Handling different Number systems
When I put 1 next to another 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 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 102 +2x101 +1
In base 3 it represents 32 + 2x3 + 1= 16(Note 11 represents 4 in this system)
In base 7 it represents 72 + 2x7 + 1= 64(Note 11 represents 8 in this system)
(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 but with modification it becomes 1001 because of the carry over of 1 from right to left and the number 1001 is nothing but 23 + 1= 9 (which is 3 squared and 11 represents the number 3 in binary system)
Let us check another example:
101x 101= 101
101
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11001
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In binary system it is 5x5 =25(From place values above it is 24 +23 +1)
In base 4 it becomes instead of 11001, 10201 (pl note this system has 2 in it where as binary system has no 2 in it) which is 44 + 2x 42 +1= 256+32+1=289 which is nothing but 17 square and 101 represents just 17 in that system.
You can guess what it represents in base 6.
Yes it is just 64 + 2x62 +1= 1296+72+1= 1369 which is 37 square (what 101 means in this system)
Thus it reduces to the algebraic expression (x^{2} + 1)2 = (x^{4} +2 x2 +1) and exactly what it is. This proves how the value is independent of the number system.
Take for example 5x8
In binary it is represented as 101 x 1000= 101000= 32 + 8= 40
Converted to algebraic equation it means (x^{2} +1).( x^{3} )= x^{5} + x^{3} and exactly what it is.
In base 3 it is represented as 12 x 22= 1111= 27 + 9+3 +1= 40
Converted to algebraic equation it means (x+2).(2x+2)=2x^{2} +6x + 4 = but since there is no number equal to or above 3, this splits up as follows:
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^{2} +x, and 2x^{2} is carried over2x^{2 }and the carried over 2x2 add up and split to x^{3} +x^{2} Thus it becomes finally x^{3}+x^{2}+x+1.
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.
Yes it is just (x^{2}+ 4) since the product is 5(x+2), with 5x2 splitting up to x+4 and x carried over. 5x becomes 6x with the carried over x, and becomes x^{2}
Let it be clear that the form the product appears to us varies but the value is the same.
It becomes clear to you now how accustomed we are to this decimal system. It needs a little bit of care to understand the intricacy and please try it over and over; you will find it easier than you have expected it to be. |