Core Studies 5: Experimental Section

The Internet - Number Theory - Programming

Proof By Contradiction

One technique that we sometimes use to prove a result is called a proof by contradiction. To prove a result by contradiction, we first assume that the result is not true. Then we show that if this result is really not true, then it leads to a contradiction of something that we absolutely know to be true. Therefore, since we end up with an absurd conclusion, we realize that our original asssumption (that the result was not true) could not possibly be valid, and therefore the result must be true. This technique is also called reductio ad absurdum (Latin for "reduce to the absurd").

Example

Let's prove the following statement:
If a number is doubled and still has the same original value, then the number must be zero.
 Let the variable n represent a number. Then this statement says
If 2*n=n, then n=0

Proof by contradiction

Let us assume that the result (n=0) is false. Then we are saying that
2*n=n and n is not 0
Since n is not 0, we can divide both sides of the equation by 0.
2=1
But the conclusion 2=1 is impossible! (It's a contradiction of what we know to be true). How did we arrive at this impossible conclusion? By assuming that n is not 0. So that assumption must have been wrong, and n must be 0, (which is what we were trying to prove in the first place!)
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