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Prove that 3+√2 is not a irrational number , given that √2 is not rational |
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Answer» Answer: Step-by-step explanation: From the equality √2 = a/b it follows that 2 = a2/b2, or a2 = 2 · b2. So the square of a is an even number SINCE it is TWO times something. From this we know that a itself is ALSO an even number. Why? Because it can't be odd; if a itself was odd, then a · a would be odd too. Odd number times odd number is always odd. Check it if you don't believe me! Okay, if a itself is an even number, then a is 2 times some other WHOLE number. In symbols, a = 2k where k is this other number. We don't need to know what k is; it won't matter. Soon comes the contradiction. If we substitute a = 2k into the original equation 2 = a2/b2, this is what we get: 2 = (2k)2/b2 2 = 4k2/b2 2*b2 = 4k2 b2 = 2k2 |
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