Jupiter2U School
Chapter 06

6. Numbers & Arithmetic

Middle School

Before any equation, there are numbers and the rules that govern them. This chapter builds the number systems, the operations that act on them, and the order and structure that make arithmetic reliable.

At a glance
Core ideaNumber systems nest, each extending the last: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ.
Key termPrime factorisation: every integer > 1 as a unique product of primes.
You can…Find HCF and LCM and apply the field axioms and order of operations.
Watch outOrder of operations is a convention — 2 + 3 × 4 = 14, not 20.
01 · Theory

The number systems and the laws of operation

Mathematics is built on nested sets of numbers, each extending the last to make more operations possible:

  • Natural numbers ℕ = {1, 2, 3, …} — counting.
  • Integers ℤ = {…, −2, −1, 0, 1, 2, …} — add subtraction and zero.
  • Rationals ℚ — every number expressible as a ratio a/b with b ≠ 0. Their decimals terminate or repeat.
  • Irrationals — numbers like √2 and π whose decimals never terminate or repeat.
  • Reals ℝ — rationals and irrationals together, filling the whole number line.
ℝ Reals — e.g. π, √2 ℚ Rationals — e.g. 1/2, 0.75 ℤ Integers — e.g. −3, 0 ℕ 1, 2, 3 …
Each number system sits fully inside the next: naturals are integers, integers are rationals, rationals are real numbers.

Every real number factors and combines under the field axioms. For all reals a, b, c:

Commutative:  a + b = b + a    a·b = b·a
Associative:  (a + b) + c = a + (b + c)
Distributive:  a·(b + c) = a·b + a·c
Identities:  a + 0 = a    a·1 = a
Inverses:  a + (−a) = 0    a·(1/a) = 1  (a ≠ 0)

A whole number greater than 1 is prime if its only positive divisors are 1 and itself. The Fundamental Theorem of Arithmetic states that every integer > 1 has a unique prime factorisation (up to order) — the atoms from which all integers are built.

02 · Explanation

Why the rules are the way they are

Think of each number system as a toolbox that grew because a job needed doing. Counting sheep needs only naturals. But "I owe three sheep" needs negatives, so integers appear. "Share two loaves among three people" needs fractions, so rationals appear. And the diagonal of a unit square — length √2 — cannot be any fraction at all, forcing the irrationals into existence.

The field axioms look abstract, but they are just the promises that let you rearrange arithmetic without fear. The distributive law is the single most useful: it is why 7 × 12 = 7 × 10 + 7 × 2 = 84, and later it is exactly the rule that lets you expand and factor algebra. Order of operations (often remembered as brackets, exponents, ×/÷, then +/−) is not a law of nature — it is a shared convention so that 2 + 3 × 4 means 14 to everyone, not 20.

π ≈ 3.14159
Circumference ÷ diameter
e ≈ 2.71828
Base of natural growth
√2 ≈ 1.41421
Diagonal of a unit square · irrational
φ ≈ 1.61803
The golden ratio
03 · Practical

Worked example: prime factorisation, HCF and LCM

Problem

Find the highest common factor (HCF) and lowest common multiple (LCM) of 84 and 120.

  1. Factor each into primes. 84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5.
  2. HCF = product of the lowest power of each shared prime: shared primes are 2 and 3, so HCF = 2² × 3 = 12.
  3. LCM = product of the highest power of every prime that appears: 2³ × 3 × 5 × 7 = 840.
  4. Check with the identity HCF × LCM = product: 12 × 840 = 10080 = 84 × 120. ✓

Answer: HCF = 12, LCM = 840.

04 · Q&A

Test your understanding

Is 0 a natural number, and is it even?

Conventions differ: many define ℕ to start at 1, others include 0. What is not in dispute is that 0 is an integer, and it is even — an even number is any integer of the form 2k, and 0 = 2 × 0. Zero is also the additive identity and neither positive nor negative.

Prove that √2 is irrational.

Suppose it were rational: √2 = a/b in lowest terms (no common factor). Squaring gives 2b² = a², so a² is even, hence a is even, say a = 2k. Then 2b² = 4k², so b² = 2k², making b even too. But then a and b share the factor 2, contradicting "lowest terms." The assumption fails, so √2 is irrational. This is a classic proof by contradiction.

Why is dividing by zero undefined?

a ÷ b asks "what number times b gives a?" If b = 0 and a ≠ 0, no number times 0 gives a non-zero result, so there is no answer. If a = 0 too, every number works, so the answer is not unique. Either way there is no single valid value, so division by zero is left undefined.

Convert the repeating decimal 0.4̄5̄ (0.454545…) to a fraction.

Let x = 0.454545…. Two digits repeat, so multiply by 100: 100x = 45.4545…. Subtract: 100x − x = 45, so 99x = 45 and x = 45/99 = 5/11. Every repeating decimal is rational, which is exactly why irrationals must have non-repeating decimals.

Concept mind map

How the ideas connect

Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.

IntegersRationalsPrime numbersFactorisationHCF and LCMOperation lawsNumber Systems and Laws
Infographic

The key facts, visualised

Prime
Whole number with exactly two factors
HCF
Highest common factor of two numbers
LCM
Lowest common multiple of two numbers
a+b=b+a
Commutative law of addition
Solved examples

Worked problems, step by step

Follow each solution line by line, then try to reproduce it on paper before moving on.

Example 1Find the HCF and LCM of 12 and 18.

  1. 12 = 2^2 x 3 and 18 = 2 x 3^2
  2. HCF takes lowest powers: 2 x 3 = 6
  3. LCM takes highest powers: 2^2 x 3^2 = 36

Example 2Evaluate -4 + 9 - (-3).

  1. Subtracting -3 is the same as adding 3
  2. So -4 + 9 + 3
  3. Combine: 8
Practice problem set

Now you try

Work each one out first, then tap to reveal the worked answer.

1Write 84 as a product of primes.
84 = 2^2 x 3 x 7.
2Find the HCF of 24 and 36.
Common factors give HCF = 12.
3Find the LCM of 4 and 6.
The lowest common multiple is 12.
4Is 51 prime?
No, because 51 = 3 x 17, so it has factors other than 1 and itself.
5Compute -7 x (-3).
A negative times a negative is positive, giving 21.
6Use the distributive law to expand 5(3 + 4).
5x3 + 5x4 = 15 + 20 = 35.