Proof
What do the statements 2 + 2 = 4 and "the sky is blue" have in common? One might say that they are both true. The statement about the sky can be confirmed by going outside and observing the color of the sky. How, then, can one confirm the truth of the statement 2 + 2 = 4? A statement in mathematics is considered true, or valid, on the basis of whether or not the statement can be proved within its mathematical system.
What does it mean to prove something mathematically? A mathematical proof is a convincing argument that is made up of logical steps, each of which is a valid deduction from a beginning statement that is known to be true. The reasons used to validate each step can be definitions or assumptions or statements that have been previously proved.
As an example of a proof, look at finding the sum of the first n whole numbers.
The sum of the first three whole numbers is 1 + 2 + 3, so n = 3, and the sum is 6.
The sum for n = 4 is 1 + 2 + 3 + 4, and this sum is 10.
Is there a pattern here? Can the sum be found (without doing all the addition) for n = 8?
In 1787, a teacher gave this problem to a 10-year-old boy, Carl Friedrich Gauss. Gauss pointed out the following pattern:
What is the sum of each of the number pairs indicated by the arrows? The sum is four pairs that sum to 9, and there are eight numbers in all.
What would the mean of these numbers be? The mean is (4 × 9) divided by 8. The mean of the whole set is 
which is also the mean of the first and last numbers in the set. The sum of eight numbers, whose mean is 
, is 8 
, or 4×9, which is 36. Maybe the sum of the first n whole numbers can be found by finding the mean of the first and last whole numbers, and then multiplying that mean by n.
Try the following pattern to see if it works for other values of n. What happens if an odd number is chosen for n?
Is the sum the same as the mean of 9 and 1 (
or 5) multiplied by n (which is 9 in this case)? Is the sum 5 × 9, or 45?
Is the sum of the first n whole numbers always equal to the mean of the first and last multiplied by n? This seems to be true, but in mathematics even a huge number of examples is not enough to prove the truth of the statement. Therefore, a proof is needed.
The first number in the sum is always 1. The last number in the sum is always n. The mean of n and 1 is 
according to the definition of mean. In algebra, n multiplied by this mean looks like 
. A proof must show that the sum of the first n whole numbers is always 
.
First, consider n as an even number. In that case, there is an even number of pairs, and each pair has a sum of n + 1, regardless of the size of n. The truth of this pattern does not depend on the size of n, as long as n is an even number so 
pairs can be made, each of which adds to (n + 1).
There are 
pairs, each of which adds to (n + 1), so the total is 
or 
. This means that the pattern is proved true as long as n is an even number.
Next, consider the case where n is an odd number:
The circled number in the middle will always be the mean of the first and last numbers. In algebra, the middle number will be 
. So the middle number adds one more mean to the (n - 1) means that were made by the paired numbers.
So, again, the total sum is n multiplied by the mean of the first and last numbers, or 
. These two cases, for n, an even number, and n, an odd number, together make up the proof.
There are several forms of mathematical proofs. The one just given is a direct proof. Indirect reasoning, or proof by contradiction, can also be used. A third kind of proof is called mathematical induction.
Although many examples do not prove a statement, one counterexample is enough to disprove a statement. For example, is it true that y + y = y × y? Try substituting values of 0 and then 2 for y. Although the statement is true for 0 and 2, it is not true in general. One counter-example is y = 1, since 1 + 1 is not equal to 1 × 1 because 2 is not equal to 1.
Here is a well-known proof that 0 = 1. Try to find the flaw, or mistake, in this proof.
- Assume that x = 0. Assumption
- x (x - 1)=0 Multiplying each side by (x - 1)
- (x - 1)=0 Dividing each side by x
- x = 1 Adding 1 to each side
- 0 = 1 Substitute x = 0, the original assumption
All the steps except one are valid. In Step 3, the proof divided each side by x. The reason for this is that, if a = b, then 
if c is not equal to 0. But the original assumption said that x was equal to 0, so Step 3 involved dividing by 0, which is undefined. Allowing division by 0 can lead to proving all sorts of untruths!
Induction.
Bibliography
Bergamina, David, and editors of Life. Mathematics. New York: Time Incorporated, 1963.
Hogben, Lancelot. Mathematics in the Making. New York: Crescent Books, Incorporated, 1960.
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