Practice Maths

Irrational Numbers and Surds

Key Ideas

Key Terms

rational number
a number that can be written as p⁄q where p and q are integers and q ≠ 0; includes all integers, terminating decimals, and recurring decimals.
irrational number
a number that cannot be written as a fraction; its decimal expansion is non-terminating and non-recurring (e.g. √2 ≈ 1.41421…, π ≈ 3.14159…).
surd
an irrational root that cannot be simplified to a rational number (e.g. √2, √3, √5 are surds; √9 = 3 is not).
real number
any number on the number line — every rational and every irrational number combined.
perfect square
a whole number that equals another whole number squared (1, 4, 9, 16, 25…); the square root of a perfect square is always an integer.

Simplifying Surds

Use the rule √(a × b) = √a × √b. Find the largest perfect square factor of the number under the root sign.

Hot Tip To simplify a surd, find the largest perfect square factor. For √72: 72 = 36 × 2, so √72 = √36 × √2 = 6√2. (If you used 4 × 18, you would get 2√18 and need to simplify again.)

Worked Example

Simplify: (a) √50    (b) √12    (c) Estimate √20 between two consecutive integers.

(a) √50: 50 = 25 × 2  →  √50 = √25 × √2 = 5√2

(b) √12: 12 = 4 × 3  →  √12 = √4 × √3 = 2√3

(c) √20: 42 = 16 and 52 = 25. Since 16 < 20 < 25, we have 4 < √20 < 5.

Rational vs. Irrational Numbers

A rational number is any number that can be written as a fraction p/q, where p and q are integers and q ≠ 0. This includes all integers (e.g. 5 = 5/1), all terminating decimals (e.g. 0.25 = 1/4), and all recurring decimals (e.g. 0.3̅ = 1/3).

An irrational number is any real number that cannot be written as a fraction. Its decimal expansion goes on forever with no repeating pattern. Famous irrational numbers include:

  • √2 ≈ 1.41421356... — the diagonal of a 1×1 square.
  • √3 ≈ 1.73205080... — the height of an equilateral triangle with side length 2.
  • π ≈ 3.14159265... — the circumference-to-diameter ratio of any circle.
  • e ≈ 2.71828182... — the base of natural logarithms (studied later).

What is a Surd?

A surd is an irrational root of a number. The most common surds are square roots like √2, √3, √5, √6, √7. Not all square roots are surds — √4 = 2, √9 = 3, √25 = 5 are all rational (they are perfect squares). Only roots that do not simplify to a rational number are surds.

To decide: is √n a surd? Ask whether n is a perfect square. If yes (n = 1, 4, 9, 16, 25, 36, ...), the root is rational. If no, the root is a surd.

  • √16 = 4 — rational, not a surd.
  • √17 ≈ 4.123... — irrational, a surd.
  • √100 = 10 — rational, not a surd.
  • √50 ≈ 7.071... — irrational, a surd.

Decimal Approximations of Surds

You cannot write a surd as an exact decimal, but you can approximate it. Use your calculator to find decimal approximations, and round to the required number of decimal places.

Examples:
√2 ≈ 1.414 (to 3 d.p.)
√5 ≈ 2.236 (to 3 d.p.)
√10 ≈ 3.162 (to 3 d.p.)

In exact calculations (like geometry), it is often better to leave the answer as a surd rather than rounding — for example, writing a side length as √5 cm instead of 2.236 cm keeps the answer perfectly accurate.

Key tip: When a question asks for an exact answer, leave surds as surds — do not convert to a decimal. When a question says "round to 2 decimal places" or "give a decimal answer", then use your calculator. Reading the instruction carefully determines which form of answer to give.

The Real Number System

All the numbers you have studied form a nested hierarchy. Think of it like a set of rings inside rings:

  • Natural numbers (ℕ): 1, 2, 3, 4, ... (counting numbers)
  • Integers (ℤ): ..., −3, −2, −1, 0, 1, 2, 3, ... (add negatives and zero)
  • Rational numbers (ℚ): All fractions p/q — includes all integers, terminating decimals, recurring decimals.
  • Irrational numbers: Numbers that are NOT rational — surds, π, e, etc.
  • Real numbers (ℝ): All rational AND all irrational numbers combined — every number on the number line.

