Question

Show that 2+√3 is an irrational number.​

Answer 1

Answer:

Proof by contradiction.

Assume 3–√ is rational.

It can therefore be written as 3–√=ab where a and b are integers with no common factor (i.e. the fraction is in its lowest terms).

Now we can write

a2b2=3

⟹a2=3b2⟹3|a2⟹3|a⟹9|a2⟹a2=9k

for some integer k. So now we have

a2=9k=3b2⟹3k=b2⟹3|b2⟹3|b

So 3 divides both a and b which contradicts them having no common factor.

Thus we must conclude that 3–√ is irrational.

[Note: the notation 3|a means ‘ 3 is a divisor or factor of a ’.]