Proof by contradiction is a powerful mathematical technique where you...
Maths800 views·Updated 20 Jul 2026·3 pagesProof by Contradiction in A-level Maths for Edexcel
Maya A@maya.ah
1 / 3
1
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Basic Contradiction Proofs with Integers
Ever wondered why there's no "biggest" odd number? Proof by contradiction makes this crystal clear. You start by assuming there IS a greatest odd integer n, then show this assumption creates chaos.
Here's the magic: if n is the greatest odd number, then n+2 (which is definitely bigger than n) should also be odd. Since odd + even = odd, we've just found an odd number larger than our "greatest" one - contradiction! This technique works brilliantly for proving there's no upper limit to odd integers.
The same logic applies to proving relationships between even and odd numbers. When you assume n² is even but n is odd, you can write n as 2k+1. Squaring this gives you 4k²+4k+1 = 2+1, which is clearly odd - contradicting your assumption that n² is even.
Key Insight: Always express odd numbers as 2k+1 and even numbers as 2k - this algebraic form makes contradictions obvious when you do the maths.
You'll master this technique quickly once you see the pattern: assume the opposite, do some algebra, spot the contradiction, then conclude the original statement must be true. It's particularly effective for proving properties about infinite sets like prime numbers, where direct proof would be impossible.
2
of 3
Rational and Irrational Number Proofs
Rational numbers can be written as fractions a/b where both a and b are integers, whilst irrational numbers absolutely cannot. The classic proof that √2 is irrational showcases contradiction at its finest.
Start by assuming √2 IS rational, so √2 = a/b in its simplest form. Square both sides to get 2 = a²/b², which means a² = 2b². This forces a² to be even, so a must be even too.
Since a is even, write it as 2n. Substituting back gives (2n)² = 2b², which simplifies to 4n² = 2b², then 2n² = b². Now b² is even, so b is even too. But wait - if both a and b are even, the fraction a/b wasn't in its simplest form after all!
Pro Tip: When proving irrationality, always assume the fraction is "fully simplified" - this sets up the perfect contradiction when you show both parts must share common factors.
This technique extends beautifully to other scenarios. Want to prove there's no greatest positive rational? Assume there is one (call it a/b), then consider a/b + 1 = /b. This new fraction is rational and clearly bigger than your supposed "greatest" - contradiction achieved! The method works because rational and irrational numbers have fundamentally different algebraic properties.
3
of 3
Advanced Contradiction Techniques
Sometimes contradiction proofs require factoring skills to expose the impossible. Take the equation 4p² - q² = 25 with positive integers p and q. Factor the left side as 2p+q$$2p-q = 25.
Since 25 only factors as 1×25 or 5×5, you get two cases to check. If 2p+q = 25 and 2p-q = 1, solving gives p = 6.5 and q = 12 - but p isn't an integer! The second case (both factors equal 5) gives p = 5 and q = 0, but q isn't positive.
Algebraic manipulation becomes your best friend when dealing with expressions involving cubes or higher powers. If m³ + 5 is odd, you can prove m must be even by assuming m is odd (writing it as 2p ± 1).
Remember: When expanding (2p ± 1)³, you'll get terms that factor out as 2(...), proving the result is even - contradicting the given information.
Expanding (2p ± 1)³ + 5 gives you 8p³ ± 12p² + 6p ± 1 + 5, which factors as 2. This is clearly even, contradicting the fact that m³ + 5 is odd. These advanced techniques show how powerful contradiction becomes when combined with systematic algebraic work.
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Maths800 views·Updated 20 Jul 2026·3 pagesProof by Contradiction in A-level Maths for Edexcel
Maya A@maya.ah
Proof by contradiction is a powerful mathematical technique where you assume the opposite of what you want to prove, then show this leads to a logical impossibility. It's like showing someone is lying by catching them in their own contradictions...
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Basic Contradiction Proofs with Integers
Ever wondered why there's no "biggest" odd number? Proof by contradiction makes this crystal clear. You start by assuming there IS a greatest odd integer n, then show this assumption creates chaos.
Here's the magic: if n is the greatest odd number, then n+2 (which is definitely bigger than n) should also be odd. Since odd + even = odd, we've just found an odd number larger than our "greatest" one - contradiction! This technique works brilliantly for proving there's no upper limit to odd integers.
The same logic applies to proving relationships between even and odd numbers. When you assume n² is even but n is odd, you can write n as 2k+1. Squaring this gives you 4k²+4k+1 = 2+1, which is clearly odd - contradicting your assumption that n² is even.
Key Insight: Always express odd numbers as 2k+1 and even numbers as 2k - this algebraic form makes contradictions obvious when you do the maths.
You'll master this technique quickly once you see the pattern: assume the opposite, do some algebra, spot the contradiction, then conclude the original statement must be true. It's particularly effective for proving properties about infinite sets like prime numbers, where direct proof would be impossible.
2
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Rational and Irrational Number Proofs
Rational numbers can be written as fractions a/b where both a and b are integers, whilst irrational numbers absolutely cannot. The classic proof that √2 is irrational showcases contradiction at its finest.
Start by assuming √2 IS rational, so √2 = a/b in its simplest form. Square both sides to get 2 = a²/b², which means a² = 2b². This forces a² to be even, so a must be even too.
Since a is even, write it as 2n. Substituting back gives (2n)² = 2b², which simplifies to 4n² = 2b², then 2n² = b². Now b² is even, so b is even too. But wait - if both a and b are even, the fraction a/b wasn't in its simplest form after all!
Pro Tip: When proving irrationality, always assume the fraction is "fully simplified" - this sets up the perfect contradiction when you show both parts must share common factors.
This technique extends beautifully to other scenarios. Want to prove there's no greatest positive rational? Assume there is one (call it a/b), then consider a/b + 1 = /b. This new fraction is rational and clearly bigger than your supposed "greatest" - contradiction achieved! The method works because rational and irrational numbers have fundamentally different algebraic properties.
3
of 3Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Advanced Contradiction Techniques
Sometimes contradiction proofs require factoring skills to expose the impossible. Take the equation 4p² - q² = 25 with positive integers p and q. Factor the left side as 2p+q$$2p-q = 25.
Since 25 only factors as 1×25 or 5×5, you get two cases to check. If 2p+q = 25 and 2p-q = 1, solving gives p = 6.5 and q = 12 - but p isn't an integer! The second case (both factors equal 5) gives p = 5 and q = 0, but q isn't positive.
Algebraic manipulation becomes your best friend when dealing with expressions involving cubes or higher powers. If m³ + 5 is odd, you can prove m must be even by assuming m is odd (writing it as 2p ± 1).
Remember: When expanding (2p ± 1)³, you'll get terms that factor out as 2(...), proving the result is even - contradicting the given information.
Expanding (2p ± 1)³ + 5 gives you 8p³ ± 12p² + 6p ± 1 + 5, which factors as 2. This is clearly even, contradicting the fact that m³ + 5 is odd. These advanced techniques show how powerful contradiction becomes when combined with systematic algebraic work.
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