Squares and Square Roots

                                                                 Squares
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properties of  square numbers:The numbers that have 0, 1, 4, 5, 6 or 9 in their units place maybe perfect squares where as the numbers that have 2, 3, 7 or 8 in their units place are never perfect squares.

Number	Square number
If a number has 1 or 9 in the units place	Square number ends with 1
If  a number has 4 0r 6 at units places	Square number ends with 6
If number ends with 0 at units place	Square number ends with 00
If number ends with 5 at units place	Square number ends with 5.

Numbers	Square of the number
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576

Numbers	Square of the numbers
25	625
26	676
27	729
28	784
29	841
30	900
35	1225
40	1600
45	2025
50	2500
55	3025
60	3600
65	4225
70	4900
75	5625
80	6400
85	7225
90	8100
95	9025
100	10000
125	15625
150	22500
175	30625
200	40000

Numbers  between  Square Numbers

1² =1(1)

2² = 4(2,3,4 ) between  1² and 2² we have 2 non square numbers

3²=9(5,6,7,8,9) between 2² and 3² we have 4 non square numbers

4²=16(10,11,12,13,14,15,16) between 3² and 4² we have 6 non square numbers

5²=25(17,18,19,20,21,22,23,24,25) between 4² and 5² we have 8 non square numbers

So number of non square numbers between square numbers formula :

If first number   = n

Second number=n+1

   (n+1)² - n  ²  =2n+1

For example:

 If n =5

N+1= 6

According to formula 2*5+1 =11

Verify

5² =25

6² = 36(26,27,28,29,30,31,32,33,34,35,36)

Including second digit square number.

If we need only difference  the two square numbers

Means 25 to 36 we have 10 numbers difference

So formula for difference of the two square numbers is 2

Adding  Odd Numbers: If the numbers is a square number ,it has to be the sum of successive odd numbers starting from 1

1 (one odd number)                          =1=1²

1+3 (sum of first two odd numbers)= 4=2²

1+3+5(sum of first three odd num)  =9=3²

1+3+5+7                                           =16=4²

1+3+5+7+9                                      =25=5²   

1+3+5+7+9+11                                =36=6²

1+3+5+7+9+11+13                          =49=7²

1+3+5+7+9+11+13+15                   =64=8²

1+3+5+7+9+11+13+15+17             =81=9²

1+3+5+7+9+11+13+15+17+19       =100=10²

Sum of the consecutive natural numbers

3² = 9     =4+5

5²=25    =12+13

7²=49    =24+25

9²=81   =40+41

11²=121=60+61

13²=169=84+85

15²=225=112+113

Square Number patterns:

1²           =                   1

11²         =                1 2 1

111²       =            1 2 3 2 1

1111²     =        1  2 3 4 3 2  1

Another interesting patterns

2² = 4
22² = 484
222² = 49284

_________________

9² = 81
99²= 9801
999² = 998001
9999² = 99980001
99999² = 9999800001
______________________________

101² = 10201  
1001² = 1002001
10001² = 100020001 

Finding square of the number:

 23*23=(23)²  =(20+3)²

           =20(20+3)+3(20+3)

          =20²+20*3+3*20+3²

          =400+60+60+9

          =529

1.Finding the square root number by repeated subtraction:

Square number subtract by odd numbers. First take square of one number and subtract with 1. Then you get product again take that product subtract with consecutive odd number(3). Then again take second product and subtract with consecutive odd number(5).

For example:

81 =9²

81 – 1 = 80

80 – 3 = 77

77 – 5 = 72

72 – 7 = 65

65 – 9 = 56

56 – 11 = 45

45 – 13 = 32

32 - 15  = 17

17 – 17 = 0

In above process will do  to get final product ‘0’ . count number of steps we did. That is the square root of given number.

In above we did 9 times . so that is 9²

Other example 16:

16 – 1 =15

15 – 3 = 12

12 – 5 = 7

7 – 7  = 0

So we did 4 times that means 16 is the square of 4.

2.Find out square root number through prime factorization:

Fine the square root of 256

256  =  2*2*2*2*2*2*2*2   = (2*2*2*2)²

                                                                             = 2*2*2*2

                                                  = 16

By this process we can find out that given number is whether perfect square or not.

3.Find square root my long division method.

a)place a bar on every two digits start from units place. If number of digits      are  odd leave odd numbers on left hand side.

b)Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend .divide and get the remainder .

c)bring down the numbers under the next bar to the right of the remainder.

d)Double the divisor and take that number as divisor to the new dividend.

e)guess the largest possible digit to fill the blank which will also become the new digit in the quotient .such that when the new divisor is multiplied to the new quotient and the product is less than or equal to the dividend.

f)since the remainders is 0 and no digits are left in the given number . 

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