The World of Numbers Class 9: Rational, Irrational and Real Numbers

What The World of Numbers is about

Weightage: This chapter is part of Unit I, Number System, which carries 7 of the 80 marks in the school annual exam.Source: CBSE Mathematics syllabus, Class IX 2026-27, course structure, Unit I Number System

Key concepts

From counting to integers: Brahmagupta's rules

NCERT section 3.3 Integers: Expanding the Horizon

Counting gave the natural numbers $\{1,2,3,\dots \}$. Brahmagupta (628 CE) made zero a number with rules: $a+0=a$, $a-0=a$, $a\times 0=0$, and defined it as $a-a$. He also treated debts as negative numbers. Natural numbers, zero and the negatives together form the integers, written Z.

His sign rules are the ones we still use:

• fortune + fortune = fortune; debt + debt = debt, e.g. $(-5)+(-4)=-9$
• debt × fortune = debt, e.g. $(-3)\times 4=-12$
• debt × debt = fortune, e.g. $(-3)\times (-4)=12$ (removing debts makes you richer)

Subtracting a negative is the same as adding a positive: $10-(-5)=15$.

Worked example

A trader takes a loan of ₹650, then makes a profit of ₹900, then has a loss of ₹380. Write this as one integer expression and find where he stands. Also find $(-9)\times (-6)$ and $0-(-14)$.

Try it first, then show the solutionSolution

1. Loan and loss are debts (negative); profit is a fortune (positive).
2. $-650+900-380=250-380=-130$.
3. He is still in debt by ₹130.
4. $(-9)\times (-6)=54$ (debt × debt = fortune).
5. $0-(-14)=0+14=14$.

Answer: $-650+900-380=-130$, a debt of ₹130; $(-9)(-6)=54$; $0-(-14)=14$.

Rational numbers and when two are equal

NCERT section 3.4 Filling the Spaces: Fractions and Rational Numbers

A rational number is any number that can be written as $\frac{p}{q}$ where p and q are integers and $q\ne 0$. The set is called Q (for quotient). We need $q\ne 0$ because division by zero has no meaning.

• Every integer is rational: $5=\frac{5}{1}$, $-10=\frac{-10}{1}$. So natural numbers, whole numbers and integers are all inside Q.
• One rational number has many forms: $\frac{1}{2}=\frac{2}{4}=\frac{50}{100}$. We usually use the lowest form, where p and q are co-prime (no common factor except 1).
• Equality test: $\frac{a}{b}=\frac{c}{d}$ exactly when $ad=bc$.
• The minus sign can sit with the numerator, the denominator or in front: $-\frac{1}{2}=\frac{-1}{2}=\frac{1}{-2}$.

Worked example

Show that $\frac{-18}{24}$ and $\frac{15}{-20}$ are equal, and write the number in its lowest form.

Try it first, then show the solutionSolution

1. Cross-multiply: $(-18)\times (-20)=360$ and $24\times 15=360$.
2. The products are equal, so the two rational numbers are equal.
3. $\frac{-18}{24}$: divide top and bottom by 6 to get $\frac{-3}{4}$.

Answer: They are equal; lowest form $-\frac{3}{4}$.

Operations on rational numbers and closure

NCERT section 3.4 Filling the Spaces: Fractions and Rational Numbers

The rules (also stated by Brahmagupta) are:

• Add or subtract: first make the denominators the same, then $\frac{a}{b}\pm \frac{c}{b}=\frac{a\pm c}{b}$.
• Multiply: $\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}$.
• Divide: $\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}=\frac{ad}{bc}$, with $c\ne 0$.

Addition and multiplication are commutative, and multiplication distributes over addition: $p(q+r)=pq+pr$.

Rational numbers are closed under addition, subtraction and multiplication (the answer is always rational), and under division as long as you do not divide by zero. Natural numbers are not closed under subtraction ($3-5$ is not natural), which is one reason integers were needed.

Worked example

Find (i) $-\frac{5}{6}+\frac{3}{4}$ and (ii) $\left(-\frac{2}{3}\right)÷\frac{4}{9}$.

