Square Numbers

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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.

This table is a shortcut to the most searched items in Google: In the section ahead we discuss the frequently asked questions.

Frequently Asked Questions

Click on the question which is of interest to you to see the collapsible content answer.

What is a Square Number?

A square number or perfect square is the result when a whole number has been multiplied by itself.

How to Square Numbers in Excel?

To square a number n in Excel type n^2 or n*n.

What is the Square of a Number?

The square of a number is the outcome of multiplying an integer by itself.

How to Do Square Numbers?

You can square a number by means of exponentiation with base n and index 2, or you can multiply n × n.

What Does Squared Mean in Math?

“Squared” is the past tense of the verb “to square”; in math it means that a number has been multiplied by itself.

How to Square Numbers?

Take the decimal, fraction or whole number and multiply it by itself! Note that only if you square an integer do you get a perfect square.

What is a Square Number of 100?

100^2 = 100 x 100 = 10,000.

Is 18 a Perfect Square?

√18 = 4.2426406871 is not an integer, and as such 18 cannot be a perfect square.

What Are the First 10 Square Numbers?

The first 10 square numbers are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81.

Why is 3 Not a Square Number?

A square number can not end in 2, 3, 7 and 8.

What Is the Smallest Square Number?

The smallest square number is equal to 0.

What Is the Easiest Way to Find a Square Number?

The easiest way to find a square number is multiplying a small, whole number by itself.

Why is 20 Not a Square Number?

√20 = 4.4721359549 has digits and as such can not be a square number.

If something remains unclear do not hesitate getting in touch with us.

We are constantly trying to improve our site, and truly appreciate your feedback.

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.

And should you want to give us feedback use the form at the bottom.

Websites related to squares can be found in the “recommended sites” section in the sidebar.

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And come back soon, practice makes perfect!

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Further references:

• Perfect square on Wikipedia

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