Inductive and deductive reasoning
Mathematicians use two kinds of reasoning: spotting patterns to make conjectures, and proving conclusions that must be true. Geometry is where most students first meet proof.
Inductive reasoning
Inductive reasoning draws a general conclusion from observed examples. Measuring many triangles and noticing their angles always add to 180 degrees leads to a conjecture, an unproved statement believed true. Inductive reasoning is how scientists form hypotheses, but it can never guarantee a conclusion.
Counterexamples
One counterexample shows a conjecture is false. The conjecture 'every prime number is odd' fails because 2 is prime and even. The conjecture 'n squared + n + 41 is always prime' works for n = 0 to 39 but fails at n = 40, giving 1,681 = 41 squared. Patterns can fool you.
Deductive reasoning
Deductive reasoning uses accepted facts, definitions, and logic to reach conclusions that must be true. If all right angles measure 90 degrees and angle A is a right angle, then angle A measures 90 degrees. Proofs in geometry are chains of deductive steps.
Law of detachment
The law of detachment says: if 'p implies q' is true and p is true, then q is true. If a figure is a square, then it has four right angles; this figure is a square; so it has four right angles. The error to avoid is reasoning backwards: a figure with four right angles may be a rectangle, not a square.
Law of syllogism
The law of syllogism chains conditionals: if p implies q and q implies r, then p implies r. If a shape is a square, it is a rectangle; if it is a rectangle, it is a parallelogram; so a square is a parallelogram. Chaining steps like this is the backbone of longer proofs.
Reasoning in algebra
Algebraic properties justify each step of solving an equation. The addition property of equality lets you add the same number to both sides; the substitution property lets you replace a quantity with an equal one; the reflexive property says a = a; the symmetric property says if a = b, then b = a; and the transitive property says if a = b and b = c, then a = c.
Why proof matters
Proof gives certainty that examples cannot. A result proved once holds for every case, forever. The Pythagorean theorem has been proved in hundreds of different ways; one published collection by Elisha Loomis in 1940 contains 370 proofs.
Worked example
Conjecture: the sum of two odd numbers is even. Testing 3 + 5 = 8 and 7 + 11 = 18 supports it inductively. A deductive proof: odd numbers can be written 2m + 1 and 2n + 1, and their sum is 2m + 2n + 2 = 2(m + n + 1), which is even. The proof covers every pair of odd numbers at once.
Key takeaways
Inductive reasoning generalises from examples to form conjectures but cannot guarantee them; a single counterexample disproves a conjecture. Deductive reasoning derives certain conclusions from accepted facts. The law of detachment applies a true conditional to a true hypothesis; the law of syllogism chains conditionals. Properties of equality, such as addition, substitution, reflexive, symmetric, and transitive, justify algebraic steps. Proof gives certainty for every case.
Flashcard
General conclusion from examples
Check your understanding
Which shows that the conjecture 'every prime number is odd' is false?