Define proof by contradiction .

A proof by contradiction begins by assuming the opposite of the statement to be proved. Logical steps then lead to something impossible, which rules the assumption out and leaves the original statement standing.

Why does a proof by contradiction need written explanation as well as algebra?

Because the logic is the proof, not the algebra. A reader has to be told what is being assumed and where the impossibility appears, and neither of those is visible in the working on its own.

What does prime factorisation let you write any composite number as?

A product of primes . It earns its place in proofs by contradiction because an assumption about a number often forces its prime factors into an impossible arrangement.

True or False? A single contradiction can prove a statement about infinitely many numbers.

True. The assumption covered every case at once, so one impossibility brings the whole of it down. That is what lets contradiction prove statements no amount of checking examples ever could.

International A Level (IAL)MathsEdexcelPure 4FlashcardsProofProof by Contradiction

Proof by Contradiction (Edexcel International A Level (IAL) Maths: Pure 4): Flashcards

Exam code: YMA01

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FrontProof by Contradiction
Define proof by contradiction.

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Define proof by contradiction.
A proof by contradiction begins by assuming the opposite of the statement to be proved.
Logical steps then lead to something impossible, which rules the assumption out and leaves the original statement standing.
Why does a proof by contradiction need written explanation as well as algebra?
Because the logic is the proof, not the algebra.
A reader has to be told what is being assumed and where the impossibility appears, and neither of those is visible in the working on its own.
To prove that $\sqrt{5}$ is irrational you assume it is rational. How is that assumption written down?
As $\sqrt{5} = \frac{a}{b}$ , where $a$ and $b$ are integers with no common factors and $b \neq 0$ .
The no-common-factors condition is the one the contradiction eventually attacks, so it has to be stated right at the start.
How does the proof that $\sqrt{5}$ is irrational reach its contradiction?
It shows that $a$ and $b$ must both be multiples of $5$ .
That contradicts the starting condition that they share no common factor, so $\sqrt{5}$ cannot be rational after all.
What does prime factorisation let you write any composite number as?
A product of primes.
It earns its place in proofs by contradiction because an assumption about a number often forces its prime factors into an impossible arrangement.
True or False?
A single contradiction can prove a statement about infinitely many numbers.
True.
The assumption covered every case at once, so one impossibility brings the whole of it down.
That is what lets contradiction prove statements no amount of checking examples ever could.

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