System introduced to define the numbers present from negative infinity to positive infinity is known as Number System. The numbers system is mostly represented in pictorial form on a number line, a number line is a straight line containing numbers from negative infinity to positive infinity. The number system contains number names in two ways, one is the Indian method and the other is the International method.
In the Indian method, the way of writing numbers is different. The commas are applied from right to left, the digits are defined in the following manner,
Square of a Number
Squares are defined as the number multiplied by itself. Hence, the number found in the number system when a number is multiplied by itself is known as the square of that number. For instance, the square of 5 is 25, the square of 7 is 49, the square of -5 is 25, the squares of 1.21 is 1.4641.
Perfect square
Perfect square is defined as the squares of the Natural numbers, that is, the squares of the decimals and negative numbers are not considered as Perfect squares. Examples of perfect squares are 8×8= 64, 11×11= 121, etc.
Note: All perfect squares are squares but not all squares are known as perfect squares
Properties of Square of a Number
- A Perfect square always ends with 0, 1, 4, 5, 6, 9. That means the numbers that are not found at the unit place of a perfect square are 2, 3, 7, 8.
Example, Is 157 a Perfect Square? NO. Since 157 ends with the number 7 and according to the first property, a perfect square does not have 7 at the unit place. - Squares of even numbers will always be even in nature and the squares of the odd numbers will always be odd in nature. Even numbers are divisible by 2, while odd numbers are not divisible by 2.
Example, the square of 6 is 36 which is even, square of 7 is 49, both the numbers are odd in this case. - If a number has 0 at its unit place, the square of that number will also have 0 at the unit place. Similarly, if a number has 5 at its unit place, the square of that number will also have 5 at its unit place.
Example, the square of 10 is 100, the square of 30 is 900, the square of 5 is 25, the square of 15 is 225.
Finding the Unit Digits of the Squares of a number
The most basic way to find the unit digit of the square of a number is by simply multiplying the number with itself and then looking at the unit digit obtained in the results since the result will provide the square of that number.
Another easier method or trick to find the unit digit of the square of a number without actually multiplying the number is,
- Look at the number at the unit place.
- Do the square of just the unit place’s number. The unit digit of that squared number is equal to the unit digit of the square of the number given.
- It is very clear that the square of the number (No matter how large the digit is), will give the same unit place digit as given by simply squaring the unit place number.
This trick will not help in actually finding the square, but it will help in finding the unit place digit of the square of the number.
What will be the unit digits of the square of the following numbers?
39, 297, 5125, 7286
The above question can be done in two methods,
The first one is to simply square the numbers and look at the unit digit.
| Numbers | Squares |
| 39 | 1521 |
| 297 | 88209 |
| 5125 | 26265625 |
| 7286 | 53085796 |
As it is clear that the above method is time-consuming since the numbers are very large and hence the calculations too.
The second Method, squaring the unit digits,
- 39 ⇢ Unit digit ⇢ 9, square of 9 = 81, Therefore, the unit digit of the square of 39 is 1.
- 297 ⇢ Unit digit ⇢ 7, square of 7= 49, Therefore, the unit digit of the square of 297 is 9.
- 5125 ⇢ Unit digit ⇢ 5, square of 5= 25, Therefore, the unit digit of the square of 5125 is 5.
- 7286 ⇢ Unit digit ⇢ 6, square of 6= 36, Therefore, the unit digit of the square of 7286 is 6.