Everybody is introduced to mathematics by “remembering” natural numbers. Today we will see, “How to construct natural numbers”.
Without much delay let’s construct “Natural Numbers”, as in “plane geometry” (you must have studied it in high school), we define “Euclid’s Axioms”, here we will define “Peano’s Axioms”, the axioms are:
- Axiom One: 1 is a natural number.
- Axiom Two: Every natural number has a successor.
- Axiom Three: 1 is not the successor of any natural number.
- Axiom Four: If the successor of equals the successor of , then equals .
- Axiom Five (Principle of mathematical induction): If a statement is true of 1, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.
Some texts, which swap meaning of natural numbers and whole numbers, consider natural numbers to include the number zero! In that case, we will start our construction with zero instead of one (see: http://mathworld.wolfram.com/PeanosAxioms.html). This supports a very criticized philosophical idea from Indian civilization:
Everything started from nothing.
Once we are able to construct natural numbers, we can construct all kinds of number systems (with motivation of solving certain algebraic equations):
- whole numbers: add ‘0’ to the list of natural numbers (I don’t know why, but some mathematicians use whole numbers and natural numbers interchangeable)
- integers: add negative natural numbers to the list of whole numbers
- rational numbers: make fractions from integers, where denominator is not allowed to be zero.
- irrational numbers: not all numbers can be represented as fractions
- real numbers: define “Dedekind cut”, and construct real numbers out of rational numbers (lengthy task!)
- complex numbers: any polynomial equation must have a solution
There is another classification of “real numbers” , called “algebraic numbers” and “transcendental numbers“, but it is altogether a different topic of discussion which I will discuss in some other blog post.