why did we create numbers
I mugged up tables in school without actually knowing what they meant. I did the mathematical operations on numbers without actually knowing what I was doing.
Lets say why we had to create negative numbers?
I could relate the answer to the daily problem in trading world. Person A borrows X units of money from B and so person A has -X money. And hence all the math in the money trade becomes easier.
Why did we have to create "i" - the complex number?
X^2 - 1 = 0 is a second order equation and it has clearly two roots
X = 1 and X = -1..
but how about X^2+1 = 0 ... we come up with a number "i" and say X = +i and X = -i are the two roots.
If we get back to the conceptual level it is basically X =1 and X = -1 on the imaginary line. Why did we have to have an imaginary line. It is because we would like to describe two systems (one corresponding to real axis and other to imaginary axis) at the same time.
That is when we write e^(jx) = cos(x) + i sin (x)... we would like to analyse cos(x) and sin(x) at the same time. in other words we would like to analyze the property of a general sinusoid signal measuring its phase from a cosine of same frequency and sine of the same frequency. [remember any sinusoid can be perfectly reconstructed if we know the amplitude, frequency and phase]
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more on the number "e" in the future posts.
Lets say why we had to create negative numbers?
I could relate the answer to the daily problem in trading world. Person A borrows X units of money from B and so person A has -X money. And hence all the math in the money trade becomes easier.
Why did we have to create "i" - the complex number?
X^2 - 1 = 0 is a second order equation and it has clearly two roots
X = 1 and X = -1..
but how about X^2+1 = 0 ... we come up with a number "i" and say X = +i and X = -i are the two roots.
If we get back to the conceptual level it is basically X =1 and X = -1 on the imaginary line. Why did we have to have an imaginary line. It is because we would like to describe two systems (one corresponding to real axis and other to imaginary axis) at the same time.
That is when we write e^(jx) = cos(x) + i sin (x)... we would like to analyse cos(x) and sin(x) at the same time. in other words we would like to analyze the property of a general sinusoid signal measuring its phase from a cosine of same frequency and sine of the same frequency. [remember any sinusoid can be perfectly reconstructed if we know the amplitude, frequency and phase]
*********************************
more on the number "e" in the future posts.
posted by Pilli at 1:41 PM
4 Comments:
yes sudhakar its all about levels of common sense and obviousness.
its when u don't like the air u go in search of the air u like to breathe... that's fubar :D
keep your posts coming in.
laks: thx dude.. sure it would be great if u could post your learning too....
This comment has been removed by a blog administrator.
Hello!
Really nice Blog You guys have here. I am going to be a frequent visitor! BTW my 2 cents on wht we created numbrs : I think beyond the natural numbers, everythings can be historically attributed to "geometrical" or "algebraic" reasons. So \pi started because of geometry reasons ofcourse. Some how how the way reals started is geometric..as suggested by they way we define them...looking at sequences of rational and thinking of them in between "rational" numbers...
On the other hand "i" is more algebraic; as your post described. How do we solve this equation: Create "i" . In the same way "surds" in general I think were "invented"..algebraic....
But the beauty as Sudhakar mention is the way they all come together. "i" is so much geometry and real are clealy expand our algebraic repertoire....
Sorry rambled of..... again loved your post!!
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