Square Numbers: Multiplying With The Same Number

When a number or integer (not a fraction) is multiplied by itself, the resultant is called a ‘Square Number’. For example, 3 multiplied by 3 is equal to 3-squared or 3 x 3 = 32. So, basically, the exponential form of multiplication of a number or integer by itself is called a square number. Also, if we again multiply the number by itself, then we get a cube of the integer., a x a x a = a3.


Square numbers are always positive. If negative sign is multiplied by itself, it results in positive sign (+). For example, (-4)2 = 16. So, we can say here 16 is a positive square number, whose square root is an integer again, i.e.√16 = 4.

In geometry, a square shape has all its sides equal. Therefore the area of the square is equal to the square of its side.

Area of a square = Side x Side = Side2
Therefore, we can say;
Square number = a x a = a2

Hence, area of square is best example of square of a number.

Side of square (in cm) Area of square (in cm2)
3 3 × 3 = \(3^{2}\) = 9
5 5 × 5 = \(5^{2}\) = 25
7 7 × 7 = \(7^{2}\) = 49
8 8 × 8 = \(8^{2}\) = 64

Numbers such as 1, 4, 9, 16, 25, 36, 49, 64, etc. are special numbers as these are the product of a number by itself.

If we express a number (x) in terms of the square of any natural number such as a2, then x is a square number. For example, 100 can be expressed as 10 × 10 = 102, where 10 is a natural number, therefore 100 is a square number. Whereas, the number 45 cannot be called a square number because it is the product of numbers 9 and 5. The number is not multiplied by itself. Square numbers can also be called perfect square numbers.

Examples of Square numbers

Check some examples of square numbers here.

  • 1 x 1 = 12
  • 2 x 2 = 22 = 4
  • 3 x 3 = 32 = 9
  • 4 x 4 = 42 = 16
  • 5 x 5 = 52 = 25
  • 6 x 6 = 62 = 36

List of Square Numbers (1 to 100)

A list of square numbers from 1 to 100 is provided here. Use the table given below to solve problems related to square numbers.

Squares of 1 to 50

Numbers Squares Numbers Squares Numbers Squares Numbers Squares Numbers Squares
1 1 11 121 21 441 31 961 41 1681
2 4 12 144 22 484 32 1024 42 1764
3 9 13 169 23 529 33 1089 43 1849
4 16 14 196 24 576 34 1156 44 1936
5 25 15 225 25 625 35 1225 45 2025
6 36 16 256 26 676 36 1296 46 2116
7 49 17 289 27 729 37 1369 47 2209
8 64 18 324 28 784 38 1444 48 2304
9 81 19 361 29 841 39 1521 49 2401
10 100 20 400 30 900 40 1600 50 2500

Squares of 51 to 100

Numbers Squares Numbers Squares Numbers Squares Numbers squares Numbers Squares
51 2601 61 3721 71 5041 81 6561 91 8281
52 2704 62 3844 72 5184 82 6724 92 8464
53 2809 63 3969 73 5329 83 6889 93 8649
54 2916 64 4096 74 5476 84 7056 94 8836
55 3025 65 4225 75 5625 85 7225 95 9025
56 3136 66 4356 76 5776 86 7396 96 9216
57 3249 67 4489 77 5929 87 7569 97 9409
58 3364 68 4624 78 6084 88 7744 98 9604
59 3481 69 4761 79 6241 89 7921 99 9801
60 3600 70 4900 80 6400 90 8100 100 10000

Two Digit Square numbers

The list of two-digit square numbers is 16, 25, 36, 49, 64 and 81.

Odd and Even square numbers

  • Squares of even numbers are even, i.e, (2n)2 = 4n2.
  • Squares of odd numbers are odd, i.e, (2n + 1) = 4(n2 + n) + 1.
  • Since every odd square is of the form 4n + 1, the odd numbers that are of the form 4n + 3 are not square numbers.

Properties of Square Numbers

The following are the properties of the square numbers:

  1.  A number with 2, 3, 7 or 8 at unit’s place should never be a perfect square. In other words, none of the square numbers ends in 2, 3, 7 or 8.
  2. If the number of zeros at the end is even, then the number is a perfect square number. Otherwise, we can say that number ending in an odd number of zeros is never a perfect square
  3.  If the even numbers are squared, it always gives even numbers. Also, if the odd numbers are squared, it always gives odd numbers.
  4. If the natural numbers other than one is squared, it should be either a multiple of 3 or exceeds a multiple of 3 by 1.
  5. f the natural numbers other than one is squared, it should be either a multiple of 4 or exceeds a multiple of 4 by 1.
  6. It is noted that the unit’s digit of the square of a natural number is equal to the unit’s digit of the square of the digit at unit’s place of the given natural number.
  7. There are n natural numbers, say p and q such that p 2 = 2q 2
  8. For every natural number n, we can write it as: (n + 1)2 – n 2 = ( n + 1) + n..
  9. If a number n is squared, it equals the sum of first n odd natural numbers.
  10. For any natural number, say”n” which is greater than 1, we can say that (2n, n 2 – 1, n 2 + 1) should be a Pythagorean triplet.

Square Roots

As we have already discussed in the introduction, the square numbers are generated when an integer is multiplied by itself. Now, to get back the original number, we have to find the square root of the square number. 

For example, 

7 x 7 = 49

√49 = 7

So the square root of any number gives us the value, that can be squared to get the original number. 

Here we have covered square numbers and their properties, you can refer to completing the square method which by following the link provided. To learn more about other topics download BYJU’S – The Learning App and learn the subjects in an interactive and innovative way.

Frequently Asked Questions – FAQs

What are square numbers?

Square numbers are those numbers that are produced by multiplying an integer to itself. If n is an integer, then n2 is the square of n.

What are the examples of square numbers?

The examples for square numbers are:
62.= 36
102 = 10 x 10 = 100
152 = 15 x 15 = 225
122 = 12 x 12 = 144

What are perfect square numbers?

The square of whole numbers is also called perfect square numbers.

What is the square of 100?

The square of 100 is:
1002 = 100 x 100 = 100,00

If the square of a number is 441, then what is the original number?

To find the original number, we need to find the square root of 441.
Now, prime factorising 441 we get;
441 = 3 x 3 x 7 x 7 = (3×7)2 = 212
Taking square root both the sides we get;
√441 = 21
Therefore, 21 is the original number.

Quiz on Square Numbers

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