Proof & Justification — SL 1.8 Mathematical proof and...

SECTION 1

Historical Context & Motivation

Mathematics is not just about getting the right number at the end of a calculation — it is about knowing why that number is correct. Since ancient times, mathematicians have felt the need to justify their claims rather than simply assert them. This desire for certainty gave birth to the idea of mathematical proof, one of the most powerful tools in human reasoning.

~600 BCE

Thales of Miletus

Greek philosopher Thales is credited as the first person to use logical deduction, rather than trial-and-error, to establish geometric facts — for example, proving that a circle is bisected by its diameter.

~300 BCE

Euclid's Elements

Euclid compiled The Elements, a thirteen-book masterpiece that organized geometry into axioms, definitions, and step-by-step proofs. This became the model for rigorous reasoning for over two thousand years.

~250 BCE

Archimedes' Verification

Archimedes frequently checked his results with multiple methods — an early version of the 'checking solutions' idea. He would solve a problem geometrically and then verify it numerically.

1637

Descartes & Algebraic Reasoning

René Descartes published methods for solving equations algebraically, and stressed the importance of substituting solutions back into the original equation to confirm validity — a technique you will practise in this lesson.

Modern Era

IB and 21st-Century Education

The IB curriculum explicitly includes proof and justification as a core skill, recognizing that understanding why something works is as important as computing the answer.

The central question this topic addresses is deceptively simple: How can you be sure your answer is correct? In everyday life you might say, "It looks right." In mathematics, that is not enough. You need a logical chain of reasoning — or at the very least, a reliable way to check. This lesson introduces you to both skills: building a basic logical argument and verifying solutions.

SECTION 2

Core Principles & Definitions

Before diving into examples, let's establish the foundational ideas you will use throughout this topic. Each of these concepts builds on the previous one, creating a toolkit for mathematical reasoning.

Conjecture

A conjecture is an educated guess or a statement that you believe to be true but have not yet proven. For example: "The sum of two even numbers is always even." Conjectures are the starting point of proof.

Justification

A justification is a logical reason or chain of reasons that explains why a mathematical statement is true. It can range from a single sentence ("by the commutative property") to a multi-step argument.

Proof

A proof is a complete, logically airtight argument that shows a statement must be true in every possible case — not just the ones you tested. At SL level you will encounter simple deductive proofs.

Counter-example

A counter-example is a single specific case that disproves a conjecture. To show that "all prime numbers are odd" is false, you only need one counter-example: the number 2.

Checking (Verification)

After solving an equation, checking means substituting your answer back into the original equation (or conditions) to confirm it satisfies the problem. This catches algebraic errors and extraneous solutions.

✦ KEY TAKEAWAY

Think of proof and checking like a GPS navigation system. Getting an answer is like arriving at a destination — but proof is like the route the GPS shows you, explaining step-by-step why you ended up there. Checking is like verifying the address when you arrive — making sure you didn't take a wrong turn along the way.

SECTION 3

Visual Explanation — The Logic Chain

A proof is essentially a chain of logical steps. Each link in the chain must be justified — either by a definition, an axiom (a fact we accept as true), or a previously proven result. The diagram below shows how a basic deductive argument flows from a given statement through logical steps to a conclusion, and then how checking loops back to verify everything.

The cyan box represents what you start with (the given). The violet boxes show logical steps, each justified by a rule or definition. The green box is the conclusion. The dashed amber arrow shows the checking loop — feeding your answer back into the original conditions to verify it.

Notice the two directions in the diagram. Forward reasoning (the solid arrows) is what you do when constructing a proof — moving from what you know to what you want to show. Backward checking (the dashed arrow) is what you do after solving — plugging your result back in to make sure nothing went wrong. Both directions are essential habits in IB Mathematics.

SECTION 4

Mathematical Framework — Reasoning & Verification

At SL 1.8 level, you are not expected to write formal proofs with symbols like ∀ and ∃. Instead, you are expected to use basic deductive reasoning and to verify solutions by substitution. Here are the key patterns and techniques you will use.

