Mathematical induction

form of mathematical proof

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

then

• 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:

$1+2+3+....+(n-1)+n={\tfrac {1}{2}}n(n+1)$ Proof:

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

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 $(n-2)180$ degrees.

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

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

Define $n$ th degree cousin:

• A $1$ st degree cousin is the child of a parent's sibling
• A $n+1$ st degree cousin is the child of a parent's $n$ th 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:

• $m+|=m|$
• $m+n|=(m+n)|$