In mathematics, a proof is a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true.^{[1]}^{[2]} Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproved proposition that is believed to be true is known as a conjecture.
The statement that is proved is often called a theorem.^{[1]} Once a theorem is proved, it can be used as the basis to prove further statements. A theorem may also be referred to as a lemma, especially if it is intended for use as a stepping stone in the proof of another theorem.
Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasiempiricism in mathematics, and socalled folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.
Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof.^{[3]} The development of mathematical proof is primarily the product of early Greek civilization. Thales (624–546 BCE) proved some theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known. Euclid (300 BCE) began with undefined terms and axioms (propositions regarding the undefined terms assumed to be selfevidently true, from the Greek “axios” meaning “something worthy”) and used these to prove theorems using deductive logic. Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption that axioms are "true" in any sense; this allows for parallel mathematical theories built on alternate sets of axioms (see Axiomatic set theory and NonEuclidean geometry for examples).
The word Proof comes from the Latin probare meaning "to test". Related modern words are the English "probe", "proboscis”, "probation", and "probability", the Spanish "probar" (to smell or taste, or (lesser use) touch or test),^{[4]} and the German "probieren" (to try). The early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony.^{[5]}
There are two different conceptions of mathematical proof.^{[6]} The first is an informal proof, a rigorous naturallanguage expression that is intended to convince the audience of the truth of a theorem. Because of their use of natural language, the standards of rigor for informal proofs will depend on the audience of the proof. In order to be considered a proof, however, the argument must be rigorous enough; a vague or incomplete argument is not a proof. Informal proofs are the type of proof typically encountered in published mathematics. They are sometimes called "formal proofs" because of their rigor, but logicians use the term "formal proof" to refer to a different type of proof entirely.
In logic, a formal proof is not written in a natural language, but instead uses a formal language consisting of certain strings of symbols from a fixed alphabet. This allows the definition of a formal proof to be precisely specified without any ambiguity. The field of proof theory studies formal proofs and their properties. Although each informal proof can, in theory, be converted into a formal proof, this is rarely done in practice. The study of formal proofs is used to determine properties of provability in general, and to show that certain undecidable statements are not provable.
A classic question in philosophy asks whether mathematical proofs are analytic or synthetic. Kant, who introduced the analyticsynthetic distinction, believed mathematical proofs are synthetic.
Proofs may be viewed as aesthetic objects, admired for their mathematical beauty. The mathematician Paul Erdős was known for describing proofs he found particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem. The book Proofs from THE BOOK, published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing.
In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.^{[7]} For example, direct proof can be used to establish that the sum of two even integers is always even:
This proof uses definition of even integers, as well as distribution law.
In proof by mathematical induction, first a "base case" is proved, and then an "induction rule" is used to prove a (often infinite) series of other cases.^{[8]} Since the base case is true, the infinity of other cases must also be true, even if all of them cannot be proved directly because of their infinite number. A subset of induction is infinite descent. Infinite descent can be used to prove the irrationality of the square root of two.
The principle of mathematical induction states that: Let N = { 1, 2, 3, 4, ... } be the set of natural numbers and P(n) be a mathematical statement involving the natural number n belonging to N such that
Then P(n) is true for all natural numbers n.
Mathematicians often use the term "proof by induction" as shorthand for a proof by mathematical induction.^{[9]} However, the term "proof by induction" may also be used in logic to mean an argument that uses inductive reasoning.
Proof by transposition establishes the conclusion "if p then q" by proving the equivalent contrapositive statement "if not q then not p".
In proof by contradiction (also known as reductio ad absurdum, Latin for "by reduction toward the absurd"), it is shown that if some statement were so, a logical contradiction occurs, hence the statement must be not so. This method is perhaps the most prevalent of mathematical proofs. A famous example of a proof by contradiction shows that is an Irrational number:
Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. Joseph Liouville, for instance, proved the existence of transcendental numbers by constructing an explicit example.
In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases.
