C1.1; E1.1 - Rational and irrational numbers | Cambridge IGCSE Mathematics

Some exact numbers can be written as fractions of integers, while others cannot. For Core and Extended, C1.1/E1.1 uses this difference to distinguish rational and irrational numbers. You will recognise terminating and recurring decimals, classify exact roots, and separate a number from a rounded display of it.

Rational numbers as ratios

A calculator shows $1.414214$ for $\sqrt2$ . Is $\sqrt2$ therefore a rational number because the displayed decimal ends? The answer depends on whether the display is the exact value or an approximation.

Start with fractions. The numbers $\frac34$ , $-\frac52$ and $\frac01$ can all be written as one integer divided by another nonzero integer. They belong to the same family, even though one is positive, one is negative and one is zero.

Rational number

A rational number can be written as $\frac ab$ , where $a$ and $b$ are integers and $b\ne0$ . The fraction need not initially be in its simplest form.

For example, $5=\frac51$ , so every integer is rational. Zero is rational because $0=\frac01$ . A denominator of zero is never allowed.

The word comes from “ratio”: a ratio of two integers. It does not mean “a number that seems sensible”. A fraction bar on its own is not enough: $\frac{\sqrt2}{3}$ does not exhibit a ratio of two integers because its numerator is not an integer.

A terminating decimal is rational because its digits give a whole-number count of tenths, hundredths or another finite decimal place. Its appearance as a decimal does not stop it being a fraction.

Worked example 1: showing a decimal is rational

Question. Explain why $0.48$ is rational by writing it as a fraction of integers.

Plan. Use the hundredths place, then simplify.

\begin{aligned} 0.48&=\frac{48}{100} && \text{forty-eight hundredths}\\ &=\frac{12}{25} && \text{divide both integers by four} \end{aligned}

Answer. $0.48$ is rational because $\frac{12}{25}$ is a ratio of integers with nonzero denominator.

Check. $12\div25=48\div100=0.48$ , so the fraction is exact.

Now change only the sign. Rational numbers include negative values too.

Worked example 2: a negative rational number

Question. Explain why $-0.48$ is rational by writing it as a fraction of integers.

Plan. Keep the denominator positive and attach the negative sign to the numerator.

\begin{aligned} -0.48&=\frac{-48}{100} && \text{negative forty-eight hundredths}\\ &=\frac{-12}{25} && \text{divide numerator and denominator by four} \end{aligned}

Answer. $-0.48$ is rational because $-12$ and $25$ are integers and $25\ne0$ .

Check. $\frac{-12}{25}+\frac{12}{25}=0$ , so the fraction is exactly the negative of the previous value.

An integer or mixed number is handled just as directly: $-7=\frac{-7}{1}$ and $2\frac13=\frac73$ . Rationality concerns whether such a representation exists; you do not need to list every possible equivalent fraction.

Watch out

“Rational numbers are positive proper fractions” is too narrow. Negative fractions, improper fractions, integers and zero can all be rational. The essential conditions are integer numerator, integer denominator, and nonzero denominator.

Terminating and recurring decimals

A decimal that continues forever is not automatically irrational. What matters is whether its digits eventually repeat in a fixed block.

The decimal $0.272727\ldots$ , with the block $27$ repeating forever, is recurring. So is $0.16666\ldots$ , where the $6$ repeats after an initial $1$ . The repeating block can begin after some non-repeating digits.

Recurring decimal

A recurring decimal continues forever with a fixed digit or finite block of digits repeating from some point onwards. Every recurring decimal is rational.

A rational number has a decimal expansion that terminates or eventually recurs. A decimal that neither terminates nor eventually recurs is irrational.

Why do fractions produce terminating or recurring decimals? Think about division by $7$ . Each remainder is one of $0,1,2,3,4,5,6$ . A remainder of zero ends the calculation. If zero never appears, one of the other remainders must eventually appear again. The same remainder then produces the same next digit and remainder, so a block repeats. The same reasoning works for any positive integer denominator.

This explains the behaviour of rational numbers without needing to calculate a fraction for a recurring decimal. Knowing that its block repeats forever is enough to classify it.

