Mathematical induction is a method for
proving that a statement ''P''(''n'') is true for every
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder:
A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natural numbers ''n'' ≥ ''N''.
The method can be extended to prove statements about more general
well-founded
In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s& ...
structures, such as
trees
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
; this generalization, known as
structural induction Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction over natural nu ...
, is used in
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
and
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
. Mathematical induction in this extended sense is closely related to
recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
. Mathematical induction is an
inference rule
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...
used in
formal proof
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the seque ...
s, and is the foundation of most
correctness proofs for computer programs.
Although its name may suggest otherwise, mathematical induction should not be confused with
inductive reasoning
Inductive reasoning is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' re ...
as used in
philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
(see
Problem of induction
First formulated by David Hume, the problem of induction questions our reasons for believing that the future will resemble the past, or more broadly it questions predictions about unobserved things based on previous observations. This inferen ...
). The mathematical method examines infinitely many cases to prove a general statement, but does so by a finite chain of
deductive reasoning
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fals ...
involving the
variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
''n'', which can take infinitely many values.
History
In 370 BC,
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
's
Parmenides
Parmenides of Elea (; grc-gre, Παρμενίδης ὁ Ἐλεάτης; ) was a pre-Socratic Greek philosopher from Elea in Magna Graecia.
Parmenides was born in the Greek colony of Elea, from a wealthy and illustrious family. His dates a ...
may have contained traces of an early example of an implicit inductive proof. An opposite iterated technique, counting ''down'' rather than up, is found in the
sorites paradox
The sorites paradox (; sometimes known as the paradox of the heap) is a paradox that results from vague predicates. A typical formulation involves a heap of sand, from which grains are removed individually. With the assumption that removing a sing ...
, where it was argued that if 1,000,000 grains of sand formed a heap, and removing one grain from a heap left it a heap, then a single grain of sand (or even no grains) forms a heap.
The earliest implicit proof by mathematical induction is in the ''al-Fakhri'' written by
al-Karaji around 1000 AD, who applied it to
arithmetic sequences to prove the
binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
and properties of
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
.
As Katz says
In India, early implicit proofs by mathematical induction appear in
Bhaskara's "
cyclic method".
None of these ancient mathematicians, however, explicitly stated the induction hypothesis. Another similar case (contrary to what Vacca has written, as Freudenthal carefully showed) was that of
Francesco Maurolico in his ''Arithmeticorum libri duo'' (1575), who used the technique to prove that the sum of the first ''n''
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s is ''n''
2.
The earliest
rigorous
Rigour (British English) or rigor (American English; American and British English spelling differences#-our, -or, see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, su ...
use of induction was by
Gersonides
Levi ben Gershon (1288 – 20 April 1344), better known by his Graecized name as Gersonides, or by his Latinized name Magister Leo Hebraeus, or in Hebrew by the abbreviation of first letters as ''RaLBaG'', was a medieval French Jewish philosoph ...
(1288–1344). The first explicit formulation of the principle of induction was given by
Pascal in his ''Traité du triangle arithmétique'' (1665). Another Frenchman,
Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is ...
, made ample use of a related principle: indirect proof by
infinite descent In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold f ...
.
The induction hypothesis was also employed by the Swiss
Jakob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Leib ...
, and from then on it became well known. The modern formal treatment of the principle came only in the 19th century, with
George Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ire ...
,
Augustus de Morgan,
Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism".
Educated as a chemist and employed as a scientist for t ...
,
Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
, and
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
.
Description
The simplest and most common form of mathematical induction infers that a statement involving a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
(that is, an integer or 1) holds for all values of . The proof consists of two steps:
# The base case (or initial case): prove that the statement holds for 0, or 1.
# The induction step (or inductive step, or step case): prove that for every , if the statement holds for , then it holds for . In other words, assume that the statement holds for some arbitrary natural number , and prove that the statement holds for .
The hypothesis in the induction step, that the statement holds for a particular , is called the induction hypothesis or inductive hypothesis. To prove the induction step, one assumes the induction hypothesis for and then uses this assumption to prove that the statement holds for .
Authors who prefer to define natural numbers to begin at 0 use that value in the base case; those who define natural numbers to begin at 1 use that value.
Examples
Sum of consecutive natural numbers
Mathematical induction can be used to prove the following statement ''P''(''n'') for all natural numbers ''n''.
:
This states a general formula for the sum of the natural numbers less than or equal to a given number; in fact an infinite sequence of statements:
,
,
, etc.
Proposition. For every
,
Proof. Let ''P''(''n'') be the statement
We give a proof by induction on ''n''.
''Base case:'' Show that the statement holds for the smallest natural number ''n'' = 0.
''P''(0) is clearly true:
''Induction step:'' Show that for every ''k ≥'' 0, if ''P''(''k'') holds, then ''P''(''k'' + 1) also holds.
Assume the induction hypothesis that for a particular ''k'', the single case ''n'' = ''k'' holds, meaning ''P''(''k'') is true:
It follows that:
:
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
ically, the right hand side simplifies as:
:
Equating the extreme left hand and right hand sides, we deduce that:
That is, the statement ''P''(''k'' + 1) also holds true, establishing the induction step.
''Conclusion:'' Since both the base case and the induction step have been proved as true, by mathematical induction the statement ''P''(''n'') holds for every natural number ''n''.
∎
A trigonometric inequality
Induction is often used to prove
inequalities
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
. As an example, we prove that
for any
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
and natural number
.
At first glance, it may appear that a more general version,
for any ''real'' numbers
, could be proven without induction; but the case
shows it may be false for non-integer values of
. This suggests we examine the statement specifically for ''natural'' values of
, and induction is the readiest tool.
Proposition. For any
and
,
.
Proof. Fix an arbitrary real number
, and let
be the statement
. We induct on
.
''Base case:'' The calculation
verifies
.
''Induction step:'' We show the
implication for any natural number
. Assume the induction hypothesis: for a given value
, the single case
is true. Using the
angle addition formula and the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
, we deduce:
:
The inequality between the extreme left-hand and right-hand quantities shows that
is true, which completes the induction step.
''Conclusion:'' The proposition
holds for all natural numbers
. ∎
Variants
In practice, proofs by induction are often structured differently, depending on the exact nature of the property to be proven.
All variants of induction are special cases of
transfinite induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.
Induction by cases
Let P(\alpha) be a property defined for a ...
; see
below.
Base case other than 0 or 1
If one wishes to prove a statement, not for all natural numbers, but only for all numbers greater than or equal to a certain number , then the proof by induction consists of the following:
# Showing that the statement holds when .
# Showing that if the statement holds for an arbitrary number , then the same statement also holds for .
This can be used, for example, to show that for .
In this way, one can prove that some statement holