Real Numbers
Sums and multiples of irrationals
Show that numbers like 3 + 2√5 are irrational too.
You already know √2, √3 and √5 are irrational. What happens when we mix them with ordinary fractions?
| You do this | The answer is |
|---|---|
| rational + or – irrational | always irrational (5 – √3) |
| non-zero rational × or ÷ irrational | always irrational (3√2, √2/5) |
| irrational with irrational | it depends: √2 × √2 = 2 is rational, √2 + √3 is not |
How to prove it for one number. The same idea as last time: assume the opposite and reach something impossible.
Show that 5 – √3 is irrational:
- Assume 5 – √3 = a/b, a fraction of whole numbers.
- Get the root alone on one side: √3 = 5 – a/b = (5b – a)/b.
- The right side is made only of whole numbers, so it is a fraction. That would make √3 rational.
- But √3 is irrational. A rational number can never equal an irrational one. Impossible, so 5 – √3 is irrational.
Show that 3√2 is irrational: assume 3√2 = a/b. Then √2 = a/(3b), again a fraction. Impossible, since √2 is irrational.
The routine every time: assume rational → make the root the subject → the other side is a fraction → contradiction. The only thing that changes is the algebra in the middle.
Practise this lesson
Questions that get harder as you improve, a Tutor that helps without giving the answer away, and progress you can see. Free for every learner.
No account, no email address. Opens the real product.