square numbermathematics
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Aspects of this topic are discussed in the following places at Britannica.
Assorted References
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Chinese mathematics
( in mathematics, East Asian: Square and cube roots )
In The Nine Chapters, algorithms for finding integral parts of square roots or cube roots on the counting surface are based on the same idea as the arithmetic ones used today. These algorithms are set up on the surface in the same way as is a division: at the top, the “quotient”; under it, the “dividend”; one row below, the “divisor”;...
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definition and properties
( in number game: Polygonal and other figurate numbers )
Square numbers are the squares of natural numbers, such as 1, 4, 9, 16, 25, etc., and can be represented by square arrays of dots, as shown in Figure 1. Inspection reveals that the sum of any two adjacent triangular numbers is always a square number.
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mathematical puzzles
( in number game: Coloured squares and cubes )
There is a wide variety of puzzles involving coloured square tiles and coloured cubes. In one, the object is to arrange the 24 three-colour patterns, including repetitions, that can be obtained by subdividing square tiles diagonally, using three different colours, into a 4 × 6 rectangle so that each pair of touching edges is the same colour and the entire border of the rectangle is the...
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numbers
( in number )
Other classes of numbers include square numbers—i.e., those that are squares of integers; perfect numbers, those that are equal to the sum of their proper factors; random numbers, those that are representative of random selection procedures; and prime numbers, integers larger than 1 whose only positive divisors are themselves and 1.
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Pythagorean mathematics
( in Pythagoreanism: Arithmetic )
...Aristotle’s account, gnomon numbers, represented by dots or pebbles, were arranged in the manner shown in Figure 2. If a series of odd numbers is put around the unit as gnomons, they always produce squares; thus, the members of the series 4, 9, 16, 25, . . . are “square” numbers. If even numbers are depicted in a similar way, the resulting figures (which offer infinite variations)...
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relationship to geometrical square
( in square )
...A square has four axes of symmetry, and its two finite diagonals (as with any rectangle) are equal. Bisection of a square by a diagonal results in two right triangles. If the length of the side of a square is s, then the area of the square is s2, or “s squared.” From this relation is derived the algebraic use of the term square, which denotes...
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Chinese mathematics mathematics, East Asian
In The Nine Chapters, algorithms for finding integral parts of square roots or cube roots on the counting surface are based on the same idea as the arithmetic ones used today. These algorithms are set up on the surface in the same way as is a division: at the top, the “quotient”; under it, the “dividend”; one row below, the “divisor”;...
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definition and properties number game
Square numbers are the squares of natural numbers, such as 1, 4, 9, 16, 25, etc., and can be represented by square arrays of dots, as shown in Figure 1. Inspection reveals that the sum of any two adjacent triangular numbers is always a square number.
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mathematical puzzles number game
There is a wide variety of puzzles involving coloured square tiles and coloured cubes. In one, the object is to arrange the 24 three-colour patterns, including repetitions, that can be obtained by subdividing square tiles diagonally, using three different colours, into a 4 × 6 rectangle so that each pair of touching edges is the same colour and the entire border of the rectangle is the...
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numbers number
Other classes of numbers include square numbers—i.e., those that are squares of integers; perfect numbers, those that are equal to the sum of their proper factors; random numbers, those that are representative of random selection procedures; and prime numbers, integers larger than 1 whose only positive divisors are themselves and 1.
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Pythagorean mathematics Pythagoreanism
...Aristotle’s account, gnomon numbers, represented by dots or pebbles, were arranged in the manner shown in Figure 2. If a series of odd numbers is put around the unit as gnomons, they always produce squares; thus, the members of the...
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definition number game
...arithmetic sequences but are seen to be the polygonal triangular and square numbers. Polygonal number series can also be added to form threedimensional figurate numbers; these sequences are called pyramidal numbers.
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description magic square
In the arithmetical magic squares, the numbers are generally placed in separate cells and arranged so that each column, every row, and the two main diagonals can produce the same sum, called the constant. A standard magic square of any given number contains the sequence of natural numbers from 1 to the square of that number. Thus, the magic square of 3 contains the numbers 1 to 9. If these nine...
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definition and formation number game
Another type of number pleasantry concerns multigrades; i.e., identities between the sums of two sets of numbers and the sums of their squares or higher powers—e.g.,
in mathematics, a factor of a number that, when multiplied by itself, gives the original number. For example, both 3 and –3 are square roots of 9. As early as the 2nd millennium bc, the Babylonians possessed effective methods for approximating square roots. See root.
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analytic geometry algebra
...a unit length that served as a reference for all other lengths and for all operations among them. For example, suppose that Descartes was given a segment AB and was asked to find its square root. He would draw the straight line DB (see the figure), where DA was defined as the unit length. Then he would bisect DB at C, draw the...
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