Elegance Resurrection

We the People of the United States, in Order to form a more perfect Union, establish Justice, insure domestic Tranquility, provide for the common defence, promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity, do ordain and establish this Constitution for the United States of America.

Aw shit, wrong preamble. Damn flashcards.

Here’s all the salient points thus far, with the partial fraction in red:

1We wanna find A and B. So multiply both sides by (x-a)(x-b). What does this do? Well, the left hand side just becomes -x. Simple. When you cancel the similar terms on the right hand-side, this happens:

2So now the traditional trick is to just substitute in various values of ‘x’ which will get us closer to finding out either A or B. Since we have two unknowns, we want to get rid of one, ideally. Thankfully, the equation is well-suited to that.

Finding A

Substitute a for x!

3

The B term dissapears, leaving an empty void, a Looney Tunes style B-shaped-hole! So now we just have: -a = A(a-b). We can put in the real values of a and b, and finally get out a value for A!

4Finding B

Exact same process, except you substitute b for x!

5Same deal. The (b-b) cancels to zero, leaving the useful B term, and an A-hole. Oh.

Aaaaanyhoo. This happens, same deal as with A. But with B:

6

Swish. Here’s the star players we’ve got on the bench at the moment:

7Putting all that jazz into the equation:

8

So far, so good. Now, remember how I said to keep in mind how a and b interact? a*b = -1. This comes in useful here. We can ignore the 1 over root 5 for now because it’s a constant, so both terms in parentheses are multiplied by it. So we’re just dealing with the two parenthetical bits. Anyway, we know a*b = -1, so we can multiply top and bottom by one of those, in order to turn the top part into -1. Sounds confusing, but it’s simple, here:

9Those expressions inside the square brackets are pretty sweet. Turns out they’ve got a very similar form to the sum of a geometric series. That sounds nonsensical and vague right now, but it’s a big deal. I’ll make sense of it in the next post. Which should cover some really neat ideas concerning the nature of infinity. Seriously, it’s cool stuff. I mean, how often do you get to manipulate a manifestation of infinity? Yeah, that’s what I thought.

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