Pure 2
Pure 3
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AS/A2 Pure 2 Pure 3 |
| P1 Topic 1: Proof | |
| Proof 2: Ö2 is irrational |   |
| Ö2 cannot be written as a fraction, so we say
that Ö2 is irrational. This
statement can be proved by using what is called "proof by contradiction". Such
a proof always starts by assuming the opposite of what you are trying to prove,
and then obtaining by logical steps a contradiction. Study each of the 13 steps in the following proof. You will see that it begins by assuming that Ö2 is rational, in other words that Ö2 can be expressed as a fraction. That fraction is assumed to be in fully simplified form, in other words the numerator a and denominator b have no common factors. |
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| A few more notes on some of the steps | |
| 1 | The fact that the fraction is assumed to be fully simplified is very important. It is this step which combines with step 11 to provide the contradiction needed to make the proof work. |
| 5 | If the square of a number is known to be even, then the number must itself be even. (Can you prove this statement?) |
| 6 | All even numbers are multiples of 2. |
| 7 | Steps 3 and 6 are combined, using algebra. |
| 11 | Using steps 5 and 10, you can see that the numbers a and b have been shown to be both even, so they have a common factor of 2. |
| 13 | If a mathematical proof produces a conflict or contradiction, then, logically, we must look back to any assumptions made, knowing they must be false. |
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