Try it first, then show the solutionSolution

1. (i) LCM of 6 and 4 is 12: $-\frac{10}{12}+\frac{9}{12}=-\frac{1}{12}$.
2. (ii) $-\frac{2}{3}\times \frac{9}{4}=-\frac{18}{12}=-\frac{3}{2}$.

Answer: (i) $-\frac{1}{12}$ (ii) $-\frac{3}{2}$

Rational numbers on the number line and absolute value

NCERT section 3.4 Filling the Spaces: Fractions and Rational Numbers

To mark $\frac{p}{q}$ (lowest form, q positive): split each unit into q equal parts and count p parts from 0, to the right if positive and to the left if negative. For a number bigger than 1, first write it as a mixed number: $\frac{9}{4}=2\frac{1}{4}$, so split the gap between 2 and 3 into 4 parts and take 1.

The absolute value $|x|$ is the distance of x from 0, so it is never negative: $\left|-\frac{5}{3}\right|=\frac{5}{3}$, $|0|=0$. The distance between two numbers a and b on the line is $|a-b|$.

Worked example

Describe how to mark $\frac{11}{4}$ and $-\frac{5}{3}$ on the number line, and find the distance between them.

Try it first, then show the solutionSolution

1. $\frac{11}{4}=2\frac{3}{4}$: divide the gap from 2 to 3 into 4 equal parts and move 3 parts right of 2.
2. $-\frac{5}{3}=-1\frac{2}{3}$: divide the gap from $-1$ to $-2$ into 3 equal parts and move 2 parts left of $-1$.
3. Distance $=\left|\frac{11}{4}-\left(-\frac{5}{3}\right)\right|=\frac{33}{12}+\frac{20}{12}=\frac{53}{12}$.

Answer: Distance $=\frac{53}{12}$ units $=4\frac{5}{12}$ units.

Density: there is always another rational number in between

NCERT section 3.4 Filling the Spaces: Fractions and Rational Numbers

Between any two rational numbers there is another rational number, so there are infinitely many between them. This property is called density.

Proof using the average. Let a and b be rational with $a<b$. Their average $\frac{a+b}{2}$ is rational, because rationals are closed under addition and division by 2. Also:

• $a<b$ gives $a+a<a+b$, so $a<\frac{a+b}{2}$;
• $a<b$ gives $a+b<b+b$, so $\frac{a+b}{2}<b$.

So the average lies strictly between a and b. Repeat to get as many as you like.

Quicker method for several numbers: write both numbers with a common denominator big enough that there are enough whole numbers between the numerators.

Worked example

Find three rational numbers between $\frac{2}{5}$ and $\frac{1}{2}$.

Try it first, then show the solutionSolution

1. Common denominator 10: $\frac{4}{10}$ and $\frac{5}{10}$; no whole number lies between 4 and 5.
2. Multiply top and bottom by 4: $\frac{16}{40}$ and $\frac{20}{40}$.
3. Numerators 17, 18, 19 lie between 16 and 20.

Answer: $\frac{17}{40}$, $\frac{18}{40}=\frac{9}{20}$ and $\frac{19}{40}$ (many other answers are possible).

Irrational numbers and the proof that √2 and √3 are irrational

NCERT section 3.5 Irrational Numbers

The diagonal of a unit square has length $d$ with $d^2=1^2+1^2=2$, so $d=\sqrt{2}$. Baudhāyana met such lengths; Hippasus (c. 400 BCE) first proved $\sqrt{2}$ cannot be a fraction. Numbers on the line that cannot be written as $\frac{p}{q}$ are irrational.

Proof by contradiction (assume the opposite and reach something impossible). For $\sqrt{2}$:
1. Suppose $\sqrt{2}=\frac{p}{q}$ in lowest form (p, q co-prime, $q\ne 0$).
2. Square: $2q^2=p^2$, so $p^2$ is even, so p is even. Write $p=2k$.
3. Then $2q^2=4k^2$, so $q^2=2k^2$; q is even too.
4. p and q share the factor 2, which contradicts 'lowest form'. So $\sqrt{2}$ is irrational.

