Stage 9 · Rational & Irrational Numbers
← All sheets
⏱
00:00
Score
0 / 20
Secondary 2 · Mathematics · Chapter 1.1

Rational &
Irrational Numbers

Rational Numbers · Irrational Numbers · Surds · Number Lines
16 questions · self-marked · 20 marks

📚 Watch & Learn first

Need a refresher on rational vs. irrational numbers and surds before you start?

More resources →
Rational Numbers Irrational Numbers Surds On a Number Line
ℝ
Section A

Rational Numbers

Definition: A rational number can be written as a ratio of two integers, a/b (where b ≠ 0). This includes whole numbers, negative integers, fractions, percentages, decimals that terminate (end), and decimals that repeat forever in a pattern.
Examples: 3 = 3/1 · 0.10 = 1/10 · 0.333... = 1/3 · -8 = -8/1 · 25% = 25/100
GROUP 1
1
2
3
GROUP 2
-5
-8
-12
GROUP 3
1/2
3/4
5 2/5
GROUP 4
π
√2
0.1434...
Q1
Look at the four groups of numbers above. Which group is the odd one out — the only group that is not rational?
1 mark
Group 1
Group 4
Group 3
Group 2
Q2
Which of these can be written as a ratio of two integers, like 3/1?
1 mark
3
π
√5
0.1010010001...
Q3
0.333... (recurring forever) is equivalent to which fraction?
1 mark
1/2
1/4
1/3
3/10
Q4
Which of the following is rational?
2 marks
√2
π
-8
0.1010010001...
RATIONAL NUMBERS — SECTION SCORE
0/5
π
Section B

Irrational Numbers

Definition: An irrational number cannot be written as a ratio of two integers. Its decimal expansion goes on forever without repeating in a pattern. Examples: π = 3.14159..., square roots of non-perfect squares like √2, and decimals like 0.1010010001...
Together, rational + irrational numbers make up the real number system — every point on the number line is either one or the other, never both.
Q5
Which of these is irrational?
1 mark
0.5
3/4
√9
√7
Q6
Why is π classed as irrational?
1 mark
Because its decimal expansion never terminates or repeats
Because it's negative
Because it's bigger than 1
Because it's a fraction
Q7
Which of these is a non-terminating, non-repeating decimal?
1 mark
0.25
0.666...
-3
0.1010010001...
Q8
Is √16 rational or irrational, and why?
2 marks
Rational, because 16 is an even number
Irrational, because it has a square root symbol
Rational, because 16 is a perfect square so √16 = 4
Irrational, because 4 is rational
IRRATIONAL NUMBERS — SECTION SCORE
0/5
√
Section C

Surds

Definition: A surd is a root of a number that cannot be simplified to a rational number. Roots of perfect squares (like √4 = 2) are not surds, because they simplify to a whole number.
Example: √2 is a surd (2 isn't a perfect square). √9 is not a surd, because it simplifies to the rational number 3.
Q9
Which of these is a surd?
1 mark
√4
√5
√9
√25
Q10
Which of these is NOT a surd?
1 mark
√2
√3
√7
√16
Q11
Is √(1/4) a surd?
1 mark
Yes, because it has a root symbol
No, because √(1/4) = 1/2, which is rational
Yes, because 1/4 is a fraction
No, because 1/4 is less than 1
Q12
Which is the best definition of a surd?
2 marks
A root of a number that cannot be simplified to a rational number
Any number with a root symbol in front of it
A number between 1 and 10
A decimal that repeats forever
SURDS — SECTION SCORE
0/5
↔
Section D

Irrational Numbers on a Number Line

Method: Use the decimal expansion to estimate where an irrational number sits. Find two consecutive perfect squares either side of the number under the root, then compare.
Example: √12 = 3.4641... — since 9 < 12 < 16, and √9 = 3, √16 = 4, we know √12 lies between 3 and 4 (closer to 3.5).
√12 ≈ 3.46, plotted between 3 and 4
0
1
2
3
4
5
6
√12
Q13
√20 lies between which two integers?
1 mark
4 and 5
3 and 4
5 and 6
2 and 3
Q14
√50 lies between which two integers?
1 mark
6 and 7
7 and 8
8 and 9
5 and 6
Q15
√12 lies between which two integers?
1 mark
2 and 3
4 and 5
3 and 4
1 and 2
Q16
Estimate √30 to 1 decimal place.
2 marks
NUMBER LINE — SECTION SCORE
0/5
0%
Keep Practising