Exact description	Decimal behaviour	Type
$0.125$ , with no further nonzero digits	Terminates	Rational
$0.272727\ldots$ , with $27$ repeating forever	Recurs	Rational
$0.16666\ldots$ , with $6$ repeating forever	Eventually recurs	Rational
A decimal known to continue forever with no eventually repeating block	Neither terminates nor recurs	Irrational

Be careful about what an ellipsis tells you. The prefix $0.12345\ldots$ alone does not specify the later digits. A stated repeating rule, or a known exact number, is needed before you can classify it. A screen showing a finite set of digits cannot prove that the actual number terminates.

Key idea

“Continues forever” does not distinguish rational from irrational. A fixed block repeating forever makes the decimal rational; no eventual repeating block makes it irrational.

Irrational numbers and exact roots

An exact root can be rational or irrational. A root sign alone does not decide: $\sqrt{49}=7$ , which is an integer and therefore rational. In contrast, $\sqrt{50}$ cannot be written as a ratio of integers.

Irrational number

An irrational number is a real number that cannot be written as a ratio of two integers with nonzero denominator. Its decimal expansion neither terminates nor eventually recurs.

Examples include $\sqrt2$ , $\sqrt3$ and $\pi$ . These are exact numbers, although their decimal displays must be shortened.

For a non-negative integer under a square-root sign, check whether it is a square number. Its square root is rational exactly when that integer is a square number. This includes $\sqrt0=0$ . The qualification “integer” matters: $\sqrt{2.25}=1.5$ is rational because $1.5^2=2.25$ .

Worked example 3: a root that is rational

Question. State whether $\sqrt{49}$ is rational or irrational. Give a reason.

Plan. Look for an integer whose square is $49$ , then use the rational-number definition.

\begin{aligned} 7^2&=49 && \text{recognise a square number}\\ \sqrt{49}&=7=\frac71 && \text{write an integer ratio} \end{aligned}

Answer. $\sqrt{49}$ is rational because it equals the integer $7$ .

Check. $7\times7=49$ , and the principal square root is non-negative. The answer is $7$ , not both $7$ and $-7$ .

Increase only the number under the root from $49$ to $50$ . Being close to a square does not make it a square.

Worked example 4: a root that is irrational

Question. State whether $\sqrt{50}$ is rational or irrational. Give a reason.

Plan. Check whether the non-negative integer $50$ is a square number.

\begin{aligned} 7^2&=49 && \text{the square immediately below fifty}\\ 8^2&=64 && \text{the next square}\\ 49&<50<64 && \text{fifty is not a square integer} \end{aligned}

Answer. $\sqrt{50}$ is irrational because $50$ is a non-square positive integer.

Check. Its value lies between $7$ and $8$ , and its square is exactly $50$ . This confirms its size; the non-square-integer rule, rather than its size alone, establishes irrationality.

Roots of fractions can also be rational: $\sqrt{\frac9{16}}=\frac34$ , because $\left(\frac34\right)^2=\frac9{16}$ . Do not assume that every non-integer or every number with a root sign is irrational.

Now return to the opening calculator display. The exact finite decimal $1.414214$ is rational: it equals $\frac{1414214}{1000000}$ . But $\sqrt2$ is irrational, and the display is only an approximation:

\sqrt2\approx1.414214,\qquad \sqrt2\ne1.414214.

Likewise, $\pi$ is irrational, while $3.142$ and $\frac{22}{7}$ are rational approximations to it. Use an equals sign only for exact equality.

Watch out

“The calculator's digits end, so the exact number is rational” is wrong. Classify the exact expression. A finite display may represent a rounded approximation to an irrational number.

Quick Recap

Rational numbers can be written as a ratio of integers with a nonzero denominator; they include integers and zero.
Terminating and eventually recurring decimals are rational.
Irrational decimals neither terminate nor eventually recur.
Check exact roots before classifying them: the square root of a square integer is rational.
A rounded decimal display does not determine the type of its exact source.

The opening calculator display does not make $\sqrt2$ rational. It is a rational approximation to an irrational exact number.

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C1.1; E1.1 - Rational and irrational numbers

Rational numbers as ratios

Terminating and recurring decimals

Irrational numbers and exact roots

Quick Recap

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