A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory. This is not to be confused with an argument that a theorem is 'probably' true. The latter type of reasoning can be called a 'plausibility argument' and is not a proof; in the case of the Collatz conjecture it is clear how far that is from a genuine proof.^{[10]} Probabilistic proof, like proof by construction, is one of many ways to show existence theorems.
A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a bijection between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a double counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal.
A nonconstructive proof establishes that a certain mathematical object must exist (e.g. "Some X satisfies f(X)"), without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proven to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. A famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that a^{b} is a rational number:
Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "proof without words". The lefthand picture below is an example of a historic visual proof of the Pythagorean theorem in the case of the (3,4,5) triangle.
Visual proof for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC. 
Visual proof for the Pythagorean theorem by rearrangement. 
An elementary proof is a proof which only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. For some time it was thought that certain theorems, like the prime number theorem, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.
A particular form of proof using two parallel columns is often used in elementary geometry classes in the United States.^{[11]} The proof is written as a series of lines in two columns. In each line, the lefthand column contains a proposition, while the righthand column contains a brief explanation of how the corresponding proposition in the lefthand column is either an axiom, a hypothesis, or can be logically derived from previous propositions. The lefthand column is typically headed "Statements" and the righthand column is typically headed "Reasons".^{[12]}
The expression "statistical proof" may be used technically or colloquially in areas of pure mathematics, such as involving cryptography, chaotic series, and probabilistic or analytic number theory.^{[13]}^{[14]}^{[15]} It is less commonly used to refer to a mathematical proof in the branch of mathematics known as mathematical statistics. See also "Statistical proof using data" section below.
Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity.^{[3]} However, computers are now used both to prove theorems and to carry out calculations that are too long for any human or team of humans to check; the first proof of the four color theorem is an example of a computerassisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a runtime error in its calculations calls the validity of such computerassisted proofs into question. In practice, the chances of an error invalidating a computerassisted proof can be reduced by incorporating redundancy and selfchecks into calculations, and by developing multiple independent approaches and programs.
A statement that is neither provable nor disprovable from a set of axioms is called undecidable (from those axioms). One example is the parallel postulate, which is neither provable nor refutable from the remaining axioms of Euclidean geometry.
Mathematicians have shown there are many statements that are neither provable nor disprovable in ZermeloFraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see list of statements undecidable in ZFC.
Gödel's (first) incompleteness theorem shows that many axiom systems of mathematical interest will have undecidable statements.
While early mathematicians such as Eudoxus of Cnidus did not use proofs, from Euclid to the foundational mathematics developments of the late 19th and 20th centuries, proofs were an essential part of mathematics.^{[16]} With the increase in computing power in the 1960’s, significant work began to be done investigating mathematical objects outside of the prooftheorem framework,^{[17]} in experimental mathematics. Early pioneers of these methods intended the work ultimately to be embedded in a classical prooftheorem framework, e.g. the early development of fractal geometry,^{[18]} which was ultimately so embedded.
The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with mathematical objects, such as numbers, to demonstrate something about everyday life, or when data used in an argument are numbers. It is sometime also used to mean a "statistical proof" (below), especially when used to argue from data.
"Statistical proof" from data refers to the application of statistics, data analysis, or Bayesian analysis to infer propositions regarding the probability of data. While using mathematical proof to establish theorems in statistics, it is usually not a mathematical proof in that the assumptions from which probability statements are derived require empirical evidence from outside mathematics to verify. In physics, in addition to statistical methods, "statistical proof" can refer to the specialized mathematical methods of physics applied to analyze data in a particle physics experiment or observational study in cosmology. "Statistical proof" may also refer to raw data or a convincing diagram involving data, such as scatter plots, when the data or diagram is adequately convincing without further anaylisis.
Proofs using inductive logic, while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner to probability, and may be less than one certainty. Bayesian analysis establishes assertions as to the degree of a person's subjective belief. Inductive logic should not be confused with mathematical induction.
Psychologism views mathematical proofs as psychological or mental objects. Mathematician philosophers, such as Leibnitz, Frege, and Carnap, have attempted to develop a semantics for what they considered to be the language of thought, whereby standards of mathematical proof might be applied to empirical science.
