Theorems, Conjectures And Axioms
Axioms
These are statements accepted as true without proof - they're our starting points or "ground rules". They're self-evident or agreed-upon truths that we use to build more complex mathematical ideas.
Example: "Through any two distinct points, there exists exactly one straight line" (OR) "If you add the same number to equal quantities, the results are equal"
The Zero Product Property
- If a × b =
, then either a = 0 or b = 0 (or both)
Addition Property of Equality
- If a = b, then a +
= b + c
The Parallel Postulate
- Through a point not on a line, there exists exactly
line parallel to the given line
Theorems
These are statements that can be proven true using logical deduction from axioms and other proven theorems. They require rigorous proof and must be true in all cases where their conditions are met.
Example: The Pythagorean Theorem (a² + b² = c² in right triangles). Using axioms about angles, lines, and geometric properties, we can prove this must be true for all
The Sum of Interior Angles of a Triangle = 180°
- Proof uses axioms about parallel lines and angles
- Can verify: Draw any triangle, measure angles, they'll sum to
°
The Fundamental Theorem of Arithmetic
- Every integer greater than 1 can be uniquely expressed as a product of prime numbers Example: 28 = 2² ×
Example: = 2² × 5²
The Triangle Inequality Theorem
- The sum of any two sides of a triangle must be greater than the third side
- Example: In a triangle with sides 3, 4, and 6:
3 + 4 > 6 (
Conjectures
These are statements that appear to be true based on evidence but haven't been proven yet.They're educated guesses that need proof to become theorems.
The Goldbach Conjecture
- States that every even number greater than 2 can be written as the sum of two prime numbers
- Example: 4 =
+ , 6 = + , 8 = + , 10 = + , and so on - This pattern has been verified for enormous numbers but still hasn't been proven for ALL even numbers
- Until proven, it remains a conjecture, not a theorem
The Twin Prime Conjecture
- There are infinitely many pairs of prime numbers that differ by
- Examples: (3,5), (5,7), (11,13), (17,19)
- We've found millions of such pairs but haven't proven there are infinitely many
The Collatz Conjecture
- Start with any positive integer n
- If n is even, divide by
- If n is odd, multiply by 3 and add 1
- Conjecture: This sequence always reaches 1
- Example with n = 7: 7 → 22 →
→ → → → → → → → → → → → → → 1
Perfect Number Conjecture
- Perfect numbers are those equal to sum of their proper divisors
- Example: 6 = 1 + 2 + 3
- Example: 28 = 1 + 2 + 4 + 7 + 14
- Conjecture: All even perfect numbers end in 6 or 8
- We know this is true for all discovered so far, but it's not proven for all.
Procedure for turning conjectures into theorems
- Start with Axioms (accepted truths)
- Use these to prove Theorems (proven statements)
- Form Conjectures (educated guesses) about patterns we observe
- Try to prove Conjectures to turn them into Theorems
This system of axioms, theorems, and conjectures forms the foundation of mathematical reasoning and proof.