The same method works for $\sqrt{3}$ and $\sqrt{5}$, using the fact: if $p^2$ is a multiple of 3 (or 5), then p is a multiple of 3 (or 5).

Worked example

Prove that $\sqrt{3}$ is irrational.

Try it first, then show the solutionSolution

1. Assume $\sqrt{3}$ is rational: $\sqrt{3}=\frac{p}{q}$ with p, q integers, $q\ne 0$, and p, q co-prime.
2. Squaring: $3=\frac{p^2}{q^2}$, so $p^2=3q^2$. Hence $p^2$ is a multiple of 3.
3. If p were not a multiple of 3, then $p=3m+1$ or $3m+2$, and $p^2=9m^2+6m+1$ or $9m^2+12m+4$, which leaves remainder 1 on division by 3. So p must be a multiple of 3: $p=3k$.
4. Then $9k^2=3q^2$, so $q^2=3k^2$, and by the same reasoning q is a multiple of 3.
5. Now 3 divides both p and q, contradicting that they are co-prime.
6. So the assumption is false, and $\sqrt{3}$ is irrational.

Answer: $\sqrt{3}$ is irrational (proof by contradiction).

Constructing √n on the number line and the square root spiral

Irrational lengths can be drawn exactly with a ruler and compass.

**$\sqrt{2}$:** mark $OA=1$ on the number line, draw a perpendicular at A, mark $AB=1$ on it. Then $OB=\sqrt{1^2+1^2}=\sqrt{2}$. With centre O and radius OB, draw an arc to cut the number line at P. P represents $\sqrt{2}$.

**$\sqrt{3}$ and beyond:** at B draw a perpendicular to OB and mark 1 unit to get C; then $OC=\sqrt{(\sqrt{2}{)}^2+1}=\sqrt{3}$. Repeating this builds the square root spiral: the hypotenuses are $\sqrt{2},\sqrt{3},\sqrt{4},\sqrt{5},\dots$, and in general the n-th triangle has hypotenuse $\sqrt{n+1}$.

A shortcut for $\sqrt{5}$: take legs 2 and 1, since $2^2+1^2=5$.

Worked example

Explain how to locate $\sqrt{5}$ on the number line, and list the hypotenuses of the first six triangles of the square root spiral. Which of them are rational?

Try it first, then show the solutionSolution

1. Mark $OA=2$ units on the number line and draw a perpendicular at A.
2. Mark $AB=1$ unit on the perpendicular. Then $OB=\sqrt{2^2+1^2}=\sqrt{5}$.
3. With centre O and radius OB, draw an arc cutting the number line at P; P represents $\sqrt{5}$.
4. Spiral hypotenuses: $\sqrt{2},\sqrt{3},\sqrt{4},\sqrt{5},\sqrt{6},\sqrt{7}$.
5. Only $\sqrt{4}=2$ is rational; the rest are irrational.

Answer: Legs 2 and 1 give $\sqrt{5}$; spiral hypotenuses $\sqrt{2}$ to $\sqrt{7}$, of which only $\sqrt{4}=2$ is rational.

Decimal expansions: terminating, repeating and non-repeating

NCERT section 3.6 Real Numbers: Decimals and Cyclic Patterns

Dividing p by q for a rational number gives one of two results:

• it terminates (remainder 0), e.g. $\frac{3}{8}=0.375$; or
• it repeats forever, e.g. $\frac{5}{11}=0.\overline{45}$. It must repeat because only $q-1$ non-zero remainders are possible, so one comes back.

Prediction without dividing: write $\frac{p}{q}$ in lowest form. The decimal terminates exactly when q has no prime factor other than 2 and 5, because then you can make the denominator a power of 10. If $q=2^m{5}^n$, the number of decimal places is the larger of m and n.

Irrational numbers have decimals that never end and never repeat, e.g. $\sqrt{2}=1.41421\dots$ and $\pi =3.14159\dots$. Values like $\frac{22}{7}$ or Āryabhaṭa's 3.1416 are only approximations to π; Mādhava showed π needs an infinite sum.

