Discriminants are cool

The idea of a discriminant is that it encodes information about the roots of a polynomial and allows us to get at it without actually finding those roots which can be very difficult in the general case.

Let's look at quadratic polynomials. We want a quick calculation which will tell us how many distinct roots it has and if those roots are real or complex. The general quadratic polynomial is

ax² + bx + c = 0

But we are only interested in knowing where it's zero and so we can multiply it by a constant because the result will still be zero and no new zeros will be introduced. In this case, we use 1/a, to get (Eqn 1)

x² + (b/a)x + c/a = 0

On the other hand, we know that if r₁ and r₂ are the roots, then the above factors as

(x - r₁)(x - r₂)

and multiplying it out, we get (Eqn 2)

x² - (r₁ + r₂)x + r₁r₂

So we can look at (Eqn 1) and (Eqn 2) and equate their coefficients:

b/a = - (r₁ + r₂)
c/a = r₁r₂

Now, we're trying to find a way to "cheat" and know stuff about the roots without actually finding them. Consider the difference r₁ - r₂. If the two roots are the same, then it will be zero. If the two roots are complex, then we know that they're of the form s + it, s - it and the result will be 2it which is pure imaginary. If the roots are both real, then the difference will be non-zero (either positive or negative). That's sort of messy. If we squared it, then any non-zero real (i.e., two distinct real roots) would square to a positive, zero (i.e., one real root, repeated twice) would square to zero and a pure imaginary (i.e. a pair of complex conjugates) would square to a negative number. This would be much nicer. So that's one indication that we want to shoot for an expression that's equal to

(r₁ - r₂)²

Another indication is that if we look at our expressions for b/a and c/a, we see that one is degree 1 in the roots and the other is degree 2 in the roots. So let's square b/a to get

(b/a)² = r₁² + r₂² + 2r₁r₂

Which is degree 2.

Now let's expand out our target to get

(r₁ - r₂)² = r₁² + r₂² - 2r₁r₂

But r₁² + r₂² = (b/a)² - 2(c/a) and 2r₁r₂ = 2(c/a) so we get

(r₁ - r₂)² = [(b/a)² - 2(c/a)] - 2(c/a) = (b/a)² - 4(c/a)

So we're pretty much done. The one annoying thing is that we'll have to square a and do division. But if we multiply both sides by a², then we won't change the sign and so won't lose any information We write the discriminant of f(x) = ax² + bx + c as d(f). Then we get

d(f) = a²(r₁ - r₂)² = b² - 4ac

It's true that we could have read that off of the quadratic equation (which is how it's normally introduced) but that doesn't generalize degrees higher than four. This method of finding it will work for any degree.

:eek:

I guess it's not for you :P Give it a shot though if you like math. There's nothing that's really that hard.

I only skimmed the post. Math makes me cringe. That's why I wanna go into Biology instead of Physics :P

I'm gonna reread this tomorrow, I need to sleep.

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Originally Posted by BLUELINE976

I only skimmed the post. Math makes me cringe. That's why I wanna go into Biology instead of Physics :P

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Funny story.

I always like physics but cringed at math too. So I avoided and just read popular stuff. Then when I was 22 (I'm 30 now) I decided to learn just enough math to learn some physics. I innocently bought a book called "Introduction to Algebra" off of Amazon and without thinking too hard about it. It came and it turns out that it was actually about abstract algebra and not high school algebra. I said "fuck it" and started working through it anyways. I had to work my ass off (I'm embarrassed to say that the homomorphism theorems took me a few weeks to understand) but I made it through.

I ended up liking math so much that I still haven't really made it to physics properly.

Math is really cool. In this country at least, we do a really good job of convincing people that it's not but it is.

I think math is really cool too but regarding the original post I dont even know where to start. Math is somewhat easy for me if I know what formulas to utilize but "x2 + (b/a)x + c/a = 0" is totally incomprehensible to me. Would you recommend starting out with an Abstract algebra book like you did or should I just start with basic algebra?

