A mathematical proof is a logical argument that demonstrates the truth of a mathematical statement or proposition.
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A mathematical proof is a detailed, logical argument that provides evidence for the truth of a mathematical theorem, proposition, or statement. It is constructed using a combination of deductive reasoning, mathematical axioms, and previously established results or theorems.
One of the most famous mathematicians, Euclid, defined a proof as “a statement of propositions in a sequence such that the last proposition is the thing required to be proved.” This definition emphasizes the importance of constructing a logical sequence of statements that leads to the desired conclusion.
The goal of a proof is to establish the certainty of a mathematical statement beyond any reasonable doubt. In order to achieve this, a proof must be rigorous, logical, and clear to readers with a basic understanding of the relevant mathematical concepts.
Interesting facts about mathematical proofs:
- One of the most famous mathematical proofs is Euclid’s proof that there are infinitely many prime numbers.
- The history of mathematical proofs dates back to ancient Greece, where mathematicians like Euclid, Pythagoras, and Archimedes developed some of the earliest known proofs.
- There are many different techniques and styles of mathematical proof, including algebraic, geometric, combinatorial, and topological proofs.
- Some mathematical proofs rely on clever tricks and techniques that seem counterintuitive or surprising, such as the proof that the square root of 2 is irrational.
- Modern computer technology has made it possible to produce extremely complex proofs that would have been impossible to verify by hand.
- The process of creating a mathematical proof can be challenging, but it is also a rewarding and satisfying experience for mathematicians who enjoy logical problem-solving.
Table: Types of mathematical proof
|Type of Proof||Description|
|Direct proof||A proof that establishes the truth of a proposition by applying the premises to the conclusion using logic|
|Indirect proof||A proof that establishes the truth of a proposition by showing that the negation of the proposition leads to a contradiction|
|Contrapositive||A proof that proves by contrapositive, which is to show that the statement is true if the negation of its conclusion is also true|
|Proof by contradiction||A proof that establishes the truth of a proposition by supposing that the opposite proposition is true and then showing that it leads to a contradiction|
See the answer to “What is a mathematical proof in English?” in this video
In this video, Alexander Knop explains the concept of a mathematical proof as a sequence of true statements that begins with what is already known and ends with the statement to be proven. He also emphasizes that in mathematics, an implication is true if and only if the initial statement is true and the conclusion statement is also true irrespective of the truth value of any other statements. Despite differing from everyday language, this definition of implication is useful for creating logical arguments in mathematics.
There are additional viewpoints
What is a proof in mathematics? The definition of a proof is the logical way in which mathematicians demonstrate that a statement is true. In general, these statements are known as theorems and lemmas. A theorem is a declaration that can be determined to be true using mathematical operations and arguments.
A mathematical proof is an argument that demonstrates why a mathematical statement is true, following the rules of mathematics. It is a way to show that a mathematical theorem is true. To prove a theorem is to show that it holds in all cases where it claims to hold. To prove a statement, one can either use axioms or theorems which have already been shown to be true.
proof is an argument thatdemonstrates why a conclusion is true, subject to certain standards of truth. mathematical proof is an argument that demonstrates why a mathematical statement is true, following the rules of mathematics.
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.
A mathematical proofis a way to show that a mathematicaltheoremis true. To prove a theorem is to show that theorem holds in all cases (where it claims to hold). To prove a statement, one can either use axioms, or theorems which have already been shown to be true.
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What does it mean to write a mathematical proof?
Answer: 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.
What are the 3 types of proofs? Response: There are many different ways to go about proving something, we’ll discuss 3 methods: direct proof, proof by contradiction, proof by induction. We’ll talk about what each of these proofs are, when and how they’re used.
What is mathematical proof and why is it important?
The answer is: According to Bleiler-Baxter & Pair , for a mathematician, a proof serves to convince or justify that a certain statement is true. But it also helps to increase the understanding of the result and the related concepts. That is why a proof also has the role of explanation.
Then, What are the steps in a mathematical proof?
The response is: The Structure of a Proof
- Draw the figure that illustrates what is to be proved.
- List the given statements, and then list the conclusion to be proved.
- Mark the figure according to what you can deduce about it from the information given.
- Write the steps down carefully, without skipping even the simplest one.
Also question is, What is mathematical proof and why is it important? Response to this: Mathematics is all about proving that certain statements, such as Pythagoras’ theorem, are true everywhere and for eternity. This is why maths is based on deductive reasoning. 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.
People also ask, What does mathematical proof mean?
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.
Additionally, How to make a mathematical proof? i. In a direct proof, the first thing you do is explicitly assume that the hypothesis is true for your selected variable, then use this assumption with definitions and previously proven results to show that the conclusion must be true. Direct Proof Walkthrough: Prove that if a is even, so is a2. Universally quantified implication: For all integers
In this manner, What type of reasoning does a mathematical proof use?
The response is: Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory.