Facts That Matter
We Know That
If a whole number is multiplied by itself, the product is called the square of that number. For example,
3 * 3 = 9 = 3^{2}
i.e. the square of 3 is 9.
5 * 5 = 25 = 5^{2}
i.e. the square of 5 is 25.
We also know that: A natural number is called a perfect square or a square number, if it is the square of some natural number. For example,
16 is the square of 4, therefore, 16 is a perfect square.
289 is the square of 17, therefore, 289 is a perfect square.
Remember
All natural numbers are not perfect-squares or square numbers e.g. 32 is not a square-number In general, if a natural number ‘m’ can be expressed as n^{2}, where n is also a natural number then ‘m’ is perfect-square. The numbers like 1, 4, 9, 16, 25, 36, … are called square numbers. Square numbers between 1 and 100 are:
Number | Square | Number | Squares |
1 | 1 | 6 | 36 |
2 | 4 | 7 | 49 |
3 | 9 | 8 | 64 |
4 | 16 | 9 | 81 |
5 | 25 | 10 | 100 |
Properties of A Square Number
Let us considers square of all natural numbers from 1 to 20.
Number | Square | Number | Square |
1 | 1 | 11 | 121 |
2 | 4 | 12 | 144 |
3 | 9 | 13 | 169 |
4 | 16 | 14 | 196 |
5 | 25 | 15 | 225 |
6 | 36 | 16 | 256 |
7 | 49 | 17 | 289 |
8 | 64 | 18 | 324 |
9 | 81 | 19 | 361 |
10 | 100 | 20 | 400 |
Property 1: “The ending digits (the digits in the one’s place) of a square number is 0, 1, 4, 5, 6 or 9 only.”
Some Interesting Patterns
1. Triangular numbers are: 1, 3, 6, 10, 15, 21, etc. If we combine two consecutive triangular numbers, we get a square number.
i.e 1 + 3 = 4, ‘4’ is a square number.
3 + 6 = 9, ‘9’ is a square number
6 + 10 = 16, ‘16’ is a square number
and so on.
2.
1^{2 }=1
11^{2} = 121
111^{2} = 12321
1111^{2} = 1234321
3. We have:
7^{2 }= 49
67^{2} = 4489
667^{2} = 444889
6667^{2} = 44448889 and so on.
Solved Examples:
Problem: What will be the unit’s digit in the square of the following numbers?
1. 12487
2. 1324
3. 91478
4. 1251
Solution: The unit’s digit in the square of the following is:
1. 12487 is 9 (as 72 = 49. 9 in the unit’s place).
2. 1324 is 6 (as 42 = 16. 6 in the unit’s place).
3. 91478 is 4 (as 82 = 64. 4 in the unit’s place).
4. 1251 is 1 (as 12 = 1. 1 in the unit’s place).
Problem: Comment on the square of an even number and of an odd number?
Solution: The square of an even number is always an even number and the square of an odd number is always an odd number. The square of an even number will always have 4, 6, or even number of zeros in its unit’s place. And the square of an odd number will always have 1, 5 or 9 in its unit’s place.