Every integer is also rational. Every rational number is also real. But not every real number is rational. The irrational numbers sit "between" the rationals on the number line, filling in the gaps.

Mastery Practice

  1. Classify each number as rational (R) or irrational (I). Give a brief reason. Fluency

    1. √9
    2. √2
    3. 0.3̇
    4. π
    5. √16
    6. √5
    7. 2/3
    8. √100

  2. Estimate each surd between two consecutive integers (e.g. 4 < √20 < 5). Do not use a calculator. Fluency

    1. √15
    2. √30
    3. √50
    4. √8
    5. √45
    6. √70
    7. √3
    8. √110

  3. Simplify each surd fully. Show the perfect square factor you used. Fluency

    1. √8
    2. √18
    3. √45
    4. √75
    5. √200
    6. √28
    7. √48
    8. √98

  4. Order each set of numbers from smallest to largest. Convert to decimals (to 2 d.p.) to help you. Understanding

    1. √2,   1.5,   1/2,   √3
    2. π,   3,   √10,   3.2
    3. 2√2,   3,   √7,   2.8
    4. √5,   2.25,   √6,   2.5
    5. 5√2,   7,   √50,   3√3
    6. 1/3,   √0.1,   0.3̇,   0.35

  5. State whether each statement is True or False. If false, write a corrected version. Understanding

    1. Every integer is a rational number.
    2. √4 is irrational because it involves a square root.
    3. The product of two irrational numbers is always irrational.
    4. π can be written as 22/7 exactly.
    5. Every real number is either rational or irrational, but not both.
    6. A surd always lies between two consecutive integers.
    7. Irrational numbers cannot be placed on a number line.
    8. 0.9̇ is an irrational number because it goes on forever.

  6. Apply surds and irrational numbers to real contexts. Problem Solving

    1. A square garden has an area of 50 m2. What is the exact side length? Write your answer as a simplified surd.
    2. A right-angled triangle has legs of 3 cm and 4 cm.
      1. Calculate the hypotenuse using Pythagoras’ theorem. Is your answer rational or irrational?
      2. Now try legs of 1 cm and 2 cm. Calculate the hypotenuse and simplify.
    3. The diagonal of a square with side length s is s√2. A square tile has a side length of 5 cm. What is the exact length of its diagonal? Give your answer as a simplified surd and as a decimal (to 1 d.p.).
    4. Jamie argues that √2 + √2 = √4 = 2. Is Jamie correct? Show why or why not.

  7. Sort the following numbers into the correct column of the classification table. Understanding

    Numbers:   √4,   √3,   π,   0.6̇,   √25,   √11,   22/7,   0.101001000…,   √49,   √6

    Rational Irrational
      

  8. Mark the position of each surd on the number line below. Use the integer squares you know to estimate each value. Understanding

    Mark:   √2,   √5,   √7,   √10

    1 2 3 4 1.5 2.5 3.5

    Hint: √2 ≈ 1.41, √5 ≈ 2.24, √7 ≈ 2.65, √10 ≈ 3.16. Scale: the interval from 1 to 2 spans 110 px.

  9. Complete the table by simplifying each surd. Show the perfect square factor used. Understanding

    Surd Largest perfect square factor Simplified form
    √12??
    √50??
    √72??
    √108??
    √?255√3

  10. Extended reasoning about surds and irrational numbers. Problem Solving

    1. Show that √2 is irrational by explaining why its decimal expansion cannot terminate or recur. (Hint: if √2 were rational, what would that mean about its decimal?)
    2. A student says “√4 + √9 = √13”. Is this correct? Calculate both sides and explain the correct rule.
    3. Between which two consecutive integers does 2√5 lie? Show your reasoning without a calculator.
    4. A square has a diagonal of length 10 cm. Use the relationship (diagonal = side × √2) to find the exact side length as a simplified surd, and estimate it to 1 decimal place.
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