SPI Mathematics · Number System

Irrational Numbers

An irrational number cannot be written as a fraction of two integers: its decimal never ends and never repeats.

Why this matters

Seven questions in the bank, six EASY and one MODERATE, and one is from the 2022 paper. Three ask you to spot the irrational number in a list; four ask whether a sum, product or square of such numbers is rational or irrational.

Concept 1 of 2: Which numbers are irrational

A root sign does not make a number irrational. The square root of a whole number is irrational only when the number is not a perfect square. And a decimal is irrational only when it goes on forever without settling into a repeating block.

Definition

$\sqrt{n}$ for a whole number $n$ is rational if $n$ is a perfect square and irrational otherwise.
To test for a perfect square, find the two squares on either side: $44^2 = 1936$ and $45^2 = 2025$ , so 1950 is not a perfect square.
$\pi$ is irrational. $\frac{22}{7}$ and 3.14 are only rational approximations of it.
A decimal that ends or repeats is rational; one that does neither is irrational.

Number	Type	Reason
$\sqrt{2}$ , $\sqrt{3}$ , $\sqrt{5}$	Irrational	2, 3 and 5 are not perfect squares
$\sqrt{49}$	Rational	It equals 7
$\sqrt{1296}$	Rational	It equals 36
$\pi$	Irrational	Its decimal never ends and never repeats $\frac{22}{7}$ is close to $\pi$ but is a fraction, so it is rational.
$\frac{22}{7}$ , 3.14	Rational	Each is a fraction of two integers
$0.\overline{3}$	Rational	It repeats: $\frac{1}{3}$
0.101001000100001...	Irrational	It never ends and has no repeating block

A root sign alone decides nothing: check whether the number under it is a perfect square.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q11Moderate

Example 1 · Number System · Irrational Numbers

____ is an irrational number.

Treating every root as irrational

\sqrt{2401} = 49

is a whole number. Before calling a root irrational, check whether the number under it is a perfect square.

Taking 22/7 for π

\frac{22}{7}

is a fraction, so it is rational. Only

\pi

itself is irrational.

Drill 2 more on which numbers are irrational

Concept 2 of 2: Sums and products with irrational numbers

Mixing a rational with an irrational cannot cancel the endless decimal, so the result stays irrational. Two irrationals, though, can cancel each other out.

Definition

Rational $+$ irrational is always irrational.
Non-zero rational $\times$ irrational is always irrational. Zero times anything is 0, which is rational.
Irrational $+$ irrational, or irrational $\times$ irrational, can be either: $\sqrt{2} + \sqrt{3}$ is irrational, but $\sqrt{2} \times \sqrt{8} = 4$ is rational.
To decide about an expression, simplify it first. If a root term is left over, the result is irrational.

Squares and products with a root

(a + \sqrt{b})^2 = a^2 + 2a\sqrt{b} + b \qquad (a + \sqrt{b})(a - \sqrt{b}) = a^2 - b

Worked example

(\sqrt{5} + 2)(\sqrt{5} - 2)

rational or irrational?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Number System · Irrational Numbers

(2 - 5\sqrt{3})^2

Assuming two irrationals add to an irrational

(5 + \sqrt{2}) + (1 - \sqrt{2}) = 6

, which is rational. The sum of two irrationals can be rational or irrational.

Dropping the middle term when squaring

(1 + \sqrt{5})^2

6 + 2\sqrt{5}

, not

1 + 5 = 6

. The middle term

2a\sqrt{b}

keeps the result irrational.

Forgetting the zero case

Rational times irrational is irrational only when the rational is not 0. A question that says just 'a rational number' hides this exception.

Drill 3 more on sums and products with irrational numbers

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 (1)

Sums and products with irrational numbers
Squares and products with a root
$(a + \sqrt{b})^2 = a^2 + 2a\sqrt{b} + b \qquad (a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$

Reference tables (1)

Which numbers are irrational7 rows

Number	Type	Reason
$\sqrt{2}$ , $\sqrt{3}$ , $\sqrt{5}$	Irrational	2, 3 and 5 are not perfect squares
$\sqrt{49}$	Rational	It equals 7
$\sqrt{1296}$	Rational	It equals 36
$\pi$	Irrational	Its decimal never ends and never repeats $\frac{22}{7}$ is close to $\pi$ but is a fraction, so it is rational.
$\frac{22}{7}$ , 3.14	Rational	Each is a fraction of two integers
$0.\overline{3}$	Rational	It repeats: $\frac{1}{3}$
0.101001000100001...	Irrational	It never ends and has no repeating block

A root sign alone decides nothing: check whether the number under it is a perfect square.

Watch out for (5)

Treating every root as irrational
→ Which numbers are irrational
Taking 22/7 for π
→ Which numbers are irrational
Assuming two irrationals add to an irrational
→ Sums and products with irrational numbers
Dropping the middle term when squaring
→ Sums and products with irrational numbers
Forgetting the zero case
→ Sums and products with irrational numbers

Test yourself on a real paper

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