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JindrⒶ, the kitsch kotschk · @jindra

217 followers · 111 posts · Server todon.nl

So I was thinking (I am paraphrasing, I didn't write it down at the time):
well, we can have a linear equation like this
x - 1 = 0

And the solution is x=1. There are also quadratic equations like this:
a*x^2 + x - 1 = 0.

I thought hmm, what if in the previous quadratic equation, we had $a$ really really small. Wouldn't that basically be like a linear equation?

I mean it should be right, because the equations are somehow "close". One would think we can neglect the quadratic term if $a$ is small.

But then there is a problem. The quadratic equation has two solutions that aren't identical!
Let's pick a=.01
It has a solution x_1 \approx .99, which is close to 1. That's expected. But then it has a solution x_2 \approx -100.99. That's really weird! This wasn't there before.

So my thinking was like: hm, since the linear equation is essentially the same as the quadratic equation with a small, then, in reality, every linear equation should have a "ghost" solution that's negative infinity. But like, how and why?

I was thinking surely there's something wrong with this argumentation, but it sounds so convincing. Of course we can neglect a small quadratic term.

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#Math #PerturbationTheory