Use proof by contradiction.
Assume √2 is rational.This means that √2 = p/q for some integers p and q, with q <>0.
We assume p and q are in lowest terms.
Square both sides and we get:
2 = p2/q2Cross multiply:
p2 = 2q2This means p2 must be an even number
This means p is also even since the square of an odd is odd.
So we have p = 2k for some integer k:
2q2 = p2 = (2k)2 = 4k22q2 = 4k2
Divide each side by 2
q2 = 2k2q2 is also even, therefore q must be even
Since the square of an odd number is odd.
So both p and q are even.
This contradicts are assumption that p and q were in lowest terms in p/q.
So √2 cannot be rational.
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Free Proofs Calculator - Various Proofs in Algebra
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Build a bridge using corollaries, axioms, and theorems to get to the declarative statement.
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Build a bridge using corollaries, axioms, and theorems to get to the declarative statement.
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What 5 concepts are covered in the Proofs Calculator?
- axiom
- A statement accepted as true without proof. an unprovable rule or first principle accepted as true because it is self-evident or particularly usefu
- corollary
- A proposition formed from a proven proposition
- postulate
- A statement accepted as true without proof
- proof
- an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion
- theorem
- A statement provable using logic
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