Real Numbers, Surds & Rounding — Grade 10 Maths (CAPS)

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 are examined in Term 1.

1The number system

Every number you meet in Grade 10 sits somewhere in this family. Each set contains the one before it.

Natural numbers ℕ: 1 ; 2 ; 3 ; … (the counting numbers).
Whole numbers ℕ₀: 0 ; 1 ; 2 ; 3 ; … (the naturals and 0).
Integers ℤ: … ; −2 ; −1 ; 0 ; 1 ; 2 ; … (whole numbers and their negatives).
Rational numbers ℚ: anything that can be written as a/b with a and b integers and b ≠ 0 — including every integer, every terminating decimal and every recurring decimal.
Irrational numbers ℚ′: real numbers that cannot be written as such a fraction, such as π and √2.
Real numbers ℝ: the rationals and the irrationals together — every number on the number line.

A square root of a NEGATIVE number is non-real: there is no real number whose square is negative, so √−9 is not a real number. (The cube root of a negative number IS real: ∛−8 = −2.)

2Rational or irrational?

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

Rational: 7, −3, 2/5, 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.

3Between 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.

4Rounding

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 2,3449 to 2 decimal places in one step gives 2,34, but rounding first to 2,345 and then to 2,35 gives a different answer — always round once, from the full value.

5Significant figures

Sometimes the accuracy is asked for in significant figures instead of decimal places. Counting starts at the first non-zero digit and then continues through every digit, zeros included.

Leading zeros are never significant: 0,004936 starts counting at the 4.
Zeros between digits always count: 50,7 has three significant figures.
Round the last figure you keep in the usual way, and use zeros to hold the place values.

Worked ExampleRound 0,004936 to THREE significant figures

1
Counting starts at the 4: the three figures are 4, 9 and 3.
The zeros in front only place the number — they are not significant.
2
The next digit is 6, so the 3 rounds up to 4.
6 is 5 or more.
3
0,004936 ≈ 0,00494.
The leading zeros stay, so the value is unchanged in size.

Worked ExampleRound 52 718 to THREE significant figures

1
The three figures are 5, 2 and 7.
Counting starts at the first digit, 5.
2
The next digit is 1, so the 7 stays.
1 is less than 5.
3
52 718 ≈ 52 700.
The two zeros hold the tens and units places — they are not significant figures.

💡

Read the instruction carefully: two decimal places counts places after the comma, two significant figures counts digits from the first non-zero one.

Practice exercises

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

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

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