“High School Mathematics”: Naive Foundations

1. It dawned on me I should specify what exactly I mean by “high school mathematics” mentioned in the “About” page.

Briefly: it’s all the mathematics we learned in secondary school (well, American secondary school) plus some naive set theory. Set theoretic considerations aside, we will explicitly state the operators and axioms used.

I may revise this post, later on, to use Knuth’s “naive” construction of the real numbers using integers, just to be more explicit.

2. Stuff and Structure. We work within an “ambient number system”. In other words: the basic guys we work with are numbers. We also use variables, letters which “stand in” for numbers.

What operators do we work with? We have addition and subtraction, multiplication and division, exponentiation.

3. Axioms for Addition, Subtraction. Let x, y, z be any numbers. Then we have several important axioms they satisfy:

  1. Associativity: \displaystyle (x+y)+z=x+(y+z) and \displaystyle (x-y)-z = x + (-y -z)
  2. Commutativity: \displaystyle x+y=y+x and \displaystyle x-y=-y+x
  3. Identity Element: There exists a number 0 such that \displaystyle 0+x=x+0=x and \displaystyle x-0=-0+x=x, moreover \displaystyle 0=-0
  4. Additive Inverse: For each x there exists a corresponding unique y such that \displaystyle x+y=y+x=0. We denote \displaystyle y=-x, and call it the negation of x. Moreover \displaystyle y = -1\times x.

4. Axioms for Multiplication. We will write x\cdot y or x\times y to indicate the product of x and y. Let x, y, z be arbitrary numbers. Then we have the following axioms.

  1. Commutativity: x\cdot y=y\cdot x
  2. Associativity: x(y\cdot z)=(x\cdot y)z.
  3. Identity Element: There exists a unique number 1 satisfying x\cdot 1=1\cdot x=x for any x.
  4. Multiplicative Inverse: For any nonzero number x there exists a corresponding unique number y such that x\cdot y=y\cdot x=1.
  5. Distributivity over Addition: we have (x+y)z=x\cdot z+y\cdot z.

5. Exponentiation. Again, let x, y, z be arbitrary positive numbers, none of them zero.

  1. Identity element: \displaystyle x^{1}=x.
  2. For any nonzero x\not=0 we have x^{0}=1.
  3. We have \displaystyle x^{y}\cdot x^{z}=x^{y+z}.
  4. (x^{y})^{z}=x^{y\cdot z}.
  5. \displaystyle x^{-y}=1/(x^{y}). In particular, the multiplicative inverse of x is precisely x^{-1}.
  6. (x\cdot y)^{z} = x^{z}\cdot y^{z}.

Observe exponentiation is not associative because \displaystyle (x^{y})^{z} \not= x^{(y^{z})}, take x=2, y=3, z=4 to prove it to yourself. (You’d get 2^{12}=2^{27} which is absurd.)

Also observe exponentiation is not commutative: x^{y}\not=y^{x}. Take x=2 and y=3 for a counter-example to this dangerous intuition.

References. There are a number of references one can look at, I believe the best is Euler’s Elements of Algebra. Although it is about 250 years old…

So some modern ones: John Redden’s Elementary Algebra (2011) is legally free online.

Of course, Wikipedia’s Elementary Algebra discusses…elementary algebra…


About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.
This entry was posted in Foundations of Mathematics, High School Mathematics, Naive. Bookmark the permalink.

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