Squaring numbers close to the base 10, 100, 1000 and so on (part-1)

To square numbers close to the bases of powers of 10 i.e. 10, 100, 1000 and so on easily at extremely fast speed, just follow the steps below:

Step-1: Find the surplus or deficit from the base 10, 100, 1000 & so on

Step-2: Add the surplus (if it is more than base) with or subtract the deficit (if it is less than base) from the whole number given and put the result.

Step-3: Find the square of surplus or deficit and write the result in the last places. Be sure that, since 10 has 1 zero so, 1 more digit to go for numbers near 10, since 100 has 2 zeros so, 2 more digits to go for numbers near 100, accordingly since 1000 has 3 zeros, 3 more digits to go for numbers near 1000 and so on. So, carry forward or put extra zero(s) if necessary to place the digits accurate.

Squaring numbers close to base 10

What is the square of 17? or, 17² =?

Notice that 17 is close to base 10. So,

1)       Find the surplus or deficit from base 10. It is 7 surpluses.

2)       Since 17 is 7 more than base 10 so, add 7 to the entire number, which is 17+7=24. Write 24 as the first part of the answer.

3)       Now, since 10 has 1 zero so, 1 more digit to go. Find the square of surplus 7 which is 49. Write 9 and carry forward 4 to be added to 4 making it 8.

So, our final answer is 289.
 

Squaring numbers close to base 100

Ideal for squaring numbers from 76 – 125 (ranges vary at capacity).

104² =?

Notice that 104 is close to base 100. So,

1)      Find the surplus or deficit from base 100. It is 4 surpluses (104–100=4).

2)     Since 104 is 4 more than base 100 so, add 4 to the entire number, which is 104+4 = 108. Write 108 as the first part of the answer.

3)      Now, since 100 has 2 zeros so, 2 more digits to go. Find the square of surplus 4 which is 16. Write 16 as the last 2 digits of the answer.

So, our answer is 10816

Now what if the number that you are trying to square is below the base that is below 100?

Once again, the same thing. Let’s say you want to find 97²

1)       Find the surplus or deficit from base 100. It is (100 – 97 =) 3 less than 100.

2)       Since 97 is 3 less than base 100 so, deduct 3 from the entire number, which is 97-3 = 94. Write 94 as the first part of the answer.

3)       Now, 2 more digits to go. Find the square of deficit 3 which is 9. Put 09 as the last 2 digits of the answer. Notice the extra ‘0’. An extra 0 is to be added to make it a 2 digit number.

So, our final answer is 9409

107² =?

Add the distance (7) from the base 100 to the entire number. 7 + 107 = 114.

Now, 2 more digits to go. Square of 7 is 49.

So, the answer is 11449.

113² =?

Once again, add surplus 13 that is the distance from the base 100 to the entire number 113.

So, 113+13 = 126.

Now for the last 2 digits you need to find the square of the distance that is 13.

Square of 13 is 169.

Now put 69 and add 1 to 126 as we have left 2 places to go to complete the square.

1+126 = 127

The answer is 12769.

89² =?

89 is 11 less than the base 100.

Subtract 11 from 89, which is 78.

Square of 11 is 121.

Write 21 in the last 2 places and add 1 to 78 to make 79

So, the answer would be 7921.