1.9 Rational and irrational numbers
Surds and recurring decimals
Everything on surds and recurring decimals that can come up in the WJEC GCSE Mathematics and Numeracy, in Unit 2 and Unit 3. Each skill below has the method, the mistake that costs the most marks, a worked example, and free practice marked the way WJEC marks it.
Practise surds and recurring decimals free
A worked example
Simplifying surds · [2 marks]
Write √72 in the form a√b, where a and b are whole numbers and b is as small as possible.
- B1Takes out a square factor, but not the largest one
- B1√72 = √36 × √2 = 6√2
The exam skills (7)
Simplifying surds
- Find the biggest square number that divides into the number under the root.
- Split the root: √8 = √4 × √2.
- Square root the square number: 2√2.
Watch out: Taking out a square factor that is not the biggest, which leaves it not fully simplified.
Multiplying surds
- √a × √b = √(ab).
- Multiply the numbers under the root.
- Simplify: if it is a square number, the root disappears.
Watch out: Adding the numbers under the root instead of multiplying.
Recurring decimals
- Let x = the recurring decimal.
- Multiply by 10 (one repeating digit) or 100 (two) so the repeats line up.
- Subtract to get rid of the repeating part, then divide and simplify.
Watch out: Multiplying by 10 when two digits repeat. Use 10 to the number of repeating digits.
Rational and irrational numbers
- Rational: can be written as a fraction of two whole numbers (including recurring decimals).
- Square roots of square numbers are rational: √16 = 4.
- π and roots of non-square numbers are irrational.
Watch out: Thinking 22/7 is π. It is a fraction, so it is rational.
Adding surds
- Simplify every surd first.
- You can only add surds with the same number under the root.
- Add the numbers in front: 1√2 + 2√2 = 3√2.
Watch out: Adding under the roots: √2 + √8 is not √10.
Expanding with surds
- Multiply every term in the first bracket by every term in the second (four terms).
- Remember √2 × √2 = 2.
- Collect the whole numbers and the surd terms separately.
Watch out: Forgetting that a surd times itself is a whole number.
Rationalising
- Multiply the top and bottom by the surd on the bottom.
- The bottom becomes a whole number: √2 × √2 = 2.
- Simplify, cancelling if you can.
Watch out: Only multiplying the bottom. Whatever you do to the bottom, do to the top.
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