What is math anyway?


Since I’m trying to write a blog about math, I should probably explain what math is first!

Math, roughly speaking, is a vast collection of knowledge that is rigorously established. What does that mean? Proofs. Proofs are at the heart of math.

Mathematicians go crazy with proofs. In fact, they insist on making everything rigorous and basing things off of very basic assumptions called axioms. Even the simple, seemingly obvious fact that 0 \cdot n = 0 for any number n has proof using simpler axioms about our number system.

So why do we bother proofs? Often the first step towards proving something is noticing that a certain fact is true in a lot of cases. This is a pattern. But, patterns are not enough to show a general result. For example, in the early 1800’s, the mathematicians Legendre and Gauss noticed that the expression n^2-n+41 is a prime number for n from 1 to 40. (A prime number is a number that is only divisible by 1 and itself, so 3 is a prime number because it cannot be broken up into more factors but 6 isn’t a prime number because it is 2 \cdot 3.) It seems like with so much evidence, it should be true for all positive counting numbers, and that’s what they thought too. However, in the very next case, n=41, it fails to be prime. This is easy to see because 41^2-41+41=41^2, which is not prime. Silly Legendre and Gauss!  Furthermore, often a proof gives us important insights that can then be applied to other problems.  The why is crucial.

Making guesses is how math evolves.  Something that is not proven but is believed to be true is called a conjecture. There are many, many important problems with the status of conjecture right at this moment. That is what current mathematicians keep themselves busy with. For example, one conjecture that hasn’t been resolved is whether or not there are an infinite number of positive counting numbers for that same expression n^2-n+41 where the result is prime (so even though it isn’t true for all positive counting numbers, it could still be true for an infinite number of them).  This is an open question for almost all quadratic equations (an expression with a squared term as its highest power), in fact.

In most middle and high school math classes, proofs are often left out or only partially stated. I think this is a mistake, since proofs are so key to math. Everything has a proof. So, the next time your teacher tells you something is true, try to find a proof!

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11 responses to this post.

  1. Posted by SG on August 26, 2010 at 4:47 am

    Can you please post the proof of O.n = O

    Reply

    • Yes! Sorry for the late reply. It’s actually a shorter proof than I remembered.

      Our main assumptions:
      1. 0+n=n for any number n. (If you haven’t seen it before it might be uncomfortable using letters in place of numbers. Remember that ‘n’ is like ‘anything’ so it’s the same as saying ‘0 plus any number is that number’, but for long expressions that would take way too much writing!)

      2. Distributivity, which is the property that a \cdot (b+c)=a \cdot b+a \cdot c for any numbers a, b, and c.

      Now the proof is very short. Consider the expression n \cdot (0+0). By distributivity, we get that n \cdot (0+0)=n \cdot 0+n\cdot 0 and also equals n \cdot 0 (since 0+0=0). So n \cdot 0+n\cdot 0 = n \cdot 0. Subtracting n \cdot 0 from both sides of the equation, we get that n \cdot 0=0. This shows that n \cdot 0=0 for any n.

      Reply

  2. Posted by Jin Ai on September 2, 2010 at 11:06 am

    I REMEMBER THIS FROM ROSS
    😀

    Reply

  3. Posted by Sachi on September 2, 2010 at 3:22 pm

    Another good example of ‘a bunch of cases does not a proof make’ would be the Polya conjecture. Spiked math did a comic on this– http://spikedmath.com/294.html

    Reply

  4. I like your point about patterns not necessarily implying that something is true. For more examples along that line, see the following paper by Richard Guy, which contains 35 patterns (some of which lead to true results and others of which lead to false results).
    http://www.math.sjsu.edu/~hsu/courses/126/Law-of-Small-Numbers.pdf

    Reply

  5. […] The busiest day of the year was August 31st with 152 views. The most popular post that day was What is math anyway?. […]

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