Cambridge Lower Secondary · Mathematics 0862 · Stage 8 · Number

Types of Number and Index Laws

Classify numbers as natural numbers, integers or rational numbers, and use positive and zero indices with the index laws for multiplication and division.

British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course

In the curriculum

Understand how the natural numbers, the integers and the rational numbers fit inside each other (the hierarchy of numbers) (8Ni.04)
Use positive and zero indices, and the index laws for multiplying and dividing powers of the same number (8Ni.05)

Mathematicians sort numbers into families. Each family sits inside a bigger one, like a box inside a box. In the first part of this page you will learn the three families you need at Stage 8 and how to decide which ones a number belongs to. In the second part you will learn to multiply and divide powers quickly using two index laws, and why any number to the power 0 equals 1.

Key words

Word / phrase	Meaning / use
natural numbers	الأعداد الطبيعية — the counting numbers 1, 2, 3, 4, …
integers	الأعداد الصحيحة — whole numbers: positive, negative and zero … −2, −1, 0, 1, 2 …
rational numbers	الأعداد النسبية — numbers that can be written as a fraction a/b of two integers (b ≠ 0)
hierarchy	تسلسل / ترتيب هرمي — an order of groups, each one inside the next
terminating decimal	عدد عشري منتهٍ — stops, e.g. 0.75
recurring decimal	عدد عشري دوري — repeats for ever, e.g. 0.333…
base	الأساس — the number being multiplied: in 5³ the base is 5
index / power / exponent	الأس — how many times the base is multiplied: in 5³ the index is 3
index form	صورة الأسس — 7 × 7 × 7 × 7 written as 7⁴

1. Natural numbers, integers and rational numbers

Natural numbers are the counting numbers: 1, 2, 3, 4, 5, … (In this course natural numbers start at 1. Some books also count 0 as a natural number; either way, 0 is certainly an integer and a rational number.)
Integers are all the whole numbers: the natural numbers, zero and the negative whole numbers: …, −3, −2, −1, 0, 1, 2, 3, …
Rational numbers are all numbers that can be written as a fraction a/b, where a and b are integers and b is not 0. This includes every integer (7 = 7/1), every fraction, every terminating decimal (0.75 = 3/4) and every recurring decimal (0.333… = 1/3).

The hierarchy: every natural number is an integer, and every integer is a rational number — but not the other way round.

So the families form a hierarchy: natural numbers ⊂ integers ⊂ rational numbers. To decide where a number belongs, ask three questions in order: Is it a counting number? Is it a whole number (positive, negative or zero)? Can it be written as a fraction of two integers?

Number	Natural?	Integer?	Rational?	Written as a/b
12	✓	✓	✓	12/1
0	✗ (in this course)	✓	✓	0/1
−5	✗	✓	✓	−5/1
3/8	✗	✗	✓	3/8
0.6	✗	✗	✓	6/10 = 3/5
−2.25	✗	✗	✓	−9/4
0.333… (recurring)	✗	✗	✓	1/3
√16	✓	✓	✓	√16 = 4 = 4/1
√2 = 1.41421…	✗	✗	✗	cannot be written as a fraction

Look past the appearance. √16 looks strange but equals 4, a natural number. 20/4 looks like a fraction but equals 5, a natural number. −18/6 = −3 is an integer. Always simplify first, then classify.

Why do we need bigger families?

Each family was needed because some calculations had no answer in the smaller family:

Calculation	Always stays in the family?	Example
natural + natural	Yes — always natural	4 + 9 = 13
natural × natural	Yes — always natural	4 × 9 = 36
natural − natural	No — can be zero or negative, so we need integers	4 − 9 = −5
integer ÷ integer (not ÷ 0)	No — can be a fraction, so we need rational numbers	4 ÷ 9 = 4/9

Always, sometimes or never true? (This is classifying and convincing.) "An integer is a natural number" — sometimes (true for 5, false for −5). "A natural number is a rational number" — always. "A negative number is a natural number" — never. "A rational number is an integer" — sometimes (true for 6/3, false for 1/2). To show "sometimes", give one example where it is true and one where it is false.