Rationals and irrationals together make the real numbers R, which fill the number line with no gaps.

Worked example

Without long division, decide whether $\frac{7}{40}$ and $\frac{5}{12}$ terminate. Write the terminating one as a decimal.

Try it first, then show the solutionSolution

1. $40=2^3\times 5$: only 2s and 5s, so $\frac{7}{40}$ terminates.
2. $\frac{7}{40}=\frac{7\times 25}{40\times 25}=\frac{175}{1000}=0.175$ (3 places, the larger power is 3).
3. $12=2^2\times 3$: it has the factor 3, so $\frac{5}{12}$ repeats: $0.41666\cdots =0.41\overline{6}$.

Answer: $\frac{7}{40}=0.175$ terminates; $\frac{5}{12}=0.41\overline{6}$ repeats.

Converting a repeating decimal to p/q

NCERT section 3.6 Real Numbers: Decimals and Cyclic Patterns

Call the decimal x and multiply by powers of 10 so that two equations have identical tails; then subtract.

• Pure repeating (repeats straight after the point): multiply by $10^n$ where n is the number of repeating digits. $x=0.\overline{45}$: $100x=45.\overline{45}$, so $99x=45$ and $x=\frac{5}{11}$.
• General repeating (some digits before the repeat): first multiply by $10^m$ (m non-repeating digits) to move them before the point, then by $10^n$ more for one full repeating block, and subtract the two.

Always reduce to lowest form, and check by dividing back. A curious result: $0.\overline{9}=1$, because $10x-x=9.\overline{9}-0.\overline{9}=9$ gives $x=1$.

Worked example

Express (i) $0.\overline{36}$ and (ii) $1.2\overline{3}$ in the form $\frac{p}{q}$.

Try it first, then show the solutionSolution

1. (i) $x=0.3636\dots$; two digits repeat, so $100x=36.3636\dots$.
2. $100x-x=36$, so $x=\frac{36}{99}=\frac{4}{11}$.
3. (ii) $x=1.2333\dots$; one non-repeating digit: $10x=12.333\dots$.
4. One repeating digit: $100x=123.333\dots$.
5. $100x-10x=111$, so $x=\frac{111}{90}=\frac{37}{30}$. Check: $37÷30=1.2333\dots$

Answer: (i) $\frac{4}{11}$ (ii) $\frac{37}{30}$

Maths in real life: The World of Numbers

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Decimal Expansions and Converting Recurring Decimals to p/q
• Irrational Numbers, Proofs and the Square Root Spiral

The World of Numbers formula sheet

Number sets

$\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R},\mathbb{R}=\mathbb{Q}\cup \text{irrationals}$

Natural numbers, integers, rationals, reals.

Rules for zero (Brahmagupta)

$a+0=a,a-0=a,a\times 0=0,a-a=0$

Zero as a number.

Sign rules

$(-a)\times b=-ab,(-a)\times (-b)=ab,a-(-b)=a+b$

Debt × fortune = debt; debt × debt = fortune.

Rational number

$\frac{p}{q},p,q\in \mathbb{Z},q\ne 0$

Lowest form: p and q co-prime.

Equality

$\frac{a}{b}=\frac{c}{d}⟺ad=bc$

Cross-multiply to test.

Operations

$\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd},\frac{a}{b}÷\frac{c}{d}=\frac{ad}{bc}(c\ne 0)$

Add or subtract only after making denominators equal.

Distributive law

$p(q+r)=pq+pr$

Saves work when a factor is common.

Absolute value and distance

$|x|\ge 0,\text{distance between }a\text{ and }b=|a-b|$

|x| is the distance of x from 0.

A rational between a and b

$a<\frac{a+b}{2}<b$

Proves density.

Diagonal of a unit square

$d^2=1^2+1^2=2\Rightarrow d=\sqrt{2}$

Baudhāyana–Pythagoras theorem.

Square root spiral

${\text{hypotenuse}}_n=\sqrt{n+1}$

Each new triangle has legs $\sqrt{n}$ and 1.

Terminating test

$\frac{p}{q}\text{ (lowest form) terminates}⟺q=2^m{5}^n$

Decimal places = larger of m and n.

