# Mathematical induction facts for kids

**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*. ( is the*induction variable*.) - Show that the statement is true when is 1.
- Assume that the statement is true for any natural number . (This is called the
*induction step*.)- Show then that the statement is true for the next number, .

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*:

Proof:

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

By induction on *n*,

First, for *n*=1:

- ,

so this is true.

Next, assume that for some *n*=*n _{0}* the statement is true. That is,:

Then for *n*=*n _{0}*+1:

can be rewritten

Since

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 -sided polygon is degrees.

The initial starting value is 3, and the interior angles of a triangle is degrees. Assume that the interior angles of a -sided polygon is degrees. Add on a triangle which makes the figure a -sided polygon, and that increases the count of the angles by 180 degrees 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 th degree cousin:

- A st degree cousin is the child of a parent's sibling
- A st degree cousin is the child of a parent's 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 is a natural number, then is a natural number
- If then

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

*Kiddle Encyclopedia.*