MediumMathematicsHigh School (9–10)GeneralNumber Systems › Rational and Irrational NumbersNumericalBoard Exam

Among all fractions p/q with a bottom q from 1 to 10, which one is closest to √3 = 1.7320508…?

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

Among all fractions $\dfrac{p}{q}$ with a bottom $q$ from $1$ to $10$ , which one is closest to $\sqrt{3} = 1.7320508\ldots$ ? For that fraction, work out $p^2 - 3q^2$ . Why can this never be $0$ ?

Need a reminder? The key ideaShow

To prove that $\sqrt{p}$ is irrational, for a prime $p$ : suppose $\sqrt{p} = \dfrac{a}{b}$ in lowest terms. Then $a^2 = pb^2$ , so $p$ divides $a$ . Write $a = pc$ . Then $b^2 = pc^2$ , so $p$ divides $b$ . A common factor $p$ contradicts lowest terms.

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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← 6 is not a prime, but √6 is still irrational. Prove it by contradiction, using only the prime 2.Prove that the cube root of 2, the number whose cube is 2, is irrational. →

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