Perfect Square
Check if any number is a perfect square instantly. Enter any integer to see if its square root is a whole number, with the square root and factorization shown for perfect squares.
About This Calculator
Our Perfect Square Calculator helps you instantly determine whether any number is a perfect square. A perfect square is a number that can be expressed as the square of an integer -- i.e., the result of multiplying an integer by itself. This free online tool checks whether the square root of your number is an integer and displays the square root when it is.
This calculator is useful for students learning about squares and square roots in algebra, for teachers preparing math worksheets, and for anyone who needs to quickly check if a number is a perfect square for number theory, geometry problems, or recreational mathematics. The tool uses the formula N = a^2, where N is your number and a is the potential square root. If a is an integer, N is a perfect square.
Key Concepts
- Perfect Square: A number that equals an integer raised to the power of 2.
- Square Root: The value that, when multiplied by itself, gives the original number.
- Negative numbers cannot be perfect squares in the real number system because a square is always non-negative.
- Perfect squares are also called square numbers and have the property that they can be arranged as a square grid of dots.
Common Perfect Squares
The first twelve positive perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144. Larger examples include 400 (20^2), 900 (30^2), and 10000 (100^2). All perfect squares end with the digits 0, 1, 4, 5, 6, or 9 -- never 2, 3, 7, or 8.
Quick Rules for Identifying Perfect Squares
A number ending in 2, 3, 7, or 8 is never a perfect square. If a number ends in 1, 4, or 9, its tens digit must be even for it to be a perfect square. If a number ends in 6, its tens digit must be odd. If a number ends in 5, its tens digit must be 2. The digital root of a perfect square must be 0, 1, 4, or 7.
Frequently Asked Questions
What is a perfect square?
A perfect square is a number that can be expressed as the product of an integer multiplied by itself. For example, 49 is a perfect square because it equals 7 x 7, and 144 is a perfect square because it equals 12 x 12. Perfect squares are also called square numbers.
How do you check if a number is a perfect square?
To check if a number is a perfect square, calculate its square root. If the square root is an integer (a whole number with no decimal part), then the original number is a perfect square. For example, the square root of 144 is 12, which is an integer, so 144 is a perfect square.
Can a negative number be a perfect square?
No, negative numbers cannot be perfect squares. This is because multiplying any integer by itself always gives a positive result. The square root of a negative number is an imaginary number, not a real integer. However, negative numbers like -4 can be perfect squares in the context of complex numbers where i^2 = -1.
Is 0 a perfect square?
Yes, 0 is a perfect square because 0 x 0 = 0. The square root of 0 is 0, which is an integer. Zero and one are the only two numbers where the number, its square, and its square root are all the same value.
What is the difference between a perfect square and a perfect cube?
A perfect square is a number obtained by multiplying an integer by itself (e.g., 16 = 4 x 4), while a perfect cube is a number obtained by multiplying an integer by itself three times (e.g., 64 = 4 x 4 x 4). A number can be both a perfect square and a perfect cube if it is a sixth power, like 64 (8^2 = 64 and 4^3 = 64).
Is 1 a perfect square?
Yes, 1 is a perfect square because 1 x 1 = 1. The square root of 1 is 1, which is an integer. Together with 0, 1 is one of the most fundamental perfect squares.
What are the first 12 perfect squares?
The first 12 perfect squares are: 1 (1^2), 4 (2^2), 9 (3^2), 16 (4^2), 25 (5^2), 36 (6^2), 49 (7^2), 64 (8^2), 81 (9^2), 100 (10^2), 121 (11^2), and 144 (12^2).