Pattern 1: Deductive Justification

IF–THEN REASONING

If P is true, and P ⇒ Q, then Q is true.

P = a premise (something given or previously proven). P ⇒ Q means "P implies Q" — if P happens, Q must follow. This is the foundation of every logical argument.

For example: "If n is an even number, then n = 2k for some integer k." This is a deductive step — you are expressing the definition of 'even' in algebraic form, which you can then manipulate.

Pattern 2: Verification by Substitution

CHECKING A SOLUTION

If x = a is a solution to f(x) = 0, then f(a) must equal 0.

Replace every x in the original equation with your proposed solution a. Simplify. If you get a true statement (like 0 = 0), the solution is verified. If not, there is an error somewhere.

Pattern 3: Disproof by Counter-example

COUNTER-EXAMPLE

To disprove "For all x, P(x) is true", find one x₀ such that P(x₀) is false.

You only need one counter-example to destroy a universal claim. However, a million examples that work do not prove a universal claim — that is why proof requires logical argument, not just testing cases.

📝 IB Exam Tip

In IB exams, the command term "justify" means you must provide a logical reason — not just state the answer. The command term "verify" means you should substitute your solution back in and show it works. Knowing the difference between these terms can earn you marks.

SECTION 5

Detailed Breakdown — Types of Reasoning

There are different ways to justify a mathematical claim, and it is important to understand the difference in strength between them. The diagram below maps these approaches on a spectrum from weakest to strongest.

The upper section shows the spectrum of justification strength from mere guessing (red) to formal proof (cyan). For SL 1.8, your target is the yellow–green range: informal but logical reasoning. The lower section illustrates the three-step checking loop for verifying solutions.

At the SL level, the IB expects you to operate primarily in the informal reasoning zone. This means using words and basic algebra to explain why something must be true, rather than just citing examples. You should also always perform the verification loop when solving equations, especially when the question uses command terms like "verify" or "check."

SECTION 6

Worked Example

Let's work through two complete examples that demonstrate both key skills: justifying a mathematical claim and checking a solution.

Example A: Justify that the sum of two even numbers is always even.

Proof by Deductive Reasoning

Step 1 — Define what 'even' meansAn even number is any integer that can be written as 2 times another integer. So let the two even numbers be a = 2m and b = 2n, where m and n are integers.

a = 2m, b = 2n

Step 2 — Add the two numbersCompute a + b by substituting the expressions from Step 1:

a + b = 2m + 2n

Step 3 — Factor out the common factorBoth terms share a factor of 2, so we factor:

a + b = 2(m + n)

Step 4 — ConcludeSince m and n are integers, their sum (m + n) is also an integer. Call it k. Then a + b = 2k, which is even by definition. Therefore, the sum of two even numbers is always even. ∎

a + b = 2k → Even ✓

Example B: Solve 3x − 5 = 7 and verify the solution.

Solving and Checking

Step 1 — Isolate the variable termAdd 5 to both sides of the equation 3x − 5 = 7:

3x = 12

Step 2 — Solve for xDivide both sides by 3:

x = 4

Step 3 — Check by substitutionReplace x with 4 in the original equation: LHS = 3(4) − 5 = 12 − 5 = 7. RHS = 7.

LHS = RHS = 7 ✓

Step 4 — State conclusionSince the left-hand side equals the right-hand side when x = 4, the solution is verified.

x = 4 is confirmed.

SECTION 7

Strengths & Limitations of Basic Reasoning

Basic reasoning and checking are powerful tools, but they have limits. Understanding what each technique can and cannot do will help you choose the right approach on exams and in problem-solving.

