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Mathematics Made Easy II
by NS Murty

You must have noticed when the number is above 15 and when you take 10 as base the calculation is somewhat difficult unless you have mastered the squares up to 9.

Now we have another rather easy method.

When the given number is nearer to a 10 multiple, take that 10 multiple as the base and find the square but, with a little adjustment for the new base. Thus you will find it easy to take 20 as base instead of 10 when you need to calculate the squares of numbers above 15. We have already seen when we take 10 as base we are subtracting from or adding to the given number the quantity by which it differs from 10 and putting as tens.

Similarly, when we take 20 or 30 or 70 as base, we will be getting values in 20s, 30s, and 70s. So we must multiply these 20s, 30s, and 70s appropriately by 2, 3 and 7 to convert them to 10s. That is all the adjustment we need to make. Let us see by example.

Let us calculate the square of 19.

Take 20 as base. Then 19 is less than 20 by 1. So subtract 1 from 19 and put it as 20s. So we get 18 twenties and multiplying it with 2 we get our 10s. Thus we have (19-1)= 18 twenties which is 36 tens. As usual we square the 1 and we get 361 which is the square of 19. You can see by taking 10 as base we got 19+9 =28 tens and another 8 tens from the square of 9 making 36 tens. It is likely that we forget to add these tens from 9 square. So it is always better to take the 10 multiple nearer to the given number.

Let us find the square of 39.

Take 40 as base. We get 39-1= 38 forties or 38x4= 152 tens. Answer is 1521. This you can say in a trice without multiplying by the usual long drawn method. Isn't it?

Try 78 square.

Take 80 as base. 78 is less than 80 by 2. You get 78-2 =76 eighties or 76x8 = 608 tens hence 78 square is 6084

Try 44 square

Take 40 as base. 44 is 4 greater than 40. So we get 44+4 = 48 forties or 192 tens. Squaring 4 we get another 1 ten and 6 units. Answer therefore is, 1936.

Thus you can find the square of any number between 1 to100 now. Can't you?

Well I know you ask if the number is like 25, 35, 45, 65, etc exactly in the middle of twenties, thirties etc., you do not see advantage of one method over the other. But strangely, you don't need any of these two methods at all. There is nothing simpler than such numbers having 5 in the units place to find the square for. It is the easiest of all things.

All you have to do is take the number to the left of 5 and multiply it by its next number and put the product in 100s and always put 25 to the right of it. Just that.

Let us see by example. Find the square of 25. Now the number to the left of 5 is 2. The next number to 2 is 3. Then 2x3 is 6 we got 6 hundreds. (I am talking about the place value) Putting 25 to the right of it we get 625.

35 square is 3x4= 12 therefore 1225
45 square is 4x5=20 therefore 2025
75 square is 7x8= 56 therefore 5625
95 square is 9x10=90 therefore 9025 and so on.

Now I am sure you can say what is the square of 125.

Yes.

12x 13 = 156 Therefore 125 square is just 15625.

How much time did you take to answer this?

There is a very interesting corollary, that means, which follows this operation.

For any two numbers between the sets of ten like 21-29, 31-39, 41-49 , ... 219-220.551-559...so on,

When the sum of the numbers in the units place is ten, then also the above principle holds. That is multiply the number to the left of units place by its next number and put it in hundreds' place. However the difference now is instead of 25 you will put the product of the two numbers in the units place.

Let us study by example:

Find the product of 23 and 27.

Since 3+7 is ten, the value is 621 See the following table:

21 * 29 it is 609
(don't forget that when there is no value for tens, there is 0 always.)
22 * 28 it is 616
23 * 27 it is 621
24 * 26 it is 624
25 * 25 it is 625

82 * 88 it is 7216 ( remember the carry forwards)
121 * 129 is 15609
124 * 126 it is 15624

Thus it is fun to find the products of numbers.  

12-Mar-2006
 
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