Mathematical induction

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Mathematical induction is a special way of proving a mathematical truth. It can be used to prove that something is true for all the natural numbers (all the positive whole numbers). The idea is that

  • Something is true for the first case
  • That same thing is always true for the next case


  • That same thing is true for every case

In the careful language of mathematics:

  • State that the proof will be by induction over n. (n is the induction variable.)
  • Show that the statement is true when n is 1.
  • Assume that the statement is true for any natural number n. (This is called the induction step.)
    • Show then that the statement is true for the next number, n+1.

Because it's true for 1, then it is true for 1+1 (=2, by the induction step), then it is true for 2+1 (=3), then it is true for 3+1 (=4), and so on.

An example of proof by induction:

Prove that for all natural numbers n:

1+2+3+....+(n-1)+n=\tfrac12 n(n+1)


First, the statement can be written: for all natural numbers n

2\sum_{k=1}^n k=n(n+1)

By induction on n,

First, for n=1:

2\sum_{k=1}^1 k=2(1)=1(1+1),

so this is true.

Next, assume that for some n=n0 the statement is true. That is,:

2\sum_{k=1}^{n_0} k = n_0(n_0+1)

Then for n=n0+1:

2\sum_{k=1}^{{n_0}+1} k

can be rewritten

2\left( \sum_{k=1}^{n_0} k+(n_0+1) \right)

Since 2\sum_{k=1}^{n_0} k = n_0(n_0+1),

2\sum_{k=1}^{n_0+1} k = n_0(n_0+1)+2(n_0+1) =(n_0+1)(n_0+2)

Hence the proof is correct.

Similar proofs[change | change source]

Mathematical induction is often stated with the starting value 0 (rather than 1). In fact, it will work just as well with a variety of starting values. Here is an example when the starting value is 3. The sum of the interior angles of a n-sided polygon is (n-2)180degrees.

The initial starting value is 3, and the interior angles of a triangle is (3-2)180degrees. Assume that the interior angles of a n-sided polygon is (n-2)180degrees. Add on a triangle which makes the figure a n+1-sided polygon, and that increases the count of the angles by 180 degrees (n-2)180+180=(n+1-2)180degrees. Proved.

There are a great many mathematical objects for which proofs by mathematical induction works. The technical term is a well-ordered set.

Inductive definition[change | change source]

The same idea can work to define, as well as prove.

Define nth degree cousin:

  • A 1st degree cousin is the child of a parent's sibling
  • A n+1st degree cousin is the child of a parent's nth degree cousin.

There is a set of axioms for the arithmetic of the natural numbers which is based on mathematical induction. This is called "Peano's Axioms". The undefined symbols are | and =. The axioms are

  • | is a natural number
  • If n is a natural number, then n| is a natural number
  • If n| = m| then n = m

One can then define the operations of addition and multiplication and so on by mathematical induction. For example:

  • m + | = m|
  • m + n| = (m + n)|