Pure repeating decimal

$0.\overline{a_1{a}_2\dots a_n}=\frac{a_1{a}_2\dots a_n}{\underset{n}{\underbrace{99\dots 9}}}$

e.g. $0.\overline{45}=\frac{45}{99}$.

A useful identity

$0.\overline{9}=1$

So every terminating decimal also has a form ending in repeating 9s.

Mādhava's series

$\pi =4\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots \right)$

An exact formula for π needs infinitely many terms.

How the school exam asks The World of Numbers

The question patterns to expect, and the method that works for each.

Tricks to remember The World of Numbers

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Russian dolls of numbers

  N sits inside Z, Z inside Q, Q inside R, like dolls nested one inside another. Irrationals are the other half of R, outside Q.

  Use it when: Answering 'every integer is a ...' and 'every rational is a ...' MCQs.

• Only 2 and 5 may stay

  Ten is 2 × 5, so only 2s and 5s in the denominator can be turned into a power of ten.

  Use it when: Deciding quickly whether a decimal terminates.

• Nines under the repeat, zeros under the rest

  For $0.a\overline{bc}$: the denominator has one 9 for each repeating digit followed by one 0 for each non-repeating digit (here 990).

  Use it when: Checking your algebraic conversion of a repeating decimal.

• Assume, square, share, clash

  Irrationality proof in four words: assume $\frac{p}{q}$ in lowest form, square, show p and q share a factor, clash with 'lowest form'.

  Use it when: Writing the proof in the right order in a 5-mark answer.

• Debt of a debt is wealth

  Taking away a debt makes you richer: minus a minus is a plus.

  Use it when: Sign rules for integers and rationals.

The World of Numbers: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 2 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in The World of Numbers

The slips examiners see most often, and what to write instead.

• ✗Mistake: Testing for terminating before reducing the fraction

  The factor 3 in the denominator looks like a 'repeating' sign, but it may cancel.

  Wrong working

  $\frac{6}{75}$: $75=3\times 5^2$, so it repeats.

  Right working

  $\frac{6}{75}=\frac{2}{25}$; $25=5^2$, so it terminates: $0.08$.

  ✓Do this: Always reduce to lowest form first.

  Usually costs: 1 mark per part.

• ✗Mistake: Leaving out the co-prime assumption in the irrationality proof

  Students write $\sqrt{2}=\frac{p}{q}$ without saying p and q have no common factor, so the contradiction at the end has nothing to clash with.

  Wrong working

  Let $\sqrt{2}=\frac{p}{q}$. ... so p and q are even. Hence proved.

  Right working

  Let $\sqrt{2}=\frac{p}{q}$, where p, q are co-prime integers and $q\ne 0$. ... p and q are both even, which contradicts that they are co-prime.

  ✓Do this: Write the assumption in full in the first line and name the contradiction in the last.

  Usually costs: 1 to 2 marks of a 5-mark proof.

• ✗Mistake: Converting a general repeating decimal as if it were pure

  The non-repeating digit is ignored.

  Wrong working

  $0.1\overline{6}=\frac{16}{99}$

  Right working

  $10x=1.\overline{6}$, $100x=16.\overline{6}$, so $90x=15$ and $x=\frac{1}{6}$

  ✓Do this: Move the non-repeating digits before the point first, then shift by one full repeating block.

  Usually costs: The whole 2- or 3-mark answer.

• ✗Mistake: Adding numerators and denominators

  Treating fractions like separate whole numbers.

  Wrong working

  $\frac{1}{3}+\frac{1}{4}=\frac{2}{7}$

  Right working

  $\frac{4}{12}+\frac{3}{12}=\frac{7}{12}$

  ✓Do this: Make the denominators equal first; only numerators are added.

  Usually costs: All the marks for that step.

• ✗Mistake: Treating 22/7 as the exact value of π

  It is used in calculations, so it feels exact.

  Wrong working

  π $=\frac{22}{7}$, so π is rational.

  Right working

  $\frac{22}{7}$ is rational and only approximately equal to π; π is irrational.

