Which of the following numbers are irrational?

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KollegeApply · Accepted Answer

(a) sqrt(2) (b) root(3)(6) (e) $\pi$ (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) = $\frac{7}{3}$). (e) $\pi$, Irrational. $\pi$ is a well‑known nonterminating, nonrepeating decimal (not a ratio of integers). (f) $\frac{22}{7}$, 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).

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