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Updated Mar 15, 2026
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Proof by Contradiction in A-level Maths for Edexcel
Maya A
@maya.ah
1 / 3
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.
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 , 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.
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 = 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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Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
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Thomas R
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Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
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This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
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The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
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I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Proof 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... Show more
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.
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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 , 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.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
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 = 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.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
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4.6/5
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4.7/5
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user