One way to show that a statement is true is to show that its opposite leads to a contradiction. This type of indirect reasoning is an important problem-solving tool. The two examples we give here are problems that were solved over $2000$ years ago but continue to have a profound influence in mathematics.
The Pythagoreans were fascinated by the beauty of the natural numbers: $1, 2, 3 ….$ They hoped that the length of every interval constructed in geometry would be a ratio of two natural numbers, and so the natural numbers would describe all of geometry. But what is the length of the diagonal of a square of side $1$? They knew (by the Pythagorean Theorem) that this length is $\sqrt 2$. The question was: Is $\sqrt 2$ the ratio of two natural numbers? In other words, is $\sqrt 2$ a rational number? The Pythagoreans were able to prove that $\sqrt 2$ is not rational.
Suppose $\sqrt 2$ is rational, so that $$\sqrt 2 = \frac{a}{b}$$
where $a$ and $b$ are natural numbers with no factor in common. Then
$\; a = \sqrt 2b \quad$ | $\color{#50F} {Multiply \; by \; b}$ |
$a^2 = 2b^2 \quad$ | $\color{#50F}{Square}$ |
This means that $a^2$ is an even number and so $a$ is an even number. Then $a = 2m$, for some integer $m$. So the last equation leads to
$(2m)^2 = 2b^2 \quad$ | $\color{#50F} {Because \; a = 2m}$ |
$\; \; \; 4m^2 = 2b^2 \quad$ | $\color{#50F}{Square}$ |
$\; \; \; 2m^2 = b^2 \quad$ | $\color{#50F}{Divide \; by \; 2}$ |
Thus $b$ is also even. So $a$ and $b$ have $2$ as a common factor and this contradicts our assumption that $a$ and $b$ have no factor in common. Thus the assumption that $\sqrt 2$ is rational leads to a contradiction and so $\sqrt 2$ must be irrational.
A prime number is one that has no factors other than $1$ and itself. The first few primes are $$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, . . .$$
How do we find the next prime? No formula is known that will produce the primes and no one has found a pattern for the location of the primes. One thing that is known is that there are infinitely many primes. Euclid gave a proof for this fact over $2000$ years ago by a brilliant use of indirect reasoning. Here is his famous proof:
Suppose that there are only a finite number of primes. We list them as $$p_1, p_2,…p_n$$
Then the number $$N = p_1. p_2. p_3. ….p_n+1$$
is not divisible by any prime (Why?) and so is itself prime. But $N$ is not in our list of primes (Why?). This contradicts our assumption that the list contained all the primes. Thus there are infinitely many primes.