Philosophermathematicians such as Schopenhauer have attempted to formulate philosophical arguments in an axiomatic manner, whereby mathematical proof standards could be applied to argumentation in general philosophy. Other mathematicianphilosophers have tried to use standards of mathematical proof and reason, without empiricism, to arrive at statements outside of mathematics, but having the certainty of propositions deduced in a mathematical proof, such as Descarte’s cogito argument.
Sometimes, the abbreviation "Q.E.D." is written to indicate the end of a proof. This abbreviation stands for "Quod Erat Demonstrandum", which is Latin for "that which was to be demonstrated". A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a "tombstone" or "halmos" after its eponym Paul Halmos. Often, "which was to be shown" is verbally stated when writing "QED", "□", or "∎" in an oral presentation on a board.
A mathematical proof is a way to show that a math theorem is true. One must show that the theory is true in all cases.
There are different ways of proving a mathematical theorem.
Contents 
One type of proof is called proof by induction. This is usually used to prove a theorem that is true for all numbers. There are 4 steps in a proof by induction.
1. State that the proof will be by induction, and state which variable will be used in the induction step.
2. Prove that the statement is true for some beginning case.
3. Assume that for some value n = n_{0} the statement is true and has all of the properties listed in the statement. This is called the induction step.
4. Show that the statement is true for the next value, n_{0}+1.
Once that is shown, then it means that for any value of n that is picked, the next one is true. Since it's true for some beginning case (usually n=1), then it's true for the next one (n=2). And since it's true for 2, it must be true for 3. And since it's true for 3, it must be true for 4, etc. Induction shows that it is always true, precisely because it's true for whatever comes after any given number.
An example of proof by induction:
Prove that for all natural numbers n, 2(1+2+3+....+n1+n)=n(n+1)
Proof: First, the statement can be written "for all natural numbers n, 2$\backslash sum\_\{k=1\}^n\; k$=n(n+1)
By induction on n,
First, for n=1, 2$\backslash sum\_\{k=1\}^1\; k$=2(1)=1(1+1), so this is true.
Next, assume that for some n=n_{0} the statement is true. That is, 2$\backslash sum\_\{k=1\}^\{n\_0\}\; k$ = n_{0}(n_{0}+1)
Then for n=n_{0}+1, 2$\backslash sum\_\{k=1\}^\{\{n\_0\}+1\}\; k$ can be rewritten 2(n_{0}+1) + 2$\backslash sum\_\{k=1\}^\{n\_0\}\; k$
Since 2$\backslash sum\_\{k=1\}^\{n\_0\}\; k$ = n_{0}(n_{0}+1), 2n_{0}+1 + 2$\backslash sum\_\{k=1\}^\{n\_0\}\; k$ = 2(n_{0}+1) + 2n_{0}(n_{0}+1)
So 2(n_{0}+1) + 2n_{0}(n_{0}+1)= 2(n_{0}+1)(n_{0} + 2)
Proof by contradiction is a way of proving a mathematical theorem by showing that if the statement is false, there is a problem with the logic of the proof. That is, if one of the results of the theorem is assumed to be false, then the proof does not work.
When proving a theorem by way of contradiction, it is important to note that in the beginning of the proof. This is usually abbreviated BWOC. When the contradiction appears in the proof, there is usually an X made with 4 lines instead of 2 placed next to that line.
Prove that x = b ± ( √b²  4ac ) / 2a from ax²+bx+c=0
Dividing the quadratic equation
by a (which is allowed because a is nonzero), gives:
or
The quadratic equation is now in a form in which completing the square can be done. To "complete the square" is to find some number k so that
for another number y. In order for these equations to be true,
and
so
Adding this number to equation (1) makes
The left side is now a perfect square because
The right side can be written as a single fraction, with common denominator 4a^{2}. This gives
Taking the square root of both sides gives
Getting x by itself gives
frr:Bewis
