Prove that is an irrational number .

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Rational Numbers

Question

Prove that $(2 \sqrt{3} - 1)$ is an irrational number .

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Let us assume that $(2 \sqrt 3 - 1)$ is rational .
add 1 to be above number, considering 1 is rational number.

as we know , sum of two rational numbers is rational.
Sum =

2 \sqrt{3} - 1 +, 1 = 2 \sqrt 3

which is only possible if 2 is rational and

\sqrt, 3

is rational.
But the fact is

\sqrt 3

is an irrational number which contradicts to our assumption.
So,

(2 \sqrt{3} - 1)

is an irrational.