The numbers 1,4,9,16....are called Square Numbers because you can arrange the number of counters to form a Square. The 4 Counters are in 2 rows of 2. The 9 counters are in 3 rows and 3 columns. 1 x 1 = 1, 2 x 2 = 4, 3 x 3 =9 , 4x4 = 16, So if we square a number we multiply it by itself. 3 Squared is (32) =9 ; 4 Squared is (42) =16;
Square numbers always have an odd number of factors. All other numbers have
an even number of factors. |
A -Squares of Numbers Ending in 5 |
Example 1: Find 252. We have to find out the square of the number 25For the number 25 |
- the last digit is 5 and the "previous" is 2.
- Hence 'one more than' previous is 2+1=3.
- The method is 'to multiply the previous digit 2 by one more than itself, by 3'.It becomes the LHS of the result, 2x3 = 6.The RHS of the result is 52, i.e., 25. Thus
25x25 = (2x3 = 6) / 25 = 625.(Answer)
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| Example 2: Find 852.We have to find out the square of the number 85. For the number 85, |
- the last digit is 5 and the "previous" is 8.
- Hence 'one more than' previous is 8+1=9.
- The method is 'to multiply the previous digit 8 by one more than itself, by
9'.It becomes the LHS of the result, 8x9 = 72
- The RHS of the result is (Square of 5) = 5x5, i.e., 25. Thus 85x85 = (8x9 = 72) / 25 = 7225.(Answer)
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Assignments Find the Squares Q1. 15 Q2. 125 Q3. 635 Q4. 1105 Q5. 2545. |
Assignments Answer Q1. 152 = 225 Q2. 1252 = 15635 Q3.6352 = 403225 |
Q4. 11052 = 1221025 Q5. 25452 = 6477025 |
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B. Squares of Numbers Close to Bases of Powers of 10. |
| The Method is "what ever the deficiency subtract that deficit from the number and write along side the square of that deficit". This method can be applicable to obtain squares of numbers close to bases of powers of 10 |
Method-1
Numbers near and less than the bases of powers of 10 |
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Example 1: Find 92
Sol Here Base is 10. The answer is separated in to two parts by a '/' |
| Note that deficit is 10 - 9 = 1 |
Multiply the deficit by itself or square of 1 = 1.
As the deficiency is 1, subtract it from the number i.e.,9-1 = 8. |
Now put 8 on the left and 1 on the right side of the vertical line or slash i.e., 8/1.
Hence 81 is answer. |
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Example. 2: Find 962
Sol Here Base is 100 |
| deficit is 100-96=4 Square of 4 it is 16. |
| The deficiency subtracted from the number 96 gives 96 - 4 = 92, we get the answer 92 /16 hus 96x96 = 9216. |
Example 3: Find 9942
Sol Here base is 1000. |
| Deficit is 1000 - 994 = 6. Square of 62 is 36 |
| Deficiency subtracted from 994 gives 994- 6 = 988 |
| Answer is 988 / 036 [036 since base 1000 has 3 zero's] Answer = 988036 |
Example 4: Find 99882
Sol Here Base is 10,000 |
| Deficit = 10000 - 9988 = 12 |
Square of deficit = 122 = 144.
Deficiency subtracted from number =9988 - 12 = 9976. |
| Answer is 9976 / 0144 [0144 since base 10,000 has 4 zero's ].Answer = 99760144 |
Example 5: Find 882
Sol Here Base is 100.
|
Deficit = 100 - 88 = 12. |
Square of deficit = 122 = 144.
Deficiency subtracted from number = 88 -12 = 76. |
Now answer is 76 / 144
[since base is 100, keep 44 and carry over 1 to left] Answer = (76+1)/44 = 7744 |
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Assignments: 1) 72 2) 982 3) 9872 4) 142 5) 1162 6) 10122 7)192
8) 4752 |
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| Answer :1) 49 2) 9604 3) 974169 4) 196 5) 13456 6) 1024144 7) 361 8) 225625 |
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Method-2
: Numbers near and greater than the bases of powers of 10. |
Instead of subtracting the deficiency from the number we add and proceed as
in Method-1 |
| Example1: Find 132 Base is 10, surplus is 3 |
Square of surplus =32 = 9 |
| Surplus added to the number =13 + 3 = 16. Answer is 16 / 9 = 169. |
| Example2: Find 1122. Base = 100, Surplus = 12, |
Square of surplus = 122 = 144
|
Add surplus to number = 112 + 12 = 124.
