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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 . ( 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=n0 the statement is true. That is,:

Then for n=n0+1:

can be rewritten

Since

Hence the proof is correct.

Similar proofsEdit

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 definitionEdit

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:

  •  
  •