Mathematical induction

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 (or all positive numbers from a point onwards). The idea is that if:

1. Something is true for the first case (base case);
2. Whenever that same thing is true for a case, it will be true for the next case (inductive case),

then

• That same thing is true for every case by induction.

In the careful language of mathematics, a proof by induction often proceeds as follows:

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

Because it is 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.

Examples of proof by induction

Sum of the first n natural numbers

Prove that for all natural numbers n:

<math>1+2+3+....+(n-1)+n=\tfrac12 n(n+1)</math>

Proof:

First, the statement can be written as:

<math>2\sum_{k=1}^n k=n(n+1)</math> (for all natural numbers n)

By induction on n,

First, for n=1:

<math>2\sum_{k=1}^1 k=2(1)=1(1+1)</math>,

so this is true.

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

<math>2\sum_{k=1}^{n_0} k = n_0(n_0+1)</math>

Then for n=n₀+1:

<math>2\sum_{k=1}^{{n_0}+1} k</math>

can be rewritten as

<math>2\left( \sum_{k=1}^{n_0} k+(n_0+1) \right) </math>

Since <math>2\sum_{k=1}^{n_0} k = n_0(n_0+1),</math>

<math>2\sum_{k=1}^{n_0+1} k = n_0(n_0+1)+2(n_0+1) =(n_0+1)(n_0+2)</math>

Hence the proof is complete by induction.

The sum of the interior angles of a polygon

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 <math>n</math>-sided polygon is <math>(n-2)180</math> degrees."

The initial starting value is 3, and the interior angles of a triangle is <math>(3-2)180</math> degrees. Assume that the interior angles of a <math>n</math>-sided polygon is <math>(n-2)180</math> degrees. Add on a triangle which makes the figure a <math>n+1</math>-sided polygon, and that increases the count of the angles by 180 degrees <math>(n-2)180+180=(n+1-2)180</math> degrees. Since both the base case and the inductive case are handled, the proof is now complete.

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 a set of objects, as well as to prove statements about that set of objects.

For example, we can define <math>n</math>th degree cousin as follows:

• A <math>1</math>st degree cousin is the child of a parent's sibling.
• A <math>n+1</math>st degree cousin is the child of a parent's <math>n</math>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 <math>n</math> is a natural number, then <math>n|</math> is a natural number.
• If <math>n| = m|</math> then <math>n = m</math>.

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

• <math>m + | = m|</math>
• <math>m + n| = (m + n)|</math>

Related pages

• Mathematical proof
• Proof by contradiction