||The English used in this article or section may not be easy for everybody to understand. (June 2012)|
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 . ( 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:
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
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 -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[change | change source]
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: