When I was a child I heard mathematics was tough and hard and even without attempting it, I started fearing it. I don’t exactly remember how this fear was allayed by my teacher, but one fine morning I started doing math on my own and slowly it developed into a passion. I am sure most of the children experience(d) the same way I did and I just want give few important tips from our old Vedic tradition. I don’t try to give you any Sanskrit Sutras but shall give you the essence of it in a capsule. This ranges from easy methods of multiplication, squaring, finding the products of large numbers, solutions to quadratic, simultaneous equations by mere observation etc. All these things one can do without using a calculator. If one develops the skill by constant practice, I am sure anyone can do these most of the times by mere observation and simple mental calculation.
I start with a sample of finding the squares of small numbers orally (and later expand it to finding tables upto 20X20 just using the ten fingers of both hands.)
First finding the square: ( I am confining myself to numbers up to 20 only for the present and as you shall see it can be extended to any number later)
The rule is as follows
When you are given a number to find its square, take ten as the base, and if the number is less than (more than) ten, then subtract (add) by the amount it was less (or more) from (to) the given number. Treat it as tens. Then simply square the number you subtracted from ten (or added to ten) and put it as units. If the square of this number has tens and units, carry over the tens to the tens we have already.
Explanation by example
Let us suppose we are given to find out the square of the number 8
Taking 10 as base, 8 is less than ten by 2.
So subtract 2 from 8. You get 6. Put it in tens place. Then square 2 (8 is less than 10 by 2) which is 4 and put it in units. So the square of 8 is 64.
Let us suppose we have to find the square of 12.
12 is more than ten by 2. So add 2 to 12. We get 14. Put it in tens.
Then square 2. We get 4. Put it in units place. We thus get 144 as the square of 12.
Find the square of 6.
6 is less than 10 by 4. subtract 4 from 6 and put it in tens. We have 2 tens.
Then square 4 and we get 16 as its square. This has tens and units. Carry the tens to the tens we already have. Thus we get 2 + 1 = 3 tens and 6 units. So square of 6 is 36.
Find the square of 16
16 is greater than ten by 6. So add another 6 to 16. We get 22 and put it under tens.
Square 6. We have 36. Carry 3 to tens and put 6 in units. We have 22+3 = 25 tens and 6 units. So 16 square is 256.
Find the square of 19.
19 is more than 10 by 9. Add another 9 to 19. We get 28 and put it in tens.
Square 9 and we get 81. Carry 8 to tens and keep 1 in units. Then we get 28+8 = 36 tens and 1 unit which is 361. So square of 19 is 361.
It is so easy. Isn’t it? Now you should be able to tell the square of any number between 1 and 20 without committing it to memory.