CDS Mathematics · Number System

Rational & Irrational Numbers

The one test that separates the two — a rational number's decimal expansion terminates or recurs, an irrational number's does neither — plus the mechanical way to turn any recurring decimal back into a fraction.

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Why this matters

Six PYQs, the smallest unit in the chapter and none of them HARD. Every question is either the terminate-or-recur test applied to a list, or a recurring decimal converted to a fraction. The recurring-to-fraction recipe is worth memorising outright because it appears in half of them.

Concept 1 of 3: What makes a number rational, and what the decimal expansion reveals

A rational number is a ratio of integers, and that forces its decimal expansion either to stop or to fall into a repeating block. An irrational number's expansion does neither — it runs forever without settling into a pattern, which is exactly why you can bracket it by fractions but never land on it.

Definition

A number is rational if it can be written $\dfrac pq$ with $p,q$ integers and $q \ne 0$ ; otherwise it is irrational.

A rational number's decimal expansion terminates or recurs — those are the only options, and either one is a guarantee of rationality.
An irrational number's expansion is non-terminating and non-repeating.
A terminating decimal is always rational, since it is a fraction over a power of 10.
Both kinds are dense: between any two numbers lie infinitely many of each.

Zoom as far as you like: the decimals bracket √2 ever more tightly and never land on it. That is the whole difference — a rational number's expansion terminates or recurs, so it is reachable; an irrational number's does neither.

Number	Rational?	Reason
0.5	Rational	Terminates
0.333...	Rational	Recurs, equals one third
$\sqrt{75}$	Irrational	Equals $5\sqrt3$ , and 75 is not a perfect square
$\sqrt{59049}$	Rational	Equals 243, since $59049=3^{10}$
0.12112211122211112222...	Irrational	Blocks grow, so it never repeats A visible pattern is not a repeating block. Recurrence needs a fixed block repeated forever.
$\pi$	Irrational	Non-terminating, non-repeating
$4\pi r^{2}$ with $r$ rational	Irrational	A non-zero rational multiple of $\pi$
$\sqrt2 \times \sqrt{50}$	Rational	Equals 10 — a product of irrationals can be rational

The decimal expansion is the test. Note the two rows that go against first instinct: a square root can be rational, and a product of irrationals can be too.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — Elementary Mathematics · Q5Easy

Example 1 · Number System · Rational and Irrational Numbers

Which one of the following is correct?

A square root is not automatically irrational

\sqrt{59049} = 243

because

59049 = 3^{10}

, so it is rational.

\sqrt{n}

is irrational exactly when

n

is not a perfect square. Check for squareness before calling a root irrational — CDS plants a large perfect square in the option list precisely to catch this.

A pattern is not the same as a recurring block

0.12112211122211112222\ldots

is clearly patterned, but the blocks lengthen, so no fixed block repeats and the number is irrational. Recurrence means one unchanging block forever, as in

0.\overline{45}

Drill 1 more on what makes a number rational, and what the decimal expansion reveals

Concept 2 of 3: Converting a recurring decimal to a fraction

A purely recurring decimal with a block of

k

digits equals that block over

k

nines. That is the whole recipe, and it comes straight from multiplying by

10^{k}

and subtracting so the infinite tail cancels.

Definition

For a purely recurring decimal, put the repeating block over as many 9s as it has digits:

0.\overline{d} = \frac{d}{9}, \qquad 0.\overline{d_1d_2} = \frac{d_1d_2}{99}, \qquad 0.\overline{d_1d_2d_3} = \frac{d_1d_2d_3}{999}.

Then reduce to lowest terms — that is usually where the answer lives.
Useful factorisations of the denominators: $99 = 9\times 11$ and $999 = 27\times 37$ .
If the recurrence starts later, shift it: $0.0\overline{459} = \frac{1}{10}\times 0.\overline{459}$ .
A famous consequence: $0.\overline{9} = \frac99 = 1$ exactly.

