square number

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Square numbers shown formed from consecutive triangular numbers

Diagram showing four types of polygonal numbers with colored dots arranged in geometric patterns: blue triangles for triangular numbers, green squares for square numbers, red pentagons for pentagonal numbers, and purple hexagons for hexagonal numbers, each illustrating how numbers like 1, 3, 6, 10, 4, 9, 16, 5, 12, 22, 6, 15, and 28 can be visually represented.

Figure 1: Polygonal arrays.

square number

mathematics

Also known as: square

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July 14, 2026

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Square numbers shown formed from consecutive triangular numbers — Square numbers shown formed from consecutive triangular numbers Figure 1: Square numbers shown formed from consecutive triangular numbers.

A square number is an integer that is the square of another integer. For example, 9 is a square number because its square root is 3. Square numbers can be visualized as dots arranged in a square array. If a square has a side length of s, its area is s squared (s²). This concept extends to algebra, where "squaring" means multiplying an expression by itself. In geometry, a square is a plane figure with four equal sides and four right angles.1234

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East Asian mathematics
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…
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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…
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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…
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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…
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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…
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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…
Read More

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…
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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…
Read More

Pythagoreanism
Arithmetic
…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) represent “oblong” numbers, such as those of the series 2, 6, 12, 20,…. On the…
Read More
…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) represent “oblong” numbers, such as those of the series 2, 6, 12, 20,…. On the…
Read More

square
…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 the product that results from multiplying any algebraic expression by itself.
Read More
…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 the product that results from multiplying any algebraic expression by itself.
Read More