) = 1/6 n(n + 1)(n + 2) (tetrahedral numbers)
• P₄(n) = 1/24 n(n + 1)(n + 2)(n + 3) (pentatopic numbers)

Our present terms square number and cubic number derive from their geometric representation as a square or cube. A triangular number is the sum of the n natural numbers from 1 to n. ... A pyramid with side length 5 contains 35 spheres. ... A pentatopic number is a number in the fifth cell of any row of Pascals triangle starting with the 5-term row 1 4 6 4 1 either from left to right or from right to left. ... In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. ... In arithmetic and algebra, the cube of a number n is its third power — the result of multiplying it by itself two times: n3 = n × n × n. ... For other uses, see Square. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ...

Gnomon

Figurate numbers were a concern of Pythagorean geometry, since Pythagoras is credited with initiating them, and the notion that these numbers are generated from a gnomon or basic unit. The gnomon is the piece added to a figurate number to transform it to the next bigger one. Pythagoras of Samos (Greek: ; born between 580 and 572 BC, died between 500 and 490 BC) was an Ionian Greek mathematician[1] and founder of the religious movement called Pythagoreanism. ... In geometry, a gnomon is a plane figure formed by removing a parallelogram from a corner of a larger parallelogram. ...

For example, the gnomon of the square number is the odd number, of the general form 2n + 1, n = 0, 1, 2, 3, ... . The square of size 8 composed of gnomons looks like this: In mathematics, any integer (whole number) is either even or odd. ...

To transform from the n-square (the square of size n) to the (n + 1)-square, one adjoins 2n + 1 elements: one to the end of each row (n elements), one to the end of each column (n elements), and a single one to the corner. For example, when transforming the 7-square to the 8-square, we add 15 elements; these adjunctions are the 8s in the above figure.

Note that this gnomonic technique also provides a proof that the sum of the first n odd numbers is n²; the figure illustrates 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 8². In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...

Square roots

Conversely, one can calculate the square root of any number by subtracting odd numbers. Thus, 64 − 1 = 63; 63 − 3 = 60; 60 − 5 = 55; 55 − 7 = 48; 48 − 9 = 39; 39 − 11 = 28; 28 − 13 = 15; 15 − 15 = 0. The subtraction of the first 8 odd numbers from 64 yields 0; hence, the square-root of 64 is 8. In mathematics, a square root (√) of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ...

The tedium of increasing number of subtractions as the number grows is bypassed by a method similar to the standard way of square-rooting taught in school. For example: 1225 = 35 × 35, Note the sum of the digits of this square root: 3 + 5 = 8. This square-root shortcut reduces 35 subtractions to only 8 subtractions. The shortcut involves two "tricks": a markoff trick, and resumptive trick.

The markoff trick is already known from the familiar square root algorithm. One marks off the target number in pairs of digits, from the right, as in marking 1225 as 12′25; then, calculation begins with the first digit-pair to the left. The reason is that squaring a one-digit number results in a 1- or 2-digit square. Thus, 1, 2, 3 have, respectively, the 1-digit squares of 1, 4, 9. But 4 has the 2-digit square of 16; and numbers 5, 6, 7, 8, 9 have 2-digit squares. To allow for this, one begins with two digits to provide one digit at each process-stage.

The resumptive trick (unique to this present algorithm) shifts from one pair of target number digits to its next (rightward) two digits, explained in calculating the square root of 1225.

1. Mark off 1225 as 12′25; begin calculation with left pair of digits, namely, 12.
2. Begin subtracting odd numbers: 12 − 1 = 11; 11 − 3 = 8; 8 − 5 = 3; but the next odd number, 7, cannot be subtracted from difference 3, so the resumptive trick is needed.
3. The left-most digit of the square root, 3, actually representing 30, because the second digit from the right in decimal numeration is the "tens digit".
4. To difference 3 (= 8 − 5), adjoin next two marked off digits (25), obtaining 325, and resume odd number subtraction.
5. The last "successful" subtrahend was 5; but the next odd number, 7, cannot be subtracted, so interpolate between 5 and 7 for number 6. (This is "first part" of the resumptive trick.)
6. Since (noted above) the successful 3 subtractions actually represent the 2-digit 30, treat the interpolated 6 as 60; resume odd number subtraction with the first odd number in the sixties, namely, 61. (This is "second and final part" of the resumptive trick or subalgorithm.)
7. Result: 325 − 61 = 264; 264 − 63 = 201; 201 − 65 = 136; 136 − 67 = 69; 69 − 69 = 0.
8. Having passed from 325 to 0 by five subtractions, the second digit is 5: and 30 + 5 = 35, that is, the square root of 1225 is 35, obtained in exactly 3 + 5 = 8 subtractions by applying the markoff and resumptive tricks or subalgorithms.

