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4 Dec 2025
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Understanding Mathematical Proof Methods
Alex
@alext_qjqz
1 / 3
Types of Mathematical Proofs
Mathematical proofs allow us to verify statements with absolute certainty. A statement is what you're trying to prove, a theorem is a statement proven true, and a conjecture is a statement not yet proven true or false.
Direct proof (or proof by deduction) starts with facts you know and uses logical steps to reach your conclusion. For example, to prove "if x is even, then x² is even," you begin with the definition of even numbers , calculate x² = (2k)² = 4k² = 2(2k²), and conclude that x² must be even because it equals 2 multiplied by another number.
Proof by exhaustion involves breaking your proof into separate cases and proving each one. This works brilliantly when you have a limited number of possibilities to cover. For instance, to prove that "the cube of any natural number is either a multiple of 9, or 1 more/less than a multiple of 9," you would examine three cases: numbers divisible by 3, numbers that are 1 more than a multiple of 3, and numbers that are 1 less than a multiple of 3.
Remember: When choosing a proof method, consider what would be most efficient for the particular statement. Direct proofs work well for straightforward implications, while exhaustion is perfect for statements with a finite number of cases.
More Proof Methods
Proof by exhaustion requires examining all cases thoroughly. For numbers divisible by 3, we show (3k)³ = 27k³ = 9(3k³) is a multiple of 9. For numbers 1 more than a multiple of 3, we demonstrate ³ = 9+1, which is 1 more than a multiple of 9. Similarly, numbers 1 less than a multiple of 3 give cubes that are 1 less than multiples of 9.
Proof by counterexample is wonderfully direct – you disprove a statement by finding just one example that contradicts it. To disprove "integers always have an even number of factors," we need only point to the number 1 (with exactly one factor) or any perfect square like 4 (with three factors: 1, 2, and 4).
Proof by contradiction assumes the opposite of what you want to prove and shows this leads to an impossible result. The negation of "All multiples of 5 are even" is "At least 1 multiple of 5 is odd." For another example, to prove "if n² is even, then n is even," we assume n² is even but n is odd. This leads to n² being odd – a contradiction that confirms our original statement must be true.
Pro tip: Contradiction is especially powerful for proving statements about irrational numbers and uniqueness. When direct approaches seem difficult, try assuming the opposite and look for a contradiction.
Advanced Contradiction Proofs
Proof by contradiction becomes particularly elegant when tackling challenging problems. When proving "if n² is even, then n is even," we start by assuming n² is even but n is odd. Since odd numbers have the form n = 2k+1, we calculate n² = ² = 4k² + 4k + 1 = 2 + 1, which is odd. This contradicts our assumption that n² is even, proving our statement.
The classic proof that √2 is irrational showcases contradiction beautifully. We begin by assuming √2 is rational, expressed as √2 = a/b in simplest form (a and b have no common factors). Squaring both sides gives a²/b² = 2, so a² = 2b². Since a² is even, a must be even, so a = 2c. Substituting this, we get (2c)² = 2b², which simplifies to b² = 2c², meaning b² is even, thus b is even.
The conclusion that both a and b are even contradicts our initial assumption that they have no common factors. This impossibility proves that √2 must be irrational. This proof demonstrates how contradiction can establish truths about numbers that cannot be written as fractions.
Challenge yourself: Try using contradiction to prove other famous results, like "there are infinitely many prime numbers" or "the sum of a rational and an irrational number is irrational." Contradiction often reveals elegant solutions to seemingly complex problems.
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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
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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.
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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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Best app on earth! no words because it’s too good
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
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
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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
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 THE SCHOOLGPT. 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 THE SCHOOLGPT. 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
Understanding Mathematical Proof Methods
Alex
@alext_qjqz
Mathematical proofs are elegant ways to establish the truth of statements with absolute certainty. They form the backbone of mathematical reasoning, using logical arguments to move from known facts to new conclusions. Understanding different proof methods will strengthen your analytical... Show more
Types of Mathematical Proofs
Mathematical proofs allow us to verify statements with absolute certainty. A statement is what you're trying to prove, a theorem is a statement proven true, and a conjecture is a statement not yet proven true or false.
Direct proof (or proof by deduction) starts with facts you know and uses logical steps to reach your conclusion. For example, to prove "if x is even, then x² is even," you begin with the definition of even numbers , calculate x² = (2k)² = 4k² = 2(2k²), and conclude that x² must be even because it equals 2 multiplied by another number.
Proof by exhaustion involves breaking your proof into separate cases and proving each one. This works brilliantly when you have a limited number of possibilities to cover. For instance, to prove that "the cube of any natural number is either a multiple of 9, or 1 more/less than a multiple of 9," you would examine three cases: numbers divisible by 3, numbers that are 1 more than a multiple of 3, and numbers that are 1 less than a multiple of 3.
Remember: When choosing a proof method, consider what would be most efficient for the particular statement. Direct proofs work well for straightforward implications, while exhaustion is perfect for statements with a finite number of cases.
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More Proof Methods
Proof by exhaustion requires examining all cases thoroughly. For numbers divisible by 3, we show (3k)³ = 27k³ = 9(3k³) is a multiple of 9. For numbers 1 more than a multiple of 3, we demonstrate ³ = 9+1, which is 1 more than a multiple of 9. Similarly, numbers 1 less than a multiple of 3 give cubes that are 1 less than multiples of 9.
Proof by counterexample is wonderfully direct – you disprove a statement by finding just one example that contradicts it. To disprove "integers always have an even number of factors," we need only point to the number 1 (with exactly one factor) or any perfect square like 4 (with three factors: 1, 2, and 4).
Proof by contradiction assumes the opposite of what you want to prove and shows this leads to an impossible result. The negation of "All multiples of 5 are even" is "At least 1 multiple of 5 is odd." For another example, to prove "if n² is even, then n is even," we assume n² is even but n is odd. This leads to n² being odd – a contradiction that confirms our original statement must be true.
Pro tip: Contradiction is especially powerful for proving statements about irrational numbers and uniqueness. When direct approaches seem difficult, try assuming the opposite and look for a contradiction.
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Advanced Contradiction Proofs
Proof by contradiction becomes particularly elegant when tackling challenging problems. When proving "if n² is even, then n is even," we start by assuming n² is even but n is odd. Since odd numbers have the form n = 2k+1, we calculate n² = ² = 4k² + 4k + 1 = 2 + 1, which is odd. This contradicts our assumption that n² is even, proving our statement.
The classic proof that √2 is irrational showcases contradiction beautifully. We begin by assuming √2 is rational, expressed as √2 = a/b in simplest form (a and b have no common factors). Squaring both sides gives a²/b² = 2, so a² = 2b². Since a² is even, a must be even, so a = 2c. Substituting this, we get (2c)² = 2b², which simplifies to b² = 2c², meaning b² is even, thus b is even.
The conclusion that both a and b are even contradicts our initial assumption that they have no common factors. This impossibility proves that √2 must be irrational. This proof demonstrates how contradiction can establish truths about numbers that cannot be written as fractions.
Challenge yourself: Try using contradiction to prove other famous results, like "there are infinitely many prime numbers" or "the sum of a rational and an irrational number is irrational." Contradiction often reveals elegant solutions to seemingly complex problems.
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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Is Knowunity really free of charge?
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Most popular content: Proof by Contradiction
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Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.9/5
App Store
4.8/5
Google Play
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 THE SCHOOLGPT. 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 THE SCHOOLGPT. 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