IB Maths AA HL Topic 1 — Number & Algebra Paper 1 & 2 ~9 min read

Proof by Deduction

A proof is a sequence of logical steps that shows a statement is true for all the numbers it claims to be true for — not just the few you’ve tested. Proof by deduction is the most basic style: you start from things you already know (algebra rules, definitions of even/odd, etc.) and derive the result step by step. The biggest jump from earlier maths is that “checking three examples” no longer counts. You need an argument that covers every case, all at once.

📘 What you need to know

Proof = a chain of logical steps showing a statement holds for every number in the stated set. Testing examples does not prove anything.
Be fluent with the set symbols: ℕ (naturals, including 0), ℤ (integers), ℚ (rationals), ℝ (reals). Each is a subset of the next.
Identity proofs (LHS = RHS): work from one side, manipulate algebraically, end at the other side. Don’t move things across the equals sign.
Translate words into algebra: any integer = n, even = 2n, odd = 2n + 1, multiple of k = kn, etc.
To prove an expression is even, factor out a 2 and check the bracket is an integer. To prove odd, write it as 2(integer) + 1.
End every proof with a concluding line — “as required”, or stating which property has been shown. The IB awards a method mark for the structure as much as the algebra.

What is a proof, and what isn’t?

A proof is an argument — a chain of statements, each following logically from the one before, that establishes a claim is always true. Verifying a result with a handful of examples isn’t enough, because it leaves open the possibility that the next case will fail.

Example: “n² + n + 41 is prime for all n” looks true for n = 0, 1, 2, …, 39 — but it fails at n = 40, where the value is 41². Forty examples don’t constitute a proof; one counter-example destroys one.

In IB exams, the verbs “prove that…”, “show that…”, and “verify that…” all mean different things. “Prove” requires a full argument; “show” usually wants the working laid out clearly; “verify” might allow checking with substituted values. Read the question carefully.

The notation you need to recognise

Most proof questions specify the set the result holds in (e.g. “for all x ∈ ℝ“). You need to read these symbols on sight.

Symbol

Set

Examples / what’s included

ℕ

Natural numbers

{0, 1, 2, 3, …} — non-negative integers, includes zero

ℤ

Integers

{…, −3, −2, −1, 0, 1, 2, 3, …} — adds the negatives

ℤ⁺

Positive integers

{1, 2, 3, …} — does not include 0

ℚ

Rationals

numbers of the form a/b, where a, b ∈ ℤ and b ≠ 0

ℝ

Reals

all rationals + irrationals like √2, π, e

Each set sits inside the next — ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ

🧠 The identity symbol ≡

The symbol ≡ means “identically equal to” — it holds for every value of the variable, not just specific solutions. Most IB questions just use = instead, but if you see ≡, treat it as “this is an identity, must be true for all x“.

Proving an identity (LHS = RHS)

The most common proof type starts with one side of an equation and uses algebra to transform it into the other side. The trick is to pick one side (usually the more complicated one) and only manipulate that side — never balance an equation by doing the same thing to both sides.

🧭 Recipe — proving LHS = RHS

Start from one side (usually whichever looks more complex) and write “LHS = …” at the top.
Apply standard algebra: expand brackets, collect like terms, factorise.
Aim each step at the structure of the other side — if the RHS has a factor, get a factor; if it has a fraction, combine to get one.
When you’ve matched the other side, write “= RHS, as required” or “∴ LHS ≡ RHS”.

Don’t “work from both sides at once” — that’s circular. The cleanest layout is a single column starting “LHS =” and ending at the RHS. Keep the working visible: examiners want to see the manipulation, not just the answer.

Translating integers into algebra

To prove a result about whole numbers (e.g. “the sum of any two odd integers is even”), you need to write down what an “any” integer looks like in algebraic form. Different types of integer have different conventional representations.

Integer type

Algebraic form

Notes

Any integer

where n ∈ ℤ

Two consecutive integers

n, n + 1

“one after the other”

Two unrelated integers

n, m

use a different letter — no link assumed

An even integer

divisible by 2

Two consecutive even integers

2n, 2n + 2

step of 2

An odd integer

2n + 1

2n − 1 also works

Two consecutive odd integers

2n + 1, 2n + 3

step of 2 between them

A multiple of k

e.g. multiple of 5 → 5n

A square number

n²

A rational number

where a, b ∈ ℤ and b ≠ 0

Proving an expression is odd, even, or a multiple

To prove EVEN

Show the expression equals 2 × (integer). Usually that means factoring out a 2 — and the part inside the bracket must clearly be an integer.

