HardMathematicsHigh School (9–10)GeneralNumber Systems › Rational and Irrational NumbersConceptualNumerical
Find √3 + √5 to three decimal places. Prove that √3 + √5 is irrational.
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
- Find to three decimal places.
- Prove that is irrational. Square it first. You may use the fact that is irrational.
Need a reminder? The key ideaShow
Rational plus irrational is irrational. A non-zero rational times an irrational number is irrational. Two irrational numbers can give a rational or an irrational sum or product. To prove that a number is irrational, suppose it is rational and get a known irrational root alone.
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.
Step 1. Part (a). to three places.
Step 2. Suppose the opposite. Suppose , where is rational.
Step 3. Square. , so .
Step 4. The contradiction. The right side is rational, so would be rational. But it is irrational.
Answer: (a) ; (b) is irrational. The decimal of part (a) alone could not prove this.
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