Maths Amazing Race Practice ? o.O
question:Prove that Sqaure root of two is irrational without the use of the calculator.
Solution:
First, we assume √2 is a rational number.
That means we can express √2 as an irreducible fraction a/b , where a and b are coprime numbers.
√2=a/b
2= a²/b²
a²=2b²--(1)
Therefore a² is an even number. Hence, a is also an even number ( since the sqaure of any even number is also even).
Thus a can be represent as a = 2k, where k is a constant.
subsituting a =2k into (1),
4k²=2b²
b²=2k²
Thus, b² and hence b is also even.
Since b and a are even, it contridicts the previous assumtion that a/b is irreducible(e.g. 2/4 , 4/6 , 4/2 , 6/4 are all reducible).
Therefore,√2 is thus an irrational number.
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