In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.^{[13]} For example, direct proof can be used to establish that the sum of two "even "integers is always even:
 Consider two even integers x and y. Since they are even, they can be written as x = 2a and y = 2b, respectively, for integers a and b. Then the sum x + y = 2a + 2b = 2(a+b). Therefore x+y has 2 as a factor and, by definition, is even. Hence the sum of any two even integers is even.
This proof uses the definition of even integers, the integer properties of "closure under addition and multiplication, and "distributivity.
Proof by mathematical induction[edit]
Despite its name, mathematical induction is a method of "deduction, not a form of "inductive reasoning. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case "implies the next case. Since in principle the induction rule can be applied repeatedly starting from the proved base case, we see that all (usually "infinitely many) cases are provable.^{[14]} This avoids having to prove each case individually. A variant of mathematical induction is "proof by infinite descent, which can be used, for example, to prove the "irrationality of the square root of two.
A common application of proof by mathematical induction is to prove that a property known to hold for one number holds for all natural numbers:^{[15]} 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
 (i) P(1) is true, i.e., P(n) is true for n = 1.
 (ii) P(n+1) is true whenever P(n) is true, i.e., P(n) is true implies that P(n+1) is true.
 Then P(n) is true for all natural numbers n.
For example, we can prove by induction that all positive integers of the form 2n − 1 are odd. Let P(n) represent "2n − 1 is odd":
 (i) For n = 1, 2n − 1 = 2(1) − 1 = 1, and 1 is odd, since it leaves a remainder of 1 when divided by 2. Thus P(1) is true.
 (ii) For any n, if 2n − 1 is odd (P(n)), then (2n − 1) + 2 must also be odd, because adding 2 to an odd number results in an odd number. But (2n − 1) + 2 = 2n + 1 = 2(n+1) − 1, so 2(n+1) − 1 is odd (P(n+1)). So P(n) implies P(n+1).
 Thus 2n − 1 is odd, for all positive integers n.
The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction".^{[16]}
Proof by contraposition[edit]
"Proof by contraposition "infers the conclusion "if p then q" from the premise "if not q then not p". The statement "if not q then not p" is called the "contrapositive of the statement "if p then q". For example, contraposition can be used to establish that, given an integer x, if x² is even, then x is even:
 Suppose x is not even. Then x is odd. The product of two odd numbers is odd, hence x² = x·x is odd. Thus x² is not even. Thus, if x² is even, the supposition must be false, so x has to be even.
Proof by contradiction[edit]
In proof by contradiction (also known as "reductio ad absurdum, Latin for "by reduction to the absurd"), it is shown that if some statement were true, a logical contradiction occurs, hence the statement must be false. A famous example of proof by contradiction shows that is an "irrational number:
 Suppose that were a rational number, so by definition where a and b are nonzero integers with "no common factor. (If there is a common factor, divide both numerator and denominator by that factor to remove it, and repeat until no common factor remains. By the method of infinite descent, this process must terminate.) Thus, . Squaring both sides yields 2b^{2} = a^{2}. Since 2 divides the left hand side, 2 must also divide the right hand side (otherwise an even number would equal an odd number). So a^{2} is even, which implies that a must also be even. So we can write a = 2c, where c is also an integer. Substitution into the original equation yields 2b^{2} = (2c)^{2} = 4c^{2}. Dividing both sides by 2 yields b^{2} = 2c^{2}. But then, by the same argument as before, 2 divides b^{2}, so b must be even. However, if a and b are both even, they have a common factor, namely 2. This contradicts our initial supposition, so we are forced to conclude that is an irrational number.
Proof by construction[edit]
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. It can also be used to construct a "counterexample to disprove a proposition that all elements have a certain property.
Proof by exhaustion[edit]
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 color theorem as of 2011^{[update]} still has over 600 cases.
Probabilistic proof[edit]
A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of "probability theory. Probabilistic proof, like proof by construction, is one of many ways to show "existence theorems.
This is not to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'. The work on the "Collatz conjecture shows how far plausibility is from genuine proof.^{[17]}
Combinatorial proof[edit]
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.
Nonconstructive proof[edit]
A nonconstructive proof establishes that a "mathematical object with a certain property exists 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 proved 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 is a "rational number:
 Either is a rational number and we are done (take ), or is irrational so we can write and . This then gives , which is thus a rational of the form
Statistical proofs in pure mathematics[edit]
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.^{[18]}^{[19]}^{[20]} 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.
Computerassisted proofs[edit]
Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity.^{[7]} 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. Errors can never be completely ruled out in case of verification of a proof by humans either, especially if the proof contains natural language and requires deep mathematical insight.
Undecidable statements[edit]
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.
Heuristic mathematics and experimental mathematics[edit]
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.^{[21]} With the increase in computing power in the 1960s, significant work began to be done investigating "mathematical objects outside of the prooftheorem framework,^{[22]} 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,^{[23]} which was ultimately so embedded.
Related concepts[edit]
Visual proof[edit]
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 "Zhoubi Suanjing 500–200 BC.
Some illusory visual proofs, such as the "missing square puzzle, can be constructed in a way which appear to prove a supposed mathematical fact but only do so under the presence of tiny errors (for example, supposedly straight lines which actually bend slightly) which are unnoticeable until the entire picture is closely examined, with lengths and angles precisely measured or calculated.
Elementary proof[edit]
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.
Twocolumn proof[edit]
A particular way of organising a proof using two parallel columns is often used in elementary geometry classes in the United States.^{[24]} 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".^{[25]}
Colloquial use of "mathematical proof"[edit]
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 is numerical. It is sometimes also used to mean a "statistical proof" (below), especially when used to argue from "data.
Statistical proof using data[edit]
"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 "physical 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 analysis.
