Square Numbers: Definition, List, Examples

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What Is a Square Number in Math?

A square number is a number obtained when an integer is multiplied by itself. It is also referred to as “a perfect square.” 

We know that the product of a positive integer with a positive integer is always positive. Also, the product of a negative integer and a negative integer is always positive. 

$(+) \times (+) = (+)$

$(-) \times (-) = (+)$

Thus, the square numbers obtained by multiplying an integer with itself are always positive.

Examples of square numbers:

  • $5 \times 5 = 25$
  • $(-8) \times (-8) = 64$

Square Number: Definition

A square number can be defined as the positive integer obtained by multiplying an integer with itself.

Properties of Square Numbers

  • Square numbers always have the digits 0, 1, 4, 5, 6, or 9 as their last digits. 
  • A number ending with 2, 3, 7, or 8 cannot be a square number.
  • A square number always has an even number of zeros at the end. Therefore, a number is not a square if it has an odd number of zeros as the last digits. For example, the numbers 900 and 4900 are square numbers while 20, 360, and 480 are not squares.
  • The square of a number ends with 1 when the last digit of an integer is either 1 or 9. 
  • The square number of an integer ends with 6 if its last digit is either 4 or 6. 
  • The square number of an even integer is always an even integer, while the square of an odd number is always an odd number. 
  • Square numbers are always positive. 
  • If a number’s square root is a fraction or a decimal number, it is not a perfect square number. For instance, $\sqrt{0.25} = 0.5$, hence 0.25 is not a square number.

List of Square Numbers (1 to 100)

Here is a list of all of the square numbers from 1 to 100.

NumberSquareNumber SquareNumberSquareNumber Square
1126676512601765776
2427729522704775929
3928784532809786084
41629841542916796241
52530300553025806400
63631961563136816561
749321024573249826724
864331089583364836889
981341156593481847056
10100351225603600857225
11121361296613721867396
12144371369623844877569
13169381444633969887744
14196391521644096897921
15225401600654225908100
16256411681664356918281
17289421764674489928464
18324431849684624938649
19361441936694761948836
20400452025704900959025
21441462116715041969216
22484472209725184979409
23529482304735329989604
24576492401745476999801
2562550250075562510010000

Two-digit Square Numbers

There are only six “two-digit square numbers.” 

$4 \times 4 = 16$

$5 \times 5 = 25$

$6 \times 6 = 36$

$7 \times 7 = 49$

$8 \times 8 = 64$

$9 \times 9 = 81$

Three-digit Square Numbers

There are only 22 three-digit square numbers.

100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 

441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961.

Odd and Even Square Numbers

The meaning of a square number is simple; when we multiply a number by itself we get a square number. These square numbers can be divided into two types: 

  • Even square numbers: The square of any even number is always an even square number. For example, $2^{2} = 4,\; 6^{2} = 36$
  • Odd square numbers: The square of any odd number is always an odd square number. For example, $3^{2} = 9,\; 7^{2} = 49$

Square Numbers and Multiplication Arrays

The name “square” comes from the fact that all square numbers can be represented using a square-shaped array. They fill a square.

The following figure has the number of balls arranged in a square. We can see that the number of balls that fill a square is a square number. 

If we take 4 balls, we can see that they can be arranged in an array to form a square shape. Same goes for the square numbers 9, 36, etc.

Square numbers forming a square array

Square Numbers on Multiplication Chart

Notice that the square numbers appear diagonally in a multiplication chart.

Square numbers on a multiplication chart

Facts about Square Numbers

  • The square of any number “n” is equivalent to the sum of the first “n” odd natural numbers. For example, $4^{2} = 16 = 1 + 3 + 5 + 7$
  • If an integer ends with 5, there’s a trick you can use to find its square. If the given integer is of the form $\left[n\right]5$, then the square will be $\left[n \times(n + 1)\right]\left[25\right]$.
  • Difference of squares formula $= a^{2} \;-\; b^{2} = (a + b)(a \;-\; b)$
    Example: $45 = \left[4 \times 5\right]\left[25\right] = 2025$
  • The sum of squares of the first $n$ natural numbers $\left(1,\; 2,\; 3,…, n\right)$ is given by
    $\sum_{i = 1}^{n}i = \frac{n (n + 1) (2n + 1)}{6}$

Conclusion

In this article, we learned about square numbers, their properties along with the list of squares from 1 to 100. Let’s solve a few examples and practice problems. 

Solved Examples on Square Numbers

1. Find the square of 89. 

Solution: 

To find the square of the number 89, we need to multiply 89 by 89.

$89^{2} = 89 \times 89$

$89 \times 89 = 7921$

Finding square of 89 by long multiplication

2. Find the square of -25. 

Solution: 

The square of $-\;25$ would be the product of $-\;25$. 

$(\;-\;25)^{2} = (\;-\;25) \times (\;-\;25)$

$(\;-\;25)^{2} = 625$ 

3. Is 113 a square number?

Solution: 

All square numbers end with 0, 1, 4, 5, 6, or 9.

If a number ends with 2, 3, 7, or 8, then it is not a square number.

Practice Problems on Square Numbers

Square Numbers: Definition, List, Examples

Attend this quiz & Test your knowledge.

1

Which of the following is a square number?

125
3600
51
500
CorrectIncorrect
Correct answer is: 3600
3600 is the square of the number 60.
2

What is the square of -24?

-576
416
96
576
CorrectIncorrect
Correct answer is: 576
The square of any negative number is always positive.
$(\;-\;24)^{2} = 576$
3

$(65)^{2} =$

3500
4225
3000
-3000
CorrectIncorrect
Correct answer is: 4225
$65^{2} = 4225$
4

If $m^{2} = n$, then $(\;-\;m)^{2} =$

$\;-\;n$
m
n
0
CorrectIncorrect
Correct answer is: n
If $m^{2} = n$, then $(\;-\;m)^{2} = n$

Frequently Asked Questions on Square Numbers

The first five square numbers are 1, 4, 9, 16, and 25.

Every number except 0 has two square roots, positive and negative. The positive square root is called the principal square root.

The square of 0 is 0.