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Education | Share This Page | |||

Mathematics Made Easy |
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by N. S. Murty |
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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.
Let us find the square of 39.
Thus you can find the square of any number between 1 to100 now. Can't you?
Now I am sure you can say what is the square of 125.
How much time did you take to answer this?
Thus it is fun to find the products of numbers.
Suppose you need to multiply 95 and 97 or 51 and 59 or 35 and 39 etc. We have a wonderful system in place which helps us find the product of any pair of multiplicands. Let us start with 95 and 97. The rule says that whenever you come across pairs of multiplicands like above, write the multiplicands one below the other and follow the procedure detailed below:
Yes. The process looks so long and cumbersome. But in reality it is damn easy and has a self-check for the value obtained. 95 The difference of the numbers with base looks like this: 95 -05 You can see by cross adding the multiplicands and the differences, here 97-5=92 as also 95-3=92. Since we have taken 100 as base what we got this value is 100s. Then multiplying (-5) and (-3) we get 15. Thus the product of 95 and 97 is 9215. 51 +1 and as already explained the 60 on the left side is 50s as such the real value in 10s is 60X50= 3000 or 30 100s by adjustment. So the final value is 3000+ 09= 3009 35 -5 Since the value 34 represents 40s the value in tens is 34X4 = 136. So the final value is 1365 Let us recap what we did and what our operations mean. Whenever we are subtracting the base from the given number, we are trying to find out how far is our number from the base. Thus when two such numbers are multiplied, we are getting the shortage from the product of the bases. But in the number system, this product place is nothing but the x Base: 90 82 -8 80 +16 Here base being 90, the value is (90X8 =720 in tens or 72 in hundreds) 7216 Base: 120 121 +1 130 9 Since 120 is the base here the final value is (13X12 = 156 in hundreds) 15609 8 -2 6 +4 Here no adjustment is necessary since 10 is the base. So 8x8= 64. Did you get it right? You see here another example; Base: 10 8 -2 4 +8 Here gain no adjustment necessary since 10 is the base. What about this example? Base: 10 12 +2 You can clearly see that the product is 4 short of 10 square since 10 is our base. Hence the value is 100 - 4= 96. Now, do you get it right?
When I put One (1) next to another One (1) and ask you what is that you will invariably say it is eleven. Of course, you are right because you are grown up and habituated to decimal system. But technically speaking you are only partly correct. The number represents different numbers in different systems. In binary system (formed from 0 and 1) it represents 3. In base 3 (the number system built around numbers 0, 1, and 2) it represents 4, in base 5 it is 6, in base 7 it is 8 and so on. Actually the number systems were simplified with the invention of Zero, acknowledged as the greatest contribution of Indian Mathematics. It is time to know that any number system x consists of digits from 0, 1, 2, 3… to (x-1) and the place values from right to left increase powers of x times. The beauty of these systems is that our regular multiplication tables hold good representing the appropriate values. Let me elaborate. We know 11x11= 121 in the decimal system if it represents 10 (Note: I did not mention binary system here at this stage to avoid confusion though it is true in that system also, only difference being in binary system we don’t represent any number as 121 (since binary system has only numbers 0 and 1 and there is no 2 in it) but with modification it becomes 1001 because of the carry-over of 1 from right to left. Let us check another example in different number systems: Binary System: 101x 101= 101 In binary system it is 5x5 =25 (From place values above it is 2 In base 4 it becomes instead of 11001, 10201 which is 4 Thus 101 x 101 reduces to the algebraic expression (x Take for example 5x8 In binary it is represented as 101 x 1000 = 101000= 32 + 8= 40. Converted to algebraic equation it means (x In base 3 it is represented as 12 x 22= 1111= 27 + 9+3 +1= 40 4 in the units place becomes x+1 and x is carried over. 6x becomes 7x with the carried over x, and splits up to 2x In base 6 it is represented as 5 x 12 = 104 = 36 + 4= 40 I am sure you can work it out algebraically from the above examples given. Let it be clear that the form the product appears to us varies but the value is the same. Continued to Hand Multiplication |
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15-Sep-2018 |
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VENU GOPAL NOUDURI11/04/2018 00:31 AM |
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