To see again, consider 144 = 12². The square-root is easily calculated by twelve subtractions: 144 − (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23) = 144 − 144 = 0. However, the mark-off and resumptive tricks reduce this to 1 + 2 = 3 odd number subtractions. Difference is the contrary of equality, in particular of objects. ...

1. Mark off 144 as 1′44.
2. Starting with left-most pair, start subtraction 1 − 1 = 0; so leftmost digit of square-root is 1, representing 10.
3. Bring down second pair of digits: 0 + 44 = 44 and begin subtracting odd numbers.
4. Interpolate between "successful" 1, "failed" 3, namely, 2, to represent the twenties, whose first odd number is 21, resuming the subtraction with 21.
5. 44 − 21 = 23; 23 − 23 = 0, resulting in two subtractions, so second digit is 2: 10 + 2 = 12 as square-root of 144, obtained by 1 + 2 = 3 subtracting of odd numbers.

A special case involves a "zero difference", illustrated in 10² = 100. Marking off 100 as 1′00, then 1 − 1 = 0 (difference), which combines with the second pair of digits as 000, But this in decimal notation is simply 0, resulting in 10 as square root, achieved in 1 + 0 = 1 subtraction. Another example is 20² = 400. Marking off 400 as 4′00. then 4 − (1 + 3) = 0 (difference), which combines with second pair of digits as 0000, representing 0, yielding root 20, achieved in 2 + 0 = 2 subtractions, i.e., 4 − (1 + 3).

Cubes and cube roots

Cubes of natural numbers or positive integers can be generated from S = 1, 3, 5, 7, 9,..., 2n − 1, ...; n = 1, 2, 3, ..., by "moving sums", similar to the "moving averages" of statistics: This article is about the field of statistics. ...

1. First member of S: 1 = 1³.
2. next two members of S: 3 + 5 = 8 = 2³.
3. Next three members of S: 7 + 9 + 11 = 27 = 3³.
4. Next four members of S: 13 + 15 + 17 + 19 = 64 = 4³.
5. Next five of S: 21 + 23 + 25 + 27 + 29 = 125 = 5³.
6. Next six of S: 31 + 33 + 35 + 37 + 39 + 41 = 216 = 6³.
7. Next seven of S: 43 + 45 + 47 + 49 + 51 + 53 + 55 = 343 = 7³.

Thus, "moving differences" of S yield cube-roots.

This procedure (taking many words to explain, but quickly executed) is not restricted to calculating square roots of natural numbers or positive integers. It can even be applied toward calculating the irrational square root of 2, to any number of decimal places.

Demonstration of mathematical properties

School children construct figurate numbers from pebbles, bottle caps, etc. As a bonus, children can use figurate numbers to discover the commutative law and associative law for addition and multiplication — laws usually dictated to them — by building rows and tables of dots. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, associativity is a property that a binary operation can have. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... In mathematics, multiplication is an elementary arithmetic operation. ...

For example, the additive commutativity of 2 + 3 = 3 + 2 = 5 becomes:

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And the multiplicative commutativity of 2 × 3 = 3 × 2 = 6 becomes: Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ...

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Besides the subtractive method, the additive method can also approximate square roots of positive integers and solve quadratic equations. Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ... Image File history File links GrayDot. ...

The concepts of figurate numbers and gnomon implicitly anticipate the modern concept of recursion. This article is about the concept of recursion. ...

See also

• Pick's theorem

Given a simple polygon constructed on a grid of equal-distanced points (i. ...

References

• Gazalé, Midhat J. (1999), Gnomon: from pharaohs to fractals, Princeton University Press, ISBN 978-0-691-00514-0 

The Princeton University Press is a publishing house, a division of Princeton University, that is highly respected in academic publishing. ...

External links

• On Regular Polytope numbers