  ✓Do this: Say 'π is approximately $\frac{22}{7}$' and classify π as irrational.

  Usually costs: 1-mark MCQ or assertion-reason.

• ✗Mistake: Marking an improper fraction from 0 instead of from the right integer

  The unit is split into the wrong number of parts or the count starts at 0.

  Wrong working

  To mark $\frac{9}{4}$, split 0 to 1 into 9 parts.

  Right working

  $\frac{9}{4}=2\frac{1}{4}$: split 2 to 3 into 4 parts and take 1.

  ✓Do this: Convert to a mixed number first.

  Usually costs: 1 mark for the number line.

NCERT exercise guide: The World of Numbers

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Practice questions on The World of Numbers

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   Every rational number is
   (a) a natural number
   (b) an integer
   (c) a real number
   (d) a whole number

   Hint

   Think of the nested sets.

   Answer

   (c) a real number

2. Q2 · 1 mark · MCQExam level

   The value of $(-12)-(-20)$ is
   (a) $-32$
   (b) $-8$
   (c) 8
   (d) 32

   Hint

   Subtracting a negative is adding.

   Answer

   (c) 8

3. Q3 · 1 mark · Assertion-reasonExam level

   Assertion (A): $\sqrt{2}$ can be located exactly on the number line.
   Reason (R): $\sqrt{2}$ is the length of the hypotenuse of a right triangle whose legs are 1 unit each.
   (a) Both A and R are true and R is the correct explanation of A.
   (b) Both A and R are true but R is not the correct explanation of A.
   (c) A is true but R is false.
   (d) A is false but R is true.

   Hint

   Think of the construction.

   Answer

   (a). The hypotenuse is drawn exactly and transferred to the line by an arc.

4. Q4 · 2 marks · Very short answerExam level

   Classify each as rational or irrational: $\sqrt{49}$, $0.101001000\dots$, $3.\overline{27}$, π, $-\sqrt{16}$.

   Hint

   Simplify roots first.

   Answer

   Rational: $\sqrt{49}=7$, $3.\overline{27}$, $-\sqrt{16}=-4$. Irrational: $0.101001000\dots$ and π.

5. Q5 · 3 marks · Short answerHarder exam level

   Express (i) $0.\overline{48}$ and (ii) $0.1\overline{6}$ in the form $\frac{p}{q}$.

   Hint

   (ii) is not pure repeating.

   Answer

   (i) $\frac{48}{99}=\frac{16}{33}$ (ii) $90x=15$, so $\frac{1}{6}$.

6. Q6 · 2 marks · Very short answerHarder exam level

   Find three rational numbers between 2.718 and 2.719.

   Hint

   Add another decimal place.

   Answer

   For example 2.7181, 2.7185, 2.7189.

7. Q7 · 3 marks · Short answerHarder exam level

   Without long division, state which of these terminate, and give the decimal for those that do: $\frac{11}{80}$, $\frac{9}{42}$, $\frac{17}{625}$, $\frac{6}{75}$.

   Hint

   Reduce each fraction first.

   Answer

   $\frac{11}{80}=0.1375$ (terminates; $80=2^4\times 5$). $\frac{9}{42}=\frac{3}{14}$ repeats ($14=2\times 7$). $\frac{17}{625}=0.0272$ (terminates). $\frac{6}{75}=\frac{2}{25}=0.08$ (terminates).

8. Q8 · 3 marks · Short answerHarder exam level

   Given that $\sqrt{3}$ is irrational, prove that $5-\sqrt{3}$ is irrational.

   Hint

   Assume it equals a rational r and isolate $\sqrt{3}$.

   Answer

   If $5-\sqrt{3}=r$ (rational), then $\sqrt{3}=5-r$, which is rational by closure. This contradicts the given fact, so $5-\sqrt{3}$ is irrational.

9. Q9 · 3 marks · Short answerHarder exam level

   Mark $-\frac{7}{4}$ and $\frac{8}{5}$ on a number line and find the distance between them.

   Hint

   Distance $=|a-b|$.

