| 12 = 1 | 112 = 121 | 212 = 441 | 312 = 961 | 412 = 1681 |
| 22 = 4 | 122 = 144 | 222 = 484 | 322 = 1024 | 422 = 1764 |
| 32 = 9 | 132 = 169 | 232 = 529 | 332 = 1089 | 432 = 1849 |
| 42 = 16 | 142 = 196 | 242 = 576 | 342 = 1156 | 442 = 1936 |
| 52 = 25 | 152 = 225 | 252 = 625 | 352 = 1225 | 452 = 2025 |
| 62 = 36 | 162 = 256 | 262 = 676 | 362 = 1296 | 462 = 2116 |
| 72 = 49 | 172 = 289 | 272 = 729 | 372 = 1369 | 472 = 2209 |
| 82 = 64 | 182 = 324 | 282 = 784 | 382 = 1444 | 482 = 2304 |
| 92 = 81 | 192 = 361 | 292 = 841 | 392 = 1521 | 492 = 2401 |
| 102 = 100 | 202 = 400 | 302 = 900 | 402 = 1600 | 502 = 2500 |
Squares of Negative Numbers
The squares of negative numbers give a positive value, because if we multiply two negative numbers then it will result in a positive number.
Remember that: (-) x (-) = (+)
Therefore, square of (-n), (-n)2 = (-n) x (-n) = n2
Where n is a number.
Examples:
- (-5)2 = (-5) x (-5) = 25
- (-7)2 = (-7) x (-7) = 49
Numbers between Squares
Suppose there are two square numbers n2 and (n+1)2, then total numbers between these two squares are given by 2n.
Let’s say 32 and 42 are two squares.
32 = 9 and 42 = 16
We need to find the numbers present between 9 and 16.
Here, n = 3
Therefore, total numbers between 9 and 16 = 2n = 2 x 3 = 16
Is that correct? Let us check.
9, 10, 11, 12, 13, 14, 15, 16.
As we can see, the numbers between 9 and 19 are 6. Therefore, the formula given above is applicable to all the squares.
| Numbers Between n2 and (n+1)2 = 2n, where n is any natural number |
Square Roots of Number
As we have already discussed, the square root of any number is the value which when multiplied by itself gives the original number. It is denoted by the symbol, ‘√’. If the square root of n is a, then a multiplied by a is equal to n. It can be expressed as:
√n = a then a x a = n
This is the formula for square root.
Square Roots of Perfect Squares
The perfect squares are the one whose square root gives a whole number. For example, 4 is a perfect square because when we take the square root of 4, it is equal to 2, which is a whole number. Let us see some of the perfect squares and their square roots.
| Perfect Squares | Square Root (√) |
| 0 | 0 |
| 1 | 1 |
| 4 | 2 |
| 9 | 3 |
| 16 | 4 |
| 25 | 5 |
Square Root of Imperfect Squares
Finding the square root of perfect squares is easy but to find the root of imperfect squares is difficult. The root of the perfect square can be estimated using the prime factorisation method.
The square root of imperfect squares is usually fractions. For example, 2 is an imperfect square because 2 cannot be prime factorised and its square root gives a fractional value.
Examples are:
- √2 = 1.4142
- √3 = 1.7321
- √8 = 2.8284
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