Answer is 124 / 144 = (124+1)/44= 12544 |
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Example 3: Find 100252
Base = 10000, Surplus = 25, |
| Square of surplus = 252 = 625 |
Add surplus to number = 10025 +25 = 10050. Answer: 10050 / 0625
[ since base is 10,000 ] = 100500625. |
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| Assignments Find the Squares of the following |
| 1) 7 2) 98 3) 987 4) 116 5) 1012 6) 9988 |
| Assignments Answers Find the Squares of the following |
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Method 3
This is applicable to numbers which are near to multiples of 10, 100, 1000
.... etc. [different base] |
|
Example: Find 3882
Nearest Base = 400. [400 = 4 x 100] |
As the number is less than the base we proceed as follows Number 388,
deficit = 400 - 388 = 12 |
| Since it is less than base, deduct the deficit i.e. 388 - 12 = 376. |
| Multiply this result by 4 since base is 4 X 100 = 400. |
So, 376 x 4 = 1504
Square of deficit = 122 = 144. |
Answer : 1504 / 144 = 150544
[Since we have taken multiples of 100, write down 44 and carry 1 over to left] |
Example 2: Find 4852
Nearest Base = 500 [Treat 500 as 5 x 100] |
| and proceed Number 485, deficit = 500 - 485 = 15 |
| Since it is less than base, deduct the deficit i.e. 485 - 15 = 470. |
| Multiply this result by 5 since base is 5 X 100 = 500. So, 470 x 5 = 2350 |
Square of deficit = 152 = 225. Hence answer is 2350 / 255 [since we have taken multiples
of 100]. Answer = 235255 |
|
| Example 3: Find 672 Nearest Base = 70 [Treat 10 as 7 x 70] |
| Number 67, deficit = 70 - 67 = 3 |
| Since it is less than base, deduct the deficit i.e. 67 - 03 = 64. |
| Multiply this result by 7 since base is 7 X 10 = 70. So,64 x 7 = 445 |
| Square of deficit = 32 = 9. |
Hence answer is 448 /9
[since we have taken multiples of 10]. Answer = 4489 |
|
Example 4: Find 4162
Nearest Base = 400 [400 = 4 x 100]
and Here surplus = 16 |
Number 416, deficit = 416 - 400 = 16 Since it is more than base, add the
deficit i.e. 416 + 16 = 432. |
| Multiply this result by 4 since base is 4 X 100 = 400. So, 432 x 4 = 1728 |
Square of deficit = 162 = 256 Hence answer is 1728 /256
= 173056 [since we have taken multiples of 100] |
|
Example 5: 50122
Nearest base is 5000 [ 5000 = 5 x 1000]
surplus = 12 |
| Number 5012, surplus =5012 - 5000 = 12 |
| Since it is more than base, add the deficit 5012 + 12 = 5024. |
| Multiply this result by 5 since base is 5 X 1000 = 5000. So, 5024 x 5 = 25120 |
| Square of deficit = 122 = 144 |
Hence answer is 25120 /144 = 25120144
[since we have taken multiples of 1000, write down 144 as it is]. |
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Assignments Find the Squares |
1) 7 2) 98 3)14 4) 116 5) 1012 6) 475 7) 118 8) 6014 |
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C - STRAIGHT SQUARING |
| We have already noticed methods useful to find out squares of numbers. But the methods are useful under some situations and conditions only. Now we go to a more general formula |
| The Duplex combination process is used in two different meanings. They are |
| a) by squaring b) by cross-multiplying. |
| We use both the meanings of Duplex combination in the context of finding squares of numbers as follows: We denote the Duplex of a number by the symbol D. We define |
- for a single digit 'a', D =a2
- for a two digit number of the form 'ab', D =2( a x b ).
