Question

Which of the following numbers are irrational?

Options

• `sqrt(2)`

• `root(3)(6)`

• 3.142857

• `2.bar3`

• π

• `22/7`

• 0.232332333...

• `5.27bar41`

Answer 1

(a) `sqrt(2)`

(b) `root(3)(6)`

(e) π

(g) 0.232332333...

Explanation:

(a) `sqrt(2)`, Irrational. The square root of a non‑perfect square integer cannot be written as a ratio of integers.

(b) `root(3)(6)`, Irrational. The cube root of 6 is not an integer (6 is not a perfect cube), so the real nth‑root of a non‑perfect nth power is irrational.

(c) 3.142857, Rational. This is a terminating decimal as written `(3.142857 = (3142857)/(1,000,000))`, so it is a rational number.

(d) `2.bar3` (2.333...), Rational. A repeating decimal represents a rational number `(2.bar(3) = 7/3)`.

(e) π, Irrational. π is a well‑known nonterminating, nonrepeating decimal (not a ratio of integers).

(f) `22/7`, Rational. It is explicitly a ratio of two integers.

(g) 0.232332333..., Irrational. Its decimal does not settle into a fixed repeating block (pattern grows), so it is nonterminating and nonrepeating.

(h) `5.27bar41` (5.27 with 41 repeating), Rational. Any decimal with a repeating block is rational (it equals some fraction).