   Answer

   $-\frac{7}{4}=-1\frac{3}{4}$ (3 of 4 parts left of $-1$); $\frac{8}{5}=1\frac{3}{5}$ (3 of 5 parts right of 1). Distance $=\frac{8}{5}+\frac{7}{4}=\frac{67}{20}$ units.

10. Q10 · 5 marks · Long answerHarder exam level

    Prove that $\sqrt{2}$ is irrational, and use the square root spiral to name the first triangle after the third whose hypotenuse is rational.

    Hint

    Hypotenuses are $\sqrt{n+1}$.

    Answer

    Proof: assume $\sqrt{2}=\frac{p}{q}$ (co-prime); $p^2=2q^2$ makes p even, $p=2k$; then $q^2=2k^2$ makes q even; contradiction. Spiral hypotenuses are $\sqrt{2},\sqrt{3},\sqrt{4},\dots$; the third is $\sqrt{4}=2$, and the next rational one is $\sqrt{9}=3$, in the 8th triangle.

11. Q11 · 3 marks · Short answerHOTS / competency

    Find a rational number between $\frac{1}{7}$ and $\frac{1}{6}$, and explain why there are infinitely many such numbers.

    Hint

    Use the average, then repeat.

    Answer

    $\frac{1}{2}\left(\frac{1}{7}+\frac{1}{6}\right)=\frac{13}{84}$. Between $\frac{1}{7}$ and $\frac{13}{84}$ there is another average, and so on forever, so there are infinitely many.

12. Q12 · 3 marks · Short answerHOTS / competency

    A rational number in lowest form has denominator $2^2\times 5^3$. How many decimal places does it have? Explain, and test your rule on $\frac{3}{500}$.

    Hint

    Multiply to make a power of 10.

    Answer

    Multiply top and bottom by 2 to get $2^3\times 5^3=1000$, so 3 decimal places (the larger power). $\frac{3}{500}=\frac{6}{1000}=0.006$, 3 places.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.Define a rational number.Show answer

   A number that can be written as $\frac{p}{q}$ with p, q integers and $q\ne 0$.

2. 2.Who made zero a number with rules, and in which work?Show answer

   Brahmagupta, in the Brāhmasphuṭasiddhānta (628 CE).

3. 3.What does 'rational numbers are dense' mean?Show answer

   Between any two rational numbers there is another rational number, so there are infinitely many between them.

4. 4.Which rational number lies exactly halfway between a and b?Show answer

   $\frac{a+b}{2}$.

5. 5.What kind of decimal does an irrational number have?Show answer

   Non-terminating and non-repeating.

6. 6.When does $\frac{p}{q}$ (lowest form) have a terminating decimal?Show answer

   When q has no prime factors other than 2 and 5.

7. 7.What method proves that $\sqrt{2}$ is irrational?Show answer

   Proof by contradiction.

8. 8.What are the hypotenuses in the square root spiral?Show answer

   $\sqrt{2},\sqrt{3},\sqrt{4},\sqrt{5},\dots$; the n-th is $\sqrt{n+1}$.

9. 9.Is $\frac{22}{7}$ equal to π?Show answer

   No. $\frac{22}{7}$ is rational and only an approximation; π is irrational.

10. 10.What is $0.\overline{9}$ equal to?Show answer

    Exactly 1.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you state Brahmagupta's three rules for zero?
• ☐Can you show $\frac{-21}{28}=\frac{3}{-4}$ by cross-multiplying?
• ☐Can you find five rational numbers between $\frac{1}{3}$ and $\frac{1}{2}$?
• ☐Can you prove that the average of two rationals lies between them?
• ☐Can you mark $-\frac{7}{4}$ on a number line?
• ☐Can you write the proof that $\sqrt{2}$ is irrational without looking?
• ☐Can you construct $\sqrt{3}$ on the number line?
• ☐Can you tell, without dividing, whether $\frac{13}{80}$ terminates and after how many places?
• ☐Can you convert $0.\overline{27}$ and $0.1\overline{8}$ to fractions?
• ☐Can you explain why $0.\overline{9}=1$?

The World of Numbers: frequently asked questions