🔶02 - Show that Square Root of 2 is irrational
🔶02 - Show that Square Root of 2 is irrational
In this video, we are going to prove that square root of 2 is irrational.
An irrational number is a number that cannot be expressed as a fraction as a/b , where b is not equal to 0
To solve this:
first lets assume that root 2 is rational, thus
root 2 = a/b,
we also assume that, root 2 is irreducible/simplified and a and b are co-prime, thus the only common factor between a and b is 1.
root 2 = a/b
2 = a2/b2
2b2 = a2..............(1)
here we have 2 times a certain number b2 = a2, thus 2 is a factor of a2 and a2 is even.
now even x even = even and odd x odd = odd,
which means
if a2 is even, a = even
and if a is even, then a = 2 times a certain number c, so that from equ(1)
2b2 = (2c)2
2b2 = 4c2
b2 = 2c2, here b2 is even and hence b is even.
Here is our contradiction, since a and b are even it means a and b have a common factor of 2, which makes our initial assumption false, in the sense that we assumed that both a and b have only one common factor which was 1.
therefore root 2 is not rational, it is irrational.
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