SQUARES AND SQUARE ROOTS

 

What is a Square number?

          When you multiply a whole number (or) Integer (not a fraction) by itself, the result is called the Square of that number.

Here are the Squares of 1 to 30 Numbers.,

 

SQARES 1 to 30
1 X 1 = 1	11 X 11 = 121	21 X 21 =441
2 X 2 = 4	12 X 12 = 144	22 X 22 = 484
3 X 3 = 9	13 X 13 = 169	23 X 23 = 529
4 X 4 = 16	14 X 14 = 196	24 X 24 = 576
5 X 5 = 25	15 X 15 = 225	25 X 25 = 625
6 X 6 = 36	16 X 16 = 256	26 X 26 = 676
7 X 7 = 49	17 X 17 = 289	27 X 27 = 729
8 X 8 = 64	18 X 18 = 324	28 X 28 = 784
9 X 9 = 81	19 X 19 = 361	29 X 29 = 841
10 X 10 = 100	20 X 20 = 400	30 X 30 = 900

 Note: 

1) If a number has 1 or 9 in its unit's place then its square ends with 1 in units place.   (We can cross check it by above table).

2) Square of a even number is always even, and square of an odd number is always Odd. 

        Ex: 2² = 4,  6² = 36,   7² = 49,   9² = 81

3) Three natural numbers x, y, z is called a Pythagorean triplet if X² + y² = z².

        Ex: (3, 4, 5),  (6, 8, 10), (7, 24, 25) etc., 

        since 3² + 4² = 9 + 16 = 25, and  5².= 25

For any natural number 'n' which is greater than 1, the Pythagorean triplet is  given by (2n, n²-1, n²+1 )

        Ex: Find the other two Pythagorean triplets, one number which is 10.

        Sol: For any natural number 'n' the Pythagorean triplet is = 2n, n²-1, n²+1

                consider n as 5 as per given above

                2n = 2 *5 = 10,

                n²-1 = 5² - 1 = 25 - 1 = 24, and 

                n²+1 = 5² + 1 = 25 + 1 = 26.

                So, the other two numbers are 24 and 26.

4) The square of a natural number 'n' is equal to the sum of the first n odd numbers.

        Ex: Thus   1² = 1, sum of the first odd number,

                           2² = 4 = 1 + 3, sum of the first 2 odd numbers,

                           3² = 9 = 1 + 3 + 5, sum of the first 3 odd numbers,

                           4² = 16 = 1+3+5+7 =16, sum of the first 4 odd numbers,    etc.,

5) For any natural number 'n' which is greater than 1, (n + 1) X (n - 1) = n²-1.

6) Finding the square of a number which is ending with 5 (last digit as 5) is as follows: 

Ex: 25² = 2 X 3 = 6, 5² = 25 so the answer is 625.

       What ever may be the earlier part (n),  multiply it with one more than it self (n+1) that will become over answer's first part, the second part of the number will be 25 only.  

        35² = 3 X 4 = 12, 5² = 25 so the answer is 1225.

        45² = 4 X 5 = 20, 5² = 25 so the answer is 2025.

        55² = 5 X 6 = 30, 5² = 25 so the answer is 3025.

        135² = 13 X 14 = 182, 5² = 25 so the answer is 18225.

        155² = 15 X 16 = 240, 5² = 25 so the answer is 24025.

Square Root: