Definition
Proof
A mathematical proof is a valid chain of reasons showing a claim must be true in every allowed case.
Continue from this idea into connected AXIO math.
Defining relationship
Examples say ‘it worked’; a proof says ‘it has to work.’
A mathematical proof is a logically valid argument that begins with accepted definitions, assumptions, or established facts and uses valid deductions to establish the stated claim for every case in its domain.
- 1
EXAMPLES
It worked twice
Examples test the claim but do not cover every case.
Examples - 2
GENERAL FORM
Write any odd numbers
Every odd number is two times a whole number plus one.
- 3
DEDUCTION
The sum has a factor of 2
So the sum must be even for every allowed pair.
- 4
PROOF
The claim has to work
A valid chain covers the whole domain, not just selected examples.
Examples say “it worked”; proof says “it must work.”
What it means
A proof explains why a result cannot fail, not just that it worked in several examples. Each step must follow from something already known, and the reasoning must cover all the cases named in the claim.
Remember it
Examples say ‘it worked’; a proof says ‘it has to work.’
Example
An odd number can be made from pairs of counters plus one extra counter. Adding two odd numbers joins the two extras into another pair, so every counter is paired. Therefore the sum of any two odd numbers is even.
Watch for
Checking 3 + 5 and 7 + 9 gives evidence, but it is not a proof about all pairs of odd numbers.
Common mix-up
A calculation can support one case; a proof must justify the whole claim.
Exact meaning
A mathematical proof is a logically valid argument that begins with accepted definitions, assumptions, or established facts and uses valid deductions to establish the stated claim for every case in its domain.