Probably not start with abstract algebra. That was hard and I wouldn't recommend it. It was purely an accident and I probably set myself back by doing it that way. Here's what's going on with what you asked about:


We start off with a quadratic equation:

ax² + bx + c = 0

The equation that you posted is

x² + (b/a)x + (c/a) = 0

which is just the first equation multiplied by 1/a. This clears the coefficient in front of x² and sets it equal to 1.

I knew to do this because I knew that I wanted to equate the coefficients (b/a and c/a in this case) with the coefficients of (x - r₁)(x - r₂) after it gets multiplied out. But the coefficient of x² has to be 1 for that to work. The whole point of that is to get expressions with the coefficients of the original quadratic on one side and the roots of the quadratic on the other without actually finding the roots which is what we have after we equate the coefficients.

That's some cool beans. Have you done any Galois theory? I'm looking forwards to that stuff, it sounds so elegant and powerful from what I've heard.

I think the ease of learning group theory comes from the approach of the lecturer / author. When I came to the homomorphism theorem I was actually able to guess at the correct answer and then prove it unassisted, which turned out to be entirely bookwork. I guess this was because the lecturer did all the general stuff first (which took some time to acclimatise to) and then the specifics after. For example I've seen some places where conjugacy was treated as a new object and all the theorems slavishly proven, like ccl(x) for all x partitions G, etc. However we had already studied group actions; conjugacy was just introduced as another type of action (although instead of a function G x X -> X where X is some set or other, X was = G and so the action mapped the group onto itself) and so things like the above just fell out of what we already knew (namely that the orbits partition X).

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Originally Posted by Xei

That's some cool beans. Have you done any Galois theory? I'm looking forwards to that stuff, it sounds so elegant and powerful from what I've heard.

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Only a very little. And a while ago. It's pretty but I never found it very useful for anything that I want to do. Honestly, it was over my head at the time. As part of my current project of redoing all my math education from the ground up, I'm going to look at it in depth.

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I think the ease of learning group theory comes from the approach of the lecturer / author...

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This is definitely true but there's still a bound on how easy it can be which is given by the "mathematical maturity" of the person trying to learn it. Most of the time, people coming to algebra have already studied linear algebra and calculus and those are great subjects for getting good at understanding arguments and tricks. If somebody already has that, then there's no reason that algebra has to be hard.

There's definitely a jump between the "math" that gets learned in high school and what I would call real math. Honestly, if I hadn't have found that book, I doubt that I would ever have been motivated to get good at math. The problem for me is that math before the jump is just incredibly boring (and I'm not that good at it on its own terms) so I had no foundation.

For example, it seems that a large measure of how good someone is at math in high school is how good they are at memorizing formulas. I suck at it. I can't remember the quadratic formula. I can't even remember how to complete the square. I understand how to figure out how to complete the square. Likewise I can't remember that whole mess of trig equations to save my life. But I know that e^iθ = cos(θ) + i sin(θ) and I can figure out pretty much everything I need to know from that. Leonard Susskind noted in one of his online lectures that American students take a year to memorize trig equations in high school and then they get to Stanford and still don't know that equation. So I'm pretty much fucked in a high school math class unless I happen to be able to see the relations. But if I've decided that I don't like math and that it sucks because it's all about rote memorization (which I had), then I'm not likely to engage the creative portions of my brain which make all these things make sense.

Another funny example is that my brother was in "algebra 2" and was having trouble memorizing the equation for the distance between two points in a plane. It turns out that his teacher had never explained that it's just a trivial consequence of Pythagoras theorem :wtf:. So don't memorize. It falls right out of something that even I can remember.

But somebody that wants to study algebra should have all this background and so there seems to be a catch 22. Try it without the background and pick it up as you go along or spend all the time getting the background without knowing what you're doing. I really want to write a book that teaches algebra without assuming the background and takes baby steps. Developing an instinct for manipulating equations is very important.

Discriminants are one of the things in mathematics I never quite understood, though probably just because I didn't follow class when we learned about it.

Conceptually, they're simple. Just multiply the squared differences of the roots. If there are any repeated roots, the whole thing is zero. Otherwise, It's non-zero.