Prove that 2 is an irrational number.

An irrational number is a real number that cannot be expressed as a ratio of integers. If any irrational number is represented in the form of a decimal fraction, it will be a non-terminating and non-recurring number. Given number is 2 Decimal expansion of 2 is, 1 . 4 1 4 2 1 2 . 00 00 00 00 + 1 1 24 1 0 0 + 4 9 6 281 4 0 0 + 1 2824 1 1 9 00 + 4 1 1 2 96 2282 6 5 0 400 6 564 3 8 3 6 So, the decimal expansion of 2 is a decimal fraction, which is a non-terminating and non-recurring number.

Prove that 3 is an irrational number.

An irrational number is a real number that cannot be expressed as a ratio of integers. If any irrational number is represented in the form of a decimal fraction, it will be a non-terminating and non-recurring number. Given number is 3 Decimal expansion of 3 is, 1 . 7 3 2 0 5 1 3 . 0 0 0 0 0 0 0 0 0 0 0 + 1 1 27 2 0 0 + 7 1 8 9 343 1 1 0 0 + 3 1 0 2 9 3462 7 1 0 0 + 2 6 9 2 4 346405 + 5 1 1 7 7 6 3 0 2 0 0 0 2 0 5 2 7 9 7 5 0 0 So, the decimal expansion of 3 is a decimal fraction, which is a non-terminating and non-recurring number. Thus, 3 is an irrational number.

Give an example of non-algebraic irrational number. If n ∈ N and 0 < n < 2 , then prove that π - n is an irrational number.

A non-algebraic irrational number is a number that is not the root of a non-zero polynomial with rational coefficients. The best known non-algebraic irrational numbers are π and e . Here, we have π - n where, n belongs to the natural number and n greater than 0 and less than 2 . Thus, we get π - 1 We know that 1 is a rational number but π is an irrational number. The difference of a rational number and an irrational number is irrational. Therefore, π − 1 is an irrational number.