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Comment by defen | Hacker News Reader

One of the best classes I took in college was Abstract Algebra, and I only took it because my math-major roommate kept telling me that I needed to "learn more math." My favorite part was coming to the realization that linear algebra was not just a set of tools for solving particular problems, but rather an instance of a much more general mathematical structure with a beauty and simplicity of its own.

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Was my favorite subject as well, precisely for those reasons. Not sure how to teach the beauty of mathematics to those who are uninitiated though. I think it really should come in at the elementary school level, when you can show really cool things about numbers like Fibonacci sequence or binary system or simple cryptography, and make mathematics seem magical and cool (and not wait until high school with these concepts, when it's too late to convince most non-predisposed folks of the poetic beauty of mathematics; you can absolutely teach these concepts earlier).

First day of college: "prove that 0 times x is equal to 0". That got rid of everyone who wasn't really into maths :-)

That exercise was literally the first question, on the first hour of back to school, after 5 minutes of a welcome speech.

I can't even figure out the proof anymore, although it really can't be complicated given the ridiculously small amount of tools you are given (basically x + -x is 0, and (a+b) times c equals a times c + b times c. You are given nothing else.

From the real number axioms:

Let x \in R. Then we have that

  x*0 + x*0 = 0*x + 0*x = (0+0)*x = 0*x = x*0

by commutativity of multiplication, distributivity, the existence of a neutral element 0 for addition, and the commutativity of multiplication once again. Thus, we have that

  x*0 + x*0 = x*0.

For clarity, define

  A = x*0.

Then we have that

  A + A = A
  A + A + (-A)  = A + (-A), by the existence of additive inverses.
  A + (A + (-A)) = (A + (-A)), by associativity of addition
  A + 0 = 0, by the definition of additive inverses
  A = 0, by the definiton of the neutral element 0 of addition

which gives us our desired result.

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