عائلات الأعداد

الطبيعية: أعداد العد ١، ٢، ٣، …
الصحيحة: الطبيعية + الصفر + السالبة.
النسبية: أي عدد ممكن يتكتب كسر (عدد صحيح ÷ عدد صحيح، والمقام مش صفر) — زي ٠٫٧٥ و ٠٫٣٣٣… و −٥.
كل طبيعي صحيح، وكل صحيح نسبي — لكن العكس مش دايمًا صح.

2. Powers and index form

A power is a short way to write repeated multiplication. In 2⁵ the base is 2 and the index is 5: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Also, any number to the power 1 is itself: 9¹ = 9.

Repeated multiplication	Index form	Value
7 × 7 × 7 × 7	7⁴	2401
3 × 3 × 3 × 3 × 3	3⁵	243
10 × 10 × 10 × 10 × 10 × 10	10⁶	1 000 000
a × a × a	a³	—

Careful: 2⁵ is not 2 × 5. And 2¹⁰ = 1024 while 10² = 100, so swapping the base and the index changes the value.

3. Index law for multiplication

When you multiply powers of the same base, count how many times the base is multiplied altogether:

2³ × 2⁴: three 2s and four 2s make seven 2s, so 2³ × 2⁴ = 2⁷.

Multiplication law

a^m × aⁿ = a^{m + n} — same base: add the indices. Examples: 5⁴ × 5³ = 5⁷; 10² × 10⁵ = 10⁷; x⁴ × x³ = x⁷; 6 × 6⁸ = 6¹ × 6⁸ = 6⁹.

4. Index law for division

When you divide powers of the same base, the factors on the bottom cancel with factors on the top:

Two 5s on the bottom cancel two 5s on the top, leaving four 5s.

Division law

a^m ÷ aⁿ = a^{m − n} — same base: subtract the indices. Examples: 5⁶ ÷ 5² = 5⁴; 9⁸ ÷ 9³ = 9⁵; y⁹ ÷ y⁴ = y⁵. At Stage 8 the first index is always at least as big as the second, so the answer never has a negative index.

5. The zero index

What is 3⁴ ÷ 3⁴? Any number divided by itself is 1. But the division law gives 3^{4 − 4} = 3⁰. So 3⁰ = 1. The pattern below shows the same thing: each time the index goes down by 1, you divide by 2.

Continue the pattern: 2¹ = 2, and 2 ÷ 2 = 1, so 2⁰ = 1.

Zero index

a⁰ = 1 for any number a that is not 0. Examples: 7⁰ = 1, 100⁰ = 1, (−4)⁰ = 1. It is not 0.

6. Using the laws together

Several steps: (3⁵ × 3⁴) ÷ 3⁷ = 3⁹ ÷ 3⁷ = 3² = 9.
A fraction line: 7³ × 7⁵ on top and 7⁶ on the bottom → top = 7⁸, so 7⁸ ÷ 7⁶ = 7² = 49.
Numbers in front (coefficients): multiply or divide the numbers, then use the index law for the letters. 2a³ × 5a² = (2 × 5) × a^{3 + 2} = 10a⁵. 12b⁷ ÷ 4b³ = (12 ÷ 4) × b^{7 − 3} = 3b⁴.
Missing index: 4^☐ × 4³ = 4⁹. The indices must add to 9, so ☐ = 6.
Different bases do not combine: 2³ × 5² cannot be written as one power. Work it out: 8 × 25 = 200.

Worked examples

Worked example 1 — Complete the table with ticks to show which sets each number belongs to: −5, 3/8, 12, 0.6. [2 marks]

−5: a whole number but negative → integer and rational (−5 = −5/1), not natural.
3/8: not whole → rational only.
12: a counting number → natural, integer and rational.
0.6 = 3/5 → rational only.
Mark scheme style: 2 marks for all correct; 1 mark for three numbers correct.