Inductive logic proofs and Bayesian analysis[edit]
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 full "certainty. Inductive logic should not be confused with "mathematical induction.
Bayesian analysis uses "Bayes' theorem to update a person's "assessment of likelihoods of hypotheses when new "evidence or "information is acquired.
Proofs as mental objects[edit]
Psychologism views mathematical proofs as psychological or mental objects. Mathematician "philosophers, such as "Leibniz, "Frege, and "Carnap have variously criticized this view and 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.^{["citation needed]}
Influence of mathematical proof methods outside mathematics[edit]
Philosophermathematicians such as "Spinoza 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 "Descartes' "cogito argument.
Ending a proof[edit]
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 "∎" during an oral presentation.
See also[edit]
References[edit]
 ^ "Bill Casselman. "One of the Oldest Extant Diagrams from Euclid". University of British Columbia. Retrieved 20080926.
 ^ Clapham, C. & Nicholson, JN. The Concise Oxford Dictionary of Mathematics, Fourth edition.
A statement whose truth is either to be taken as selfevident or to be assumed. Certain areas of mathematics involve choosing a set of axioms and discovering what results can be derived from them, providing proofs for the theorems that are obtained.
 ^ Cupillari, Antonella. The Nuts and Bolts of Proofs. Academic Press, 2001. Page 3.
 ^ Gossett, Eric. Discrete Mathematics with Proof. John Wiley and Sons, 2009. Definition 3.1 page 86. "ISBN 0470457937
 ^ New Shorter Oxford English Dictionary, 1993, OUP, Oxford.
 ^ The Emergence of Probability, Ian Hacking
 ^ ^{a} ^{b} The History and Concept of Mathematical Proof, Steven G. Krantz. 1. February 5, 2007
 ^ Kneale, p. 2
 ^ Howard Eves, An Introduction to the History of Mathematics, Saunders, 1990, "ISBN 0030295580 p. 141: "No work, except "The Bible, has been more widely used...."
 ^ Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", "Annals of the New York Academy of Sciences, 500: 253–277 [260], "doi:10.1111/j.17496632.1987.tb37206.x
 ^ Eder, Michelle (2000), Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam, "Rutgers University, retrieved 20080123
 ^ "Buss, Samuel R. (1998), "An introduction to proof theory", in "Buss, Samuel R., Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, 137, Elsevier, pp. 1–78, "ISBN "9780080533186. See in particular p. 3: "The study of Proof Theory is traditionally motivated by the problem of formalizing mathematical proofs; the original formulation of firstorder logic by Frege [1879] was the first successful step in this direction."
 ^ Cupillari, page 20.
 ^ Cupillari, page 46.
 ^ Examples of simple proofs by mathematical induction for all natural numbers
 ^ Proof by induction, University of Warwick Glossary of Mathematical Terminology
 ^ While most mathematicians do not think that probabilistic evidence ever counts as a genuine mathematical proof, a few mathematicians and philosophers have argued that at least some types of probabilistic evidence (such as Rabin's "probabilistic algorithm for testing primality) are as good as genuine mathematical proofs. See, for example, Davis, Philip J. (1972), "Fidelity in Mathematical Discourse: Is One and One Really Two?" American Mathematical Monthly 79:25263. Fallis, Don (1997), "The Epistemic Status of Probabilistic Proof." Journal of Philosophy 94:16586.
 ^ "in number theory and commutative algebra... in particular the statistical proof of the lemma." [1]
 ^ "Whether constant π (i.e., pi) is normal is a confusing problem without any strict theoretical demonstration except for some statistical proof"" (Derogatory use.)[2]
 ^ "these observations suggest a statistical proof of Goldbach's conjecture with very quickly vanishing probability of failure for large E" [3]
 ^ "What to do with the pictures? Two thoughts surfaced: the first was that they were unpublishable in the standard way, there were no theorems only very suggestive pictures. They furnished convincing evidence for many conjectures and lures to further exploration, but theorems were coins of the realm ant the conventions of that day dictated that journals only published theorems", "David Mumford, Caroline Series and David Wright, "Indra's Pearls, 2002
 ^ "Mandelbrot, working at the IBM Research Laboratory, did some computer simulations for these sets on the reasonable assumption that, if you wanted to prove something, it might be helpful to know the answer ahead of time."A Note on the History of Fractals,
 ^ "... brought home again to Benoit [Mandelbrot] that there was a 'mathematics of the eye', that visualization of a problem was as valid a method as any for finding a solution. Amazingly, he found himself alone with this conjecture. The teaching of mathematics in France was dominated by a handful of dogmatic mathematicians hiding behind the pseudonym 'Bourbaki'... ", Introducing Fractal Geometry, Nigel LesmoirGordon
 ^ Patricio G. Herbst, Establishing a Custom of Proving in American School Geometry: Evolution of the TwoColumn Proof in the Early Twentieth Century, Educational Studies in Mathematics, Vol. 49, No. 3 (2002), pp. 283312,
 ^ Introduction to the TwoColumn Proof, Carol Fisher
Sources[edit]
 "Pólya, G. (1954), "Mathematics and Plausible Reasoning, Princeton University Press.
 Fallis, Don (2002), "What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians", Logique et Analyse, 45: 373–388.
 "Franklin, J.; Daoud, A. (2011), Proof in Mathematics: An Introduction, Kew Books, "ISBN "0646545094.
 Solow, D. (2004), How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, "Wiley, "ISBN "0471680583.
 Velleman, D. (2006), How to Prove It: A Structured Approach, Cambridge University Press, "ISBN "0521675995.
External links[edit]
""  Look up proof in Wiktionary, the free dictionary. 
 Proofs in Mathematics: Simple, Charming and Fallacious
 A lesson about proofs, in a course from Wikiversity