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139
•
5 Apr 2023
•
Yusra
@yusra_pmb
Mathematical proofs help us understand why mathematical statements are true... Show more
A mathematical proof is a rigorous demonstration that establishes the truth of a mathematical statement based on previously established statements and accepted rules of reasoning. When constructing proofs, mathematicians use known facts to build logical pathways leading to definitive conclusions.
Definition: A mathematical proof is a detailed explanation that demonstrates why a mathematical statement must be true, using logical reasoning and established mathematical facts.
The three fundamental types of mathematical proofs include deductive proof, proof by exhaustion in mathematics, and mathematical proof by counter example explanation. Each method serves different purposes and follows distinct approaches to verify mathematical claims.
Deductive proof, commonly used at the GCSE level, relies on logical reasoning and known mathematical facts. When constructing a deductive proof, it's essential to clearly state all assumptions, present logical steps in sequence, and ensure all cases are thoroughly covered before reaching a conclusion.
Example: Consider proving that = x³+8x²-101x-70. This requires:
Examples of proof by exhaustion in mathematics involve examining all possible cases within a finite set of possibilities. This method is particularly effective when dealing with a limited number of scenarios that can be systematically checked.
Highlight: Proof by exhaustion works best when:
When proving statements about square numbers, for instance, we can split numbers into even and odd cases. For even numbers ², we get 4n², which is always a multiple of 4. For odd numbers ², we get 4n²+4n+1, which is always one more than a multiple of 4.
Counter examples provide a powerful way to disprove general statements. By finding just one case where the statement fails, we can definitively show that the statement cannot be universally true.
How to use completing the square for proof is a sophisticated technique that helps establish properties about quadratic expressions. This method involves rewriting quadratic expressions in the form ²+k, which can reveal important characteristics about the function.
Vocabulary: Completing the square transforms a quadratic expression ax²+bx+c into the form a²+k, where h represents the x-coordinate of the vertex.
When applying completing the square in proofs, follow these steps:
This technique is particularly useful for proving properties about quadratic functions, such as minimum values, maximum values, and the nature of roots.
This page contains various algebraic manipulations and examples related to prime numbers.
Several examples of algebraic manipulations are provided, including:
Example: Simplifying the expression ² is shown step-by-step.
The page also includes an example related to prime numbers:
Example: Checking if 3² - 1 + 3 = 9 is a prime number.
Vocabulary: A prime number is a natural number greater than 1 that is only divisible by 1 and itself.
This final page covers more advanced algebraic techniques and problem-solving.
Examples include:
Example: Factoring the expression 2x³ + x² - 43x - 60 into .
The page also demonstrates how to solve equations involving parameters, such as x² - kx + k = 0.
Highlight: These examples showcase the application of mathematical proof by counter example explanation and examples of proof by exhaustion in mathematics in more complex algebraic scenarios.
This collection of problems and solutions provides a comprehensive overview of various proof techniques and their applications in algebra and number theory.
Mathematical proofs are essential for demonstrating the truth of conjectures using logical reasoning and known facts. This document explores various types of proofs and provides examples of their application in algebra and geometry.
Definition: A mathematical proof is a logical pathway built from known facts to demonstrate a conjecture's truth under stated conditions.
Highlight: The hallmark of a good proof is that no additional clarification is required once completed.
App Store
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
Yusra
@yusra_pmb
Mathematical proofs help us understand why mathematical statements are true or false through logical reasoning.
Proof by counter exampleis a powerful method where we disprove a statement by finding just one case where it fails. For instance, to disprove... Show more
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
A mathematical proof is a rigorous demonstration that establishes the truth of a mathematical statement based on previously established statements and accepted rules of reasoning. When constructing proofs, mathematicians use known facts to build logical pathways leading to definitive conclusions.
Definition: A mathematical proof is a detailed explanation that demonstrates why a mathematical statement must be true, using logical reasoning and established mathematical facts.
The three fundamental types of mathematical proofs include deductive proof, proof by exhaustion in mathematics, and mathematical proof by counter example explanation. Each method serves different purposes and follows distinct approaches to verify mathematical claims.
Deductive proof, commonly used at the GCSE level, relies on logical reasoning and known mathematical facts. When constructing a deductive proof, it's essential to clearly state all assumptions, present logical steps in sequence, and ensure all cases are thoroughly covered before reaching a conclusion.
Example: Consider proving that = x³+8x²-101x-70. This requires:
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Examples of proof by exhaustion in mathematics involve examining all possible cases within a finite set of possibilities. This method is particularly effective when dealing with a limited number of scenarios that can be systematically checked.
Highlight: Proof by exhaustion works best when:
When proving statements about square numbers, for instance, we can split numbers into even and odd cases. For even numbers ², we get 4n², which is always a multiple of 4. For odd numbers ², we get 4n²+4n+1, which is always one more than a multiple of 4.
Counter examples provide a powerful way to disprove general statements. By finding just one case where the statement fails, we can definitively show that the statement cannot be universally true.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
How to use completing the square for proof is a sophisticated technique that helps establish properties about quadratic expressions. This method involves rewriting quadratic expressions in the form ²+k, which can reveal important characteristics about the function.
Vocabulary: Completing the square transforms a quadratic expression ax²+bx+c into the form a²+k, where h represents the x-coordinate of the vertex.
When applying completing the square in proofs, follow these steps:
This technique is particularly useful for proving properties about quadratic functions, such as minimum values, maximum values, and the nature of roots.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This page contains various algebraic manipulations and examples related to prime numbers.
Several examples of algebraic manipulations are provided, including:
Example: Simplifying the expression ² is shown step-by-step.
The page also includes an example related to prime numbers:
Example: Checking if 3² - 1 + 3 = 9 is a prime number.
Vocabulary: A prime number is a natural number greater than 1 that is only divisible by 1 and itself.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This final page covers more advanced algebraic techniques and problem-solving.
Examples include:
Example: Factoring the expression 2x³ + x² - 43x - 60 into .
The page also demonstrates how to solve equations involving parameters, such as x² - kx + k = 0.
Highlight: These examples showcase the application of mathematical proof by counter example explanation and examples of proof by exhaustion in mathematics in more complex algebraic scenarios.
This collection of problems and solutions provides a comprehensive overview of various proof techniques and their applications in algebra and number theory.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Mathematical proofs are essential for demonstrating the truth of conjectures using logical reasoning and known facts. This document explores various types of proofs and provides examples of their application in algebra and geometry.
Definition: A mathematical proof is a logical pathway built from known facts to demonstrate a conjecture's truth under stated conditions.
Highlight: The hallmark of a good proof is that no additional clarification is required once completed.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
App Store
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
