What Are Squares of Integers?

Definition

The square of an integer is defined as the product of the integer multiplied by itself. For example, the square of the integer 2 is 4 because 2 x 2 = 4. The square of -2 is also 4 because -2 x -2 = 4. The square of an integer is always an integer and the square of a non-integer is never an integer. Each positive integer n has a unique square number associated with it (written n^2). Since there are an infinite number of positive integers, there are an infinite number of integer squares as well.

Square numbers are an important concept in nearly every branch of mathematics. As implied by their name, square numbers are found in geometry in the formula for the area of a square. The formula for the square with a side length s is A = s^2. A square with dimensions of 3 inches by 3 inches therefore has an area of 3^2 square inches, or 9 square inches. Square numbers also appear in quadratic equations in algebra. Quadratic equations are polynomials of the form Ax^2 + Bx + C = 0.

Squaring is an example of raising a number to a power, which means multiplying the number by itself a certain number of times. The square of an integer is equal to the integer raised to the power of 2. Raising the integer to a different power will yield a different value. For example, 4 raised to the second power is 4 x 4 or 16, while 4 raised to the third power is 4 x 4 x 4 or 64. You can also raise numbers to a negative power: the value x^(-n) is equal to the value 1 / (x^n). 4^(-3) is equal to 1 / (4^3) or 1/64.

Like all other basic mathematical operations, squaring has an inverse operation, which is called taking the square root. The inverse of an operation is the operation that will completely reverse its effect. For example, division is the inverse of multiplication because dividing by 2 will reverse the effect of multiplying by 2: 4 x 2 = 8, 8 / 2 = 4. The square root of a number reverses the effect of squaring a number. For example: the square of 4 is 16 (4^2 = 16), so the square root of 16 is 4.