Show: expression = 2(integer)

To prove ODD

Show the expression equals 2 × (integer) + 1.

Show: expression = 2(integer) + 1

To prove a multiple of k

Show the expression equals k × (integer).

Show: expression = k(integer)

🤔 Why does the bracket need to be an integer?

Because the definition of “even” is “exactly divisible by 2” — and that’s only the case if you’re multiplying 2 by a whole number. If you wrote 2(n + 1/3), the result wouldn’t be an even integer at all (because 1/3 isn’t an integer). The “2 ×” alone doesn’t prove evenness; the contents of the bracket matter.

Worked examples

WE 1

Prove an identity by expanding

Prove that (x + 3)² − 6x = x² + 9 for all x ∈ ℝ.

Start with LHS LHS = (x + 3)² − 6x Expand the bracket = x² + 6x + 9 − 6x Collect like terms = x² + 9 = RHS ∴ LHS ≡ RHS, as required

WE 2

Prove a divisibility result

Prove that the sum of any three consecutive integers is divisible by 3.

Step 1: Represent three consecutive integers let them be n, n + 1, n + 2 (where n ∈ ℤ) Step 2: Find their sum sum = n + (n + 1) + (n + 2) = 3n + 3 Step 3: Factor out 3 = 3(n + 1) Step 4: Conclude since n + 1 is an integer, 3(n + 1) is divisible by 3 ✓

WE 3

Prove an expression is odd

Prove that the difference between the squares of two consecutive integers is always odd.

Step 1: Set up let two consecutive integers be n and n + 1 Step 2: Compute the difference of squares (n + 1)² − n² = n² + 2n + 1 − n² = 2n + 1 Step 3: Conclude 2n + 1 is by definition odd, as required ✓ notice we ended in the form 2(integer) + 1 — that’s the structure for “odd”

WE 4

Sum of an odd and an even integer

Prove that the sum of any odd integer and any even integer is always odd.

Step 1: Use different letters since they’re unrelated odd integer = 2m + 1 even integer = 2n Step 2: Add them sum = (2m + 1) + 2n = 2m + 2n + 1 = 2(m + n) + 1 Step 3: Conclude since m + n ∈ ℤ, the sum has the form 2(integer) + 1, so it is odd ✓ two unrelated integers → use two different letters

WE 5

Product of an odd and an even integer

Prove that the product of any odd integer and any even integer is always even.

Step 1: Set up using two different letters odd integer = 2m + 1 even integer = 2n Step 2: Compute the product product = (2m + 1)(2n) = 2n(2m + 1) = 2 × n(2m + 1) Step 3: Conclude since n(2m + 1) is an integer, the product is 2 × integer, so it is even ✓ factoring the 2 out is what lets you finish the proof — keep an eye out for that

💡 Top tips

Translate before you compute. Write down what each integer in the question looks like algebraically before you start manipulating.
For unrelated integers, use different letters (m, n) — don’t reuse one letter and assume they’re equal.
Aim for the right shape. “Even” needs 2(integer); “odd” needs 2(integer) + 1; “multiple of k” needs k(integer). End the algebra in that form.
Always say “since [bracket] is an integer” at the end. This is the key sentence the IB looks for.
For identity proofs, work from one side only. Don’t shuffle terms across the equals sign — that proves nothing.
Keep brackets visible. (n + 1)² means n² + 2n + 1, not n² + 1.
End with a concluding line — “as required”, “QED”, or “since …, the result is even/odd/divisible by …”. Don’t leave the proof hanging.

⚠ Common mistakes

Treating examples as proof. “It works for n = 1, 2, 3″ is not enough. The argument has to cover every case.
Using the same letter for unrelated integers. Two odd numbers should be 2m + 1 and 2n + 1, not 2n + 1 and 2n + 1 (which means they’re equal).
Forgetting the “+1” in odd. An odd integer is 2n + 1, not just 2n.
Putting a non-integer inside the bracket. 2(n + ½) is not even — the bracket must be a whole number.
Working from both sides at once in identity proofs. This is circular reasoning. Pick a side and stick with it.
Skipping the conclusion. Reaching “= 2(n + 1)” without saying “therefore even” loses the final mark.
Confusing “consecutive even” with “any two even”. Consecutive evens differ by 2; arbitrary even integers don’t.

Proof questions reward structure as much as algebra. A clean, short proof with a clear conclusion will score full marks; a tangled chain of working that ends “= 2n + 2″ without saying what that means won’t. Always ask at the end: “what have I shown?” — and write that line down.

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