Square Numbers

Square Numbers Definition

A square number is the product of some integer with itself.

Therefore, it is expressed as n² = n × n; n is a whole number.

As you can see, a square number is an exponentiation involving the base n and the exponent 2, sometimes written as n^2.

In other words, it means n to the power of 2.

It follows from that that n can be positive or negative (If you square a negative number the result will be a positive), but usually the meaning is n > 0.

If n represents the length of a side, then n^2 expresses the area of the shape of a square with side length n.

This answers the question how do you square a number?

Square Numbers Formula

Every square number equals the sum of the first n odd numbers: .

Next, let’s look at some examples.

Square Numbers Examples

• Example 1 – The square number of 3: 3 × 3 = 9.
• Example 2 – The square number of 4: 4 × 4 = 16.
• Example 3 – The square number of 12: 12 × 12 = 144.

If n² is the square number, then n is the principal square root.

• Example 4: The square root of 169 (13 × 13) is ± 13.

To find out if a given integer is a perfect square, calculate its square roots.

If the square root is a whole number (has no decimal places), then the number is indeed a square number.

• Example 5: Is 18 a square number? No, √18 = ±4.2426406871 (has decimals).
• Example 6: Is 25 a square number? Yes, √25 = ±5 (has no decimals).
• Example 7: If the square of a number is y, then what is the original number x? By definition, the product y = x², so x = √y.

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Ahead is the table in the context of this site.

Square Numbers List

The table below contains the square numbers up to 100:

As follows, the sequence of square numbers is 0 (0 x 0), 1 (1 x 1), 4 (2 x 2), 9 (3 x 3), 16 (4 x 4), 25 (5 x 5), 36 (6 x 6), 49 (7 x 7), 64 (8 x 8), 81 (9 x 9), …

Square Numbers Calculator

Here we inform you that our tool at the top of this page works both ways; you can either fill in the upper or the lower input field.

Our app works with all real numbers, and you don’t need to press a button unless you want to start over.

You may use the up and down arrows (called spinners) to increase or decrease the input value.

Frequently calculated terms include, for example:

• Square Number of 5
• Square Number of 12
• Square Number of 3

We now show you two recursive formulas plus an identity which are sometimes useful to come up with n^2.

Additional Information

In addition to the formula discussed above, any product n² can be produced recursively:

1. n² = (n − 1)² + (n − 1) + n = (n − 1)² + (2n − 1)
2. n² = 2 x (n − 1)² − (n − 2)² + 2

This is the difference between two perfect squares:

• n² − (n − 1)² = 2n − 1

Squares of even integers are even:

• (2n)² = 4n²

Squares of odd integers are odd:

• (2n + 1) = 4(n² + n) + 1

Last digit:

• In base 10, a square number cannot end in digits 2, 3, 7, 8.

Note that you can always employ the search form located in the menu and in the sidebar of this site to locate information about a particular term.

Or simply consult our square numbers chart.

Next is the summary of our article.

Bottom Line

A square number n² means n × n, n is an integer and × is the multiplication symbol.

The exponentiation form n² or n^2 is mostly used to express a square number.

For a number to be a perfect square it’s last digit must be 0, 1, 4, 5, 6 or 9; else it is an imperfect square.

If the second root of a squared number has no decimal places, then the squared number is a square number!

If this article about the square of numbers has been of help to you, then please share it by means of the social buttons below.

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And come back soon, practice makes perfect!

Thanks for your visit.Further references:

• Perfect square on Wikipedia

– Article written by Mark, last updated on November 26th, 2023

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