Real Numbers, Surds & Rounding (Grade 10)

Rational versus irrational numbers, working out which two integers a surd lies between, and rounding to a given number of decimal places or significant figures — the opening skills of the Grade 10 syllabus.

Grade 10 opens by tightening up what a number is. You must be able to classify any number, place a surd between two integers without a calculator, and round sensibly — all three appear in the first exam.

1Rational or irrational?

Rational: can be written as ab with a, b integers, b ≠ 0.
Irrational: cannot — its decimal never ends and never repeats.

Rational: 7, −3, 25, 0,25 (terminates), 0,333… (recurs), 49 = 7 (exact root).
Irrational: π, 2, 20, 10 — roots that do not come out exactly.

⚠️

A root is not automatically irrational. Work it out first: 64 = 8 is rational, but 65 is irrational.

2Between which two integers does a surd lie?

This is a standard exam question and no calculator is allowed. The trick is to find the perfect squares on either side.

√20 lies between 4 and 5 because 20 lies between the perfect squares 16 and 25.

Worked ExampleBetween which two integers does 20 lie?

1
Find the perfect squares just below and above 20: 16 and 25.
16 = 4² and 25 = 5².
2
So 16 < 20 < 25.
3
Taking roots: 4 < 20 < 5.
Square roots keep the order, so the answer is between 4 and 5.

Worked ExampleBetween which two integers does 54 lie?

1
7² = 49 and 8² = 64.
49 < 54 < 64.
2
So 7 < 54 < 8.

💡

Learn your squares to 15² = 225. Almost every surd question in Grade 10 sits inside that range.

3Rounding

Look at the digit immediately after the place you are rounding to.
If it is 5 or more, round up. If it is 4 or less, leave the digit as it is.

Worked ExampleRound 3,14159 to 2 decimal places

1
The 2nd decimal is 4; the next digit is 1.
1 is less than 5.
2
3,14159 ≈ 3,14.
The 4 stays as it is.

Worked ExampleRound 7,3962 to 2 decimal places

1
The 2nd decimal is 9; the next digit is 6.
6 is 5 or more, so round up.
2
9 rounds up to 10, so it carries: 7,3962 ≈ 7,40.
Keep the trailing zero — it shows the accuracy asked for.

⚠️

Do not round in stages. Rounding 7,3962 to 7,396 then 7,40 can give a different answer to rounding once from the original — always round from the full value.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Is 81 rational or irrational?
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Rational — 81 = 9 exactly.
2
Is 30 rational or irrational?
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Irrational — 30 is not a perfect square.
3
Between which two integers does 20 lie?
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16 < 20 < 25 → 4 and 5
4
Between which two integers does 54 lie?
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49 < 54 < 64 → 7 and 8
5
Between which two integers does 110 lie?
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100 < 110 < 121 → 10 and 11
6
Round 3,14159 to 3 decimal places.
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3,142
7
Round 7,3962 to 2 decimal places.
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7,40
8
Round 0,04567 to 3 decimal places.
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0,046
9
Is 0,272727… rational?
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Yes — it recurs, so it can be written as a fraction.