Comparison of basic reasoning techniques at SL level

Technique	Strengths	Limitations
Testing examples	Quick way to form conjectures; easy to find counter-examples that disprove false claims.	No number of examples can prove a statement true for ALL cases. Missing one exception means the conclusion is wrong.
Informal deductive reasoning	Uses logical steps that cover every case at once; accessible at SL level; shows understanding of why, not just what.	Can be too vague if reasoning steps are not clearly stated; may overlook hidden assumptions.
Checking by substitution	Catches algebraic errors; identifies extraneous solutions (e.g., from squaring both sides); simple and mechanical.	Only confirms or denies a specific answer — does not prove the answer is the only solution or explain why it works.
Counter-example	Instantly disproves false universal claims; a single example is enough.	Cannot prove a claim true; requires creativity to find the right example; doesn't explain why the claim fails in general.

✦ KEY TAKEAWAY

Think of it like quality control in a factory. Testing examples is like inspecting a few products off the assembly line — useful, but you can't check every single one. Deductive proof is like redesigning the machine so that it can't produce a defective product. Checking is like scanning the final product with a barcode reader to make sure it matches the order.

SECTION 8

Connection to Advanced Proof Techniques

The basic reasoning you learn in SL 1.8 is the entry point to a much larger world of proof. As you progress through mathematics — whether in IB HL, university courses, or other advanced study — you will encounter increasingly powerful proof methods. The table below gives you a preview.

How SL 1.8 skills connect to advanced proof techniques

SL 1.8 (What you learn now)	Advanced (What comes later)
Informal deductive reasoning: use definitions and logical steps in plain language.	Direct proof: write a formal chain of implications using algebraic notation and precise definitions.
Counter-example to disprove a claim.	Proof by contradiction: assume the claim is false and show this leads to a logical impossibility.
Testing specific values to spot patterns.	Proof by induction: prove a base case, then prove that if it works for n, it works for n + 1 — like an infinite chain of dominoes.
Checking solutions by substituting back into original equation.	Existence & uniqueness theorems: proving that a solution exists and that it is the only one, using formal analysis.

Don't feel overwhelmed by the advanced column — you don't need any of that for SL. The point is that every technique you learn now is a foundation for more sophisticated methods. Mastering the habit of asking "Why is this true?" and "Does my answer actually work?" will serve you well no matter how far you go in mathematics.

SECTION 9

Practice Problems

Try these five problems to solidify your understanding. They increase in difficulty, so take your time and show your reasoning for each one.

PROBLEM 1 — CONCEPTUAL

Maria claims: "All square numbers are positive." Is this conjecture true or false? Justify your answer.

PROBLEM 2 — BASIC CALCULATION

Solve the equation 5x + 2 = 17, and then verify your solution by substitution.

PROBLEM 3 — INTERMEDIATE

Justify that the product of two odd numbers is always odd. Use algebraic reasoning (let the two odd numbers be 2m + 1 and 2n + 1).

PROBLEM 4 — APPLIED

A phone plan costs $20 per month plus $0.05 per text message. Aisha's bill one month was $27.50. She calculates that she sent 150 texts. Set up the equation, solve it, check her answer, and state whether she is correct.

PROBLEM 5 — CRITICAL THINKING

Ben solves the equation x² = 9 and gets x = 3. He checks: 3² = 9 ✓. He concludes that x = 3 is the only solution. Is his reasoning complete? Explain what is wrong and what he should do instead.

SUMMARY

Lesson Summary

In this lesson you explored the foundations of mathematical proof and justification as outlined in IB SL 1.8. You learned that a conjecture is an unproven claim, a justification provides logical reasons for why something is true, and a proof is a complete argument that covers all cases. You also saw that a single counter-example is enough to destroy a false universal claim.

Equally important is the skill of checking solutions by substitution: after solving any equation, substitute your answer back into the original equation to verify that LHS = RHS. This simple habit catches algebraic mistakes and identifies extraneous solutions. Remember: testing examples can suggest a pattern, but only deductive reasoning can prove it for all cases. These skills form the bedrock upon which all higher-level proof techniques are built.