What is a Square Number and why do we call it a Square Number?
What is a square number? Square number definition can be defined as numbers that are multiplied by itself. In other words, if a natural number is multiplied by itself it is known as a natural number. For example, 2 multiplied to itself (2x2) is a square number.
Square numbers are named after a square because it forms an area of a square. Imagine that you have 9 items of 3 different categories. If we arrange it into a grid of 3 columns and 3 rows(3*3=9), placed vertically and horizontally. This grid thus will take the form of a square.
Squaring Decimals
Just like squaring whole numbers (integers) is easy, it is also very easy to square decimals and both are done in the same way.
For example, 1.53 = 1.53 x 1.53 = 2.3409
7.19 x 7.19 = 51.6961
Properties of Square Numbers
Property 1: A number that has 2, 3, 7, or 8 at its unit's place can never be a perfect square. In other words, square numbers never end in 2, 3, 7, or 8.
Example: 152, 7693, 14357, 88888, 798328 can never be perfect square as the numbers in their unit digit ends with 2,3,7 or 8
Property 2: The number of zeros at the end of a number determines if it is a perfect square or not. If a number ends with an even number of zeros then it can be a perfect square but if a number ends with an odd number of zeros then it might not be a perfect square.
Example: 250000 is a perfect square as it has an even number of zeros.
25000 is not a perfect square as it ends with an odd number of zeros.
Property 3: Squares of even numbers results in even numbers and squares of odd numbers result in odd numbers always.
Example : 8 2 = 8 x 8 = 64. (both 8 and 64 are even numbers)
7 2 = 7 x 7 = 49 (both 7 and 49 are odd numbers)
Property 4: On Squaring a natural number other than one the multiple will either be a multiple of 3 or will exceed a multiple of 3 by 1.
Example: 635,98,122 are not perfect squares because they leave the remainder 2 when divided by 3.
Property 5: On Squaring of a natural number other than one the multiple will either be a multiple of 4 or exceeds a multiple of 4 by 1.
Example: 67,146,10003 are not perfect squares because they leave the remainder 3,2,3 respectively when divided by 4.
Property 6: The unit’s number of the square of a natural number is the unit’s digit of the square of the digit at the unit's place of the given natural number.
Example :
1) Unit digit of square of 146.
Solution: Unit digit of 6 2 = 36 and also the unit digit of 36 is 6, therefore, the unit digit of square of 146 is 6.
2) Unit digit of square of 321.
Solution: Unit digit of 1 2 = 1, therefore, the unit digit of square of 321 is also 1.
Property 7: There are n natural numbers p and q so that p 2 = 2q 2.
Property 8: For each natural number n, (n + 1)2- n2 is equal to ( n + 1) + n.
Property 9: The square of a number n = to the sum of first n odd natural numbers.
1 2 = 1
2 2 = 1 + 3
3 2 = 1 + 3 + 5
4 2 = 1 + 3 + 5 + 7 and so on.
Property 10: If a natural number m is greater than 1,
(2m, m 2 - 1, m 2 + 1) is a Pythagorean triplet.
1 x 1 | 1 | 21 x 21 | 441 | 41 x 41 | 1681 | ||
2 x 2 | 4 | 22 x 22 | 484 | 42 x 42 | 1764 | ||
3 x 3 | 9 | 23 x 23 | 529 | 43 x 43 | 1849 | ||
4 x 4 | 16 | 24 x 24 | 576 | 44 x 44 | 1936 | ||
5 x 5 | 25 | 25 x 25 | 625 | 45 x 45 | 2025 | ||
6 x 6 | 36 | 26 x 26 | 676 | 46 x 46 | 2116 | ||
7 x 7 | 49 | 27 x 27 | 729 | 47 x 47 | 2209 | ||
8 x 8 | 64 | 28 x 28 | 784 | 48 x 48 | 2304 | ||
9 x 9 | 81 | 29 x 29 | 841 | 49 x 49 | 2401 | ||
10 x 10 | 100 | 30 x 30 | 900 | 50 x 50 | 2500 | ||
11 x 11 | 121 | 31 x 31 | 961 | 51 x 51 | 2601 | ||
12 x 12 | 144 | 32 x 32 | 1024 | 52 x 52 | 2704 | ||
13 x 13 | 169 | 33 x 33 | 1089 | 53 x 53 | 2809 | ||
14 x 14 | 196 | 34 x 34 | 1156 | 54 x 54 | 2916 | ||
15 x 15 | 225 | 35 x 35 | 1225 | 55 x 55 | 3025 | ||
16 x 16 | 256 | 36 x 36 | 1296 | 56 x 56 | 3136 | ||
17 x 17 | 289 | 37 x 37 | 1369 | 57 x 57 | 3249 | ||
18 x 18 | 324 | 38 x 38 | 1444 | 58 x 58 | 3364 | ||
19 x 19 | 361 | 39 x 39 | 1521 | 59 x 59 | 3481 | ||
20 x 20 | 400 | 40 x 40 | 1600 | 60 x 60 | 3600 |
61 x 61 | 3721 | 81 x 81 | 6561 | |
62 x 62 | 3844 | 82 x 82 | 6724 | |
63 x 63 | 3969 | 83 x 83 | 6889 | |
64 x 64 | 4096 | 84 x 84 | 7056 | |
65 x 65 | 4225 | 85 x 85 | 7225 | |
66 x 66 | 4356 | 86 x 86 | 7396 | |
67 x 67 | 4489 | 87 x 87 | 7569 | |
68 x 68 | 4624 | 88 x 88 | 7744 | |
69 x 69 | 4761 | 89 x 89 | 7921 | |
70 x 70 | 4900 | 90 x 90 | 8100 | |
71 x 71 | 5041 | 91 x 91 | 8281 | |
72 x 72 | 5184 | 92 x 92 | 8464 | |
73 x 73 | 5329 | 93 x 93 | 8649 | |
74 x 74 | 5476 | 94 x 94 | 8836 | |
75 x 75 | 5625 | 95 x 95 | 9025 | |
76 x 76 | 5776 | 96 x 96 | 9216 | |
77 x 77 | 5929 | 97 x 97 | 9409 | |
78 x 78 | 6084 | 98 x 98 | 9604 | |
79 x 79 | 6241 | 99 x 99 | 9801 | |
80 x 80 | 6400 | 100 x 100 | 10000 |
We can also make a square numbers list for easier reference.
Q1. What is the Relation Between Square Numbers and Square Root?
Ans. When a square root is multiplied, it produces a square number. Thus, a square root has an inverse operation than the square number. Let’s take an example, the square root of 36 is 6 because 6 x 6 = 36. Finding the square root of a number can be much trickier than calculating the square number. Therefore, the modern calculator has the square root button to make the calculation easier.
Q2. How do we Square Negative Numbers?
Ans. We all know that when two negative or two positive numbers are multiplied to each other then the result will always be a positive number. For example, if we multiple -5 to -5 then the result will be 10. If 5 will be multiplied to 5, the result will be 10 as well. But what if a negative number is multiplied to a positive number or vice versa? The result would be a negative number. For example, if -3 is multiplied to 3, the result would be -9. A number cannot be a square number if it is a negative number because -3 and 3 are very different from each other.
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