MediumMathematicsHigh School (9–10)GeneralNumber Systems › Rational and Irrational Numbers Number Systems › Exponents and RadicalsConceptual

Give two irrational numbers whose sum is rational.

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

Give two irrational numbers whose sum is rational.

Give two irrational numbers whose product is rational.

Give two irrational numbers whose product is irrational.

Need a reminder? The key ideaShow

Rational plus irrational is irrational, while two irrational numbers can give either kind. $\sqrt{a}\,\sqrt{b} = \sqrt{ab}$ , but $\sqrt{a} + \sqrt{b} \neq \sqrt{a + b}$ . $(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$ and $(\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab}$ .

The worked solution

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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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