Common mistake: forgetting that every integer is also rational, so leaving the rational box empty for −5 and 12.

Worked example 2 — Write down (a) a number that is an integer but not a natural number (b) a rational number that is not an integer. [2 marks]

(a) Any negative whole number or 0: for example −4.
(b) Any fraction or decimal that is not whole: for example 2/3 or 0.5.

Common mistake: giving 8/4 for (b) — it equals 2, which is an integer.

Worked example 3 — Simplify, giving each answer in index form: (a) 6⁴ × 6⁵ (b) 9⁸ ÷ 9³. Then (c) work out (2⁵ × 2³) ÷ 2⁶. [3 marks]

(a) Same base, multiply → add indices: 6^{4 + 5} = 6⁹.
(b) Same base, divide → subtract indices: 9^{8 − 3} = 9⁵.
(c) Top: 2^{5 + 3} = 2⁸. Then 2^{8 − 6} = 2² = 4.

Common mistake: multiplying the indices (6²⁰) or multiplying the bases (36⁹).

Worked example 4 — Work out 7⁰ + 4⁰ × 5. [1 mark]

7⁰ = 1 and 4⁰ = 1.
Order of operations: 1 + 1 × 5 = 1 + 5 = 6.

Common mistake: thinking a⁰ = 0 and getting 0, or adding first and getting 10.

Worked example 5 — Find the value of n. (a) 3ⁿ × 3⁴ = 3¹⁰ (b) 5⁸ ÷ 5ⁿ = 1 [2 marks]

(a) n + 4 = 10, so n = 6.
(b) 1 = 5⁰, so 8 − n = 0 and n = 8. (A number divided by itself is 1.)

Worked example 6 — Hassan says, "2³ × 4² = 8⁵." Explain why Hassan is wrong, and find the correct value. [2 marks]

The index laws only work when the bases are the same. 2 and 4 are different bases, and the bases are never multiplied together.
Correct value: 2³ × 4² = 8 × 16 = 128.
(Extension: 4² = 4 × 4 = 2² × 2² = 2⁴, so 2³ × 4² = 2³ × 2⁴ = 2⁷ = 128 ✓. But 8⁵ = 32 768, which is much bigger.)

Common mistakes

Thinking 0 or a negative number is a natural number, or forgetting that integers and natural numbers are also rational.
Classifying by appearance: √16 = 4 and 20/4 = 5 are natural numbers.
Working out 2⁵ as 2 × 5 = 10 instead of 32.
Multiplying the indices when you multiply powers: 3⁴ × 3² = 3⁶, not 3⁸.
Multiplying the bases: 3⁴ × 3² = 3⁶, not 9⁶. The base stays the same.
Using the index laws with different bases: 2³ × 5² is 8 × 25 = 200.
Writing a⁰ = 0. Any non-zero number to the power 0 is 1.

How Checkpoint asks this

"Tick the boxes to show which numbers are integers / rational numbers" — a table to complete.
"Write a number that is rational but not an integer" — give a simple example such as 1/2.
"Is this statement always, sometimes or never true?" — give examples as evidence.
"Write as a single power" or "Give your answer in index form" — leave it as, e.g., 5⁷; do not work it out unless asked.
"Find the value of n" in an index equation — write an equation for the indices, e.g. n + 4 = 10.
"Ring the expressions equal to 1" — look for zero indices and equal powers divided, such as 9⁰ or 4³ ÷ 4³.
"Explain the mistake" — name the wrong rule (multiplied the bases / multiplied the indices) and give the correct answer.

Practice

Remember

Natural numbers ⊂ integers ⊂ rational numbers. Every natural number is an integer; every integer is rational (n = n/1).
A rational number can be written as a fraction of two integers: terminating and recurring decimals are rational.
Same base, multiply → add the indices: a^m × aⁿ = a^{m + n}.
Same base, divide → subtract the indices: a^m ÷ aⁿ = a^{m − n}.
a⁰ = 1 (a ≠ 0). Different bases do not combine — work out each power.

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