A mathematical proof is a logical argument that shows a statement is true based on previously established axioms and/or theorems. It must be complete, clear, and concise, with each step logically following from the previous one.
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A mathematical proof is a fundamental concept in mathematics that underpins the validity and accuracy of mathematical arguments and conclusions. It is an explanation based on logical reasoning that demonstrates the truth of a particular statement, proposition, or theorem. Proofs are the backbone of mathematical research and provide the necessary rigor that distinguishes mathematics from other subjects.
“Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost.” – W.S. Anglin
To prove a mathematical statement, a mathematician must follow a specific set of steps or rules. These rules are based on logic and establish a formal system of reasoning that can be used to prove the truth or falsehood of any mathematical statement. Famous mathematician Euclid’s book, “The Elements,” is an excellent example of a mathematical proof.
Interesting Facts:
- The shortest-known proof is a two-word proof known as the Cayley–Hamilton theorem. The proof is “Cayley–Hamilton.”
- Gödel’s incompleteness theorem implies that any mathematical system will inevitably contain statements that cannot be either proved or disproved.
- The first recorded proof in mathematics was in Ancient Greece around 300 BC.
- In 2018, a mathematical proof by British mathematician Sir Michael Atiyah was challenged by the mathematical community for its incompleteness.
- The “proof” that 1 equals 2 is a well-known incorrect mathematical proof that relies on an invalid algebraic step.
Below is a table of some common proof techniques used in mathematics:
| Proof Technique | Description |
|---|---|
| Direct Proof | Starts with a premise and then applies logic to reach the conclusion. |
| Indirect Proof | Begins by assuming the opposite of the conclusion and then showing that this assumption leads to a contradiction. |
| Contrapositive Proof | Shows the contrapositive of a statement is true, which is equivalent to the original statement. |
| Proof by Induction | A proof technique that is used for statements that involve natural numbers. |
| Proof by Exhaustion | Involves checking a finite number of cases to prove a statement is true. |
Video answer
The video discusses the importance of going back and identifying key results and techniques to solve math proofs. The speaker advises spending time on a problem, but if it becomes too challenging, one should analyze the solution step by step and rewrite it on their own to understand every step. By doing so, individuals can learn more techniques for future problems and understand the significance of each step in the solution.
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A mathematical proof shows a statement to be true using definitions, theorems, and postulates. Just as with a court case, no assumptions can be made in a mathematical proof. Every step in the logical sequence must be proven.
A proof in mathematics is a convincing argument that some mathematical statement is true. It should contain enough mathematical detail to be convincing to the person(s) to whom the proof is addressed. In essence, a proof is an argument that communicates a mathematical truth to another person who has the appropriate mathematical background. A mathematical proof is an argument that deduces the statement that is meant to be proven from other statements that you know for sure are true.
A proof in mathematics is a convincing argument that some mathematical statement is true. A proof should contain enough mathematical detail to be convincing to the person (s) to whom the proof is addressed. In essence, a proof is an argument that communicates a mathematical truth to another person (who has the appropriate mathematical background).
A mathematical proof is an argument that deduces the statement that is meant to be proven from other statements that you know for sure are true. For example, if you are given two of the angles in a triangle, you can deduce the value of the third angle from the fact that the angles in all triangles drawn in a plane always add up to 180 degrees.
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Beside above, What is mathematical proof short? Answer: A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.
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Also to know is, What is math proof by example?
The answer is: In logic and mathematics, proof by example (sometimes known as inappropriate generalization) is a logical fallacy whereby the validity of a statement is illustrated through one or more examples or cases—rather than a full-fledged proof.
What is a math proof without words?
In reply to that: In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text.
What is the concept of proof?
As an answer to this: A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community.
Consequently, How to write a proof math? The reply will be: proving, you should begin the proof itself with the notation Proof: or Pf:. End with notation like QED , qed, or # . Example: The question tells you to “Prove that if x is a non-zero element of R , then x has a multiplicative inverse.”
Secondly, What does mathematical proof mean?
As a response to this: Here are all the possible meanings and translations of the word mathematical proof. A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.
People also ask, What is the longest mathematical proof? [r/threadkillers] TIL that the longest mathematical proof is 15000 pages long, involved more than 100 mathematicians and took 30 years just to complete it. u/DIRTY_CRAPPED_BRIEFS] If you follow any of the above links, please respect the rules of reddit and don’t vote in the other threads. (Info / ^Contact) 11. Share.
Keeping this in view, How to write a proof math? proving, you should begin the proof itself with the notation Proof: or Pf:. End with notation like QED , qed, or # . Example: The question tells you to “Prove that if x is a non-zero element of R , then x has a multiplicative inverse.”
Consequently, What does mathematical proof mean?
In reply to that: Here are all the possible meanings and translations of the word mathematical proof. A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.
Then, What is the longest mathematical proof?
Response to this: [r/threadkillers] TIL that the longest mathematical proof is 15000 pages long, involved more than 100 mathematicians and took 30 years just to complete it. u/DIRTY_CRAPPED_BRIEFS] If you follow any of the above links, please respect the rules of reddit and don’t vote in the other threads. (Info / ^Contact) 11. Share.