- for a 3 digit number like 'abc', D =2( a x c ) + b2.
- for a 4 digit number 'abcd', D = 2( a x d ) + 2( b x c ) and so on.
|
| If the digit is single central digit, D represents 'square'. |
| Consider the examples: |
Number |
Duplex D
|
| 3 | 32 = 9 |
| 6 | 62 = 36 |
| 23 | 2 (2 x 3) = 12 |
| 64 | 2 (6 x 4) = 48 |
| 128 | 2 (1 x 8) + 22 = 16 + 4 = 20 |
| 305 | 2 (3 x 5) + 02 = 30 + 0 = 30 |
| 4231 | 2 (4 x 1) + 2 (2 x 3) = 8 + 12 = 20 |
| 7346 | 2 (7 x 6) + 2 (3 x 4) = 84 + 24 = 108 |
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| For a n- digit number, the square of the number contains 2n or 2n-1 digits. Thus in this process, we take extra Zeros to the left one less than the number of digits in the given numbers. |
|
Examples:1 Find 622
Since number of digits = 2, we take one extra ZERO to the left. Thus |
| 062 |
| For 2, D = 22 = 4, Write down 4 as the right most digit 4 |
| For 62, D = 2 ( 6 x 2) = 24 , write down 4 and carry over 2 to the left 244 |
For 062, D = 2 (0 x 2) + 62 = 36 36244
Finally answer 622 = 3844. |
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| Examples:2 Find 2342 |
| Number of digits = 3. Extra ZEROS added to the left =Number of digits -1 = 2 Thus |
| 00234 |
| For 4, D = 42 = 16, Write down 6 as the right most digit and carry 1 over to left 16 |
| For 34, D = 2 ( 3 x 4) = 24 , write down 4 and carry over 2 to the left 2416 = 256 |
| For 234, D = 2 (2 x 4) + 32 = 16 + 9 = 25, write down 5 and carry over 2 to the left 25 = 25256 =2756 |
For 0234, D = 2 (0 x 4) + 2 (2 x 3)= 0 + 12 =12, write down 2 and carry over 1
to the left 12 = 14756 = (12252416) |
| For 00234, D = 2 (0 x 4) + 2 (0 x 3) + 22 = 0 + 0 + 4 =4, write down 4 as it is = 54756 ( 412252416) |
| Finally answer = 54756 |
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Examples:3 14262.
Number of digits = 4 Extra ZEROS =Number of digits -1 = 3 Thus |
| 0001426 |
| For 6, D = 62 = 36, Write down 6 as the right most digit and carry 3 over to left 36 |
| For 26, D = 2 ( 2 x 6) = 24 , write down 4 and carry over 2 to the left 2436 |
| For 426, D = 2 (6 x 4) + 22 = 48 + 4 =52, write down 2 and carry over 5 to the left = 522436 |
| For 1426, D = 2 (1 x 6) + 2 (2 x 4)= 12 + 16 =28, write down 8 and carry over 2 to the left = 28522436 |
| For 01426, D = 2 (0 x 6) + 2 (1 x 2) + 42 = 0 + 4 + 16 =20, write down 0 and carry over 2 to the left = 2028522436 |
| For 001426, D = 2 (0 x 6) + 2 (0 x 2) +2 (1 x 4)= 0 + 0 + 8 =8, write down 8 as it is 82028522436 |
For 0001426, D = 2 (0 x 6) + 2 (0 x 2) +2 (0 x 4) +12= 1, write down 1 as it is 182028522436
= 2033476 |
| Answer = 14262 = 2033476 |
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Assignments Find the Squares of the following
Q1. 54 Q2. 123 Q3. 2051 Q4. 3146 |
| Assignments Answers Find the Squares of the following |
| Q1. 2916 Q2. 15129 Q3. 4206601 Q4. 9897316 |
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