Purely recurring decimal

0.\overline{d_1d_2\cdots d_k}=\frac{d_1d_2\cdots d_k}{\underbrace{99\cdots9}_{k}}

Worked example

Express

0.\overline{45}

as a fraction in lowest terms.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — Elementary Mathematics · Q34Moderate

Example 2 · Number System · Rational and Irrational Numbers

Which one of the following decimal numbers is a rational number with denominator 37?

The recurring bar changes the value, and the question turns on it

0.\overline{9} = 1

exactly, so

0.\overline{9} - 0.9 = 0.1

, not

0.0999\ldots

. Likewise

0.\overline{459}

\frac{17}{37}

while the terminating

0.459459459

is a fraction over a power of 10 and has no 37 in its denominator. Read whether the bar is present before converting.

Drill 1 more on converting a recurring decimal to a fraction

Concept 3 of 3: Deciding irrationality of roots, sums and products

For a root, the test is whether the number under it is a perfect power. For combinations, the useful facts are asymmetric: adding a rational to an irrational always stays irrational, but multiplying two irrationals can land back on a rational.

Definition

Working rules:

$\sqrt{n}$ is rational exactly when $n$ is a perfect square; simplify first, since $\sqrt{75} = 5\sqrt3$ .
rational $+$ irrational is always irrational.
non-zero rational $\times$ irrational is always irrational — which settles every question about $\pi$ .
irrational $+$ irrational and irrational $\times$ irrational may be either: $\sqrt2+(-\sqrt2)=0$ and $\sqrt2\times\sqrt{50}=10$ are both rational.

So a question asking which of several expressions is irrational is answered by elimination, and the ones to check hardest are the products.

Root test

\sqrt{n} \in \mathbb{Q} \iff n \text{ is a perfect square}

Worked example

\sqrt{18}

rational or irrational?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (II) 2019 — Elementary Mathematics · Q30Moderate

Example 3 · Number System · Rational and Irrational Numbers

Which one of the following is an irrational number ?

Simplify the surd before judging it

\sqrt{75}

looks irreducible but equals

5\sqrt3

;

\sqrt{59049}

looks irrational but equals 243. Pull out every square factor first. The 2019 statement question hinges on exactly this:

\sqrt{75}

being called rational is the planted falsehood.

Drill 1 more on deciding irrationality of roots, sums and products

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Converting a recurring decimal to a fraction
Purely recurring decimal
$0.\overline{d_1d_2\cdots d_k}=\frac{d_1d_2\cdots d_k}{\underbrace{99\cdots9}_{k}}$
Deciding irrationality of roots, sums and products
Root test
$\sqrt{n} \in \mathbb{Q} \iff n \text{ is a perfect square}$

Reference tables (1)

What makes a number rational, and what the decimal expansion reveals8 rows

Number	Rational?	Reason
0.5	Rational	Terminates
0.333...	Rational	Recurs, equals one third
$\sqrt{75}$	Irrational	Equals $5\sqrt3$ , and 75 is not a perfect square
$\sqrt{59049}$	Rational	Equals 243, since $59049=3^{10}$
0.12112211122211112222...	Irrational	Blocks grow, so it never repeats A visible pattern is not a repeating block. Recurrence needs a fixed block repeated forever.
$\pi$	Irrational	Non-terminating, non-repeating
$4\pi r^{2}$ with $r$ rational	Irrational	A non-zero rational multiple of $\pi$
$\sqrt2 \times \sqrt{50}$	Rational	Equals 10 — a product of irrationals can be rational

The decimal expansion is the test. Note the two rows that go against first instinct: a square root can be rational, and a product of irrationals can be too.

Watch out for (4)

A square root is not automatically irrational
→ What makes a number rational, and what the decimal expansion reveals
A pattern is not the same as a recurring block
→ What makes a number rational, and what the decimal expansion reveals
The recurring bar changes the value, and the question turns on it
→ Converting a recurring decimal to a fraction
Simplify the surd before judging it
→ Deciding irrationality of roots, sums and products

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