lesson
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If you test a mathematical pattern one hundred times and it works every single time, does that mean it will work forever?
In geometry, spotting a pattern gives you a strong lead, but you cannot declare it a universal law until you back it up with airtight logic.
What Is a Conjecture?
A conjecture is an unproven statement or educated guess formed by observing patterns, a process called inductive reasoning.
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Can we simply test thousands of cases and declare a conjecture true, or can a single surprise destroy it?
The Power of a Counterexample
A counterexample is a single specific case that proves a conjecture false.
Consider the conjecture: "Every prime number is odd." While 3, 5, 7, and 11 support it, the number 2 is both prime and even—a single counterexample that instantly disproves the entire conjecture.
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What happens when a conjecture survives every test, and we want to turn it into an unbreakable rule of geometry?
From Conjecture to Theorem
A theorem is a mathematical statement that has been formally proven true using deductive reasoning—logical steps built on definitions, postulates, and established facts.