Question

How would you visualize the operation and conversion of partial fractions and more compounded fractions?

Answer 1

Ex: \[\frac{ 1 }{ (x+1)(x+2)(x+9) } = \frac{ A }{ x+1 } + \frac{ B }{ x+2 } + \frac{ C }{ x+9 }\]

Answer 2

you mean how u solve it right?

Answer 3

No. How to visualize it. How these changes occur, or what they could mean in the applicational world. Scaling A,B,C makes the function to the left match the one to the right. I just wonder what these characteristics are.

Answer 4

@ArifL

Answer 5

I never really got a good grasp on the justification for partial fraction decomp myself.
This answer is kind of neat though,
http://math.stackexchange.com/questions/20963/integration-by-partial-fractions-how-and-why-does-it-work

Answer 6

@zepdrix that's a neat structure of computations, indeed. It did not discuss any theoretical insight, but it offered some ideas... such as the covering term technique. I do like to imagine this as breaking a huge, "complicated" window into smaller and simpler blocks. I just need to be able to see this in the numbers themselves and find a good analogy to confirm. But thank you. The question remains as to how to interpret them, though, or what analogy would be useful to describe this very process.

Answer 7

the justification is; how would you add rational functions (glorified fractions) together to start with?

Answer 8

@amistre64 By looking at each individual piece/ fraction and scaling them up to a common "whole piece" or picture. That's true. But how exactly does it work backwards?
Breaking down your "whole picture" or denominator into smaller pieces in which the part (or the numerator) decomposes too. A

Answer 9

Partial fraction decomposition is used so we can integrate an otherwise very awkward function or also used to find the inverse Laplace transform

Answer 10

\[\frac56=\frac A2+\frac B3\]
\[6(\frac56=\frac A2+\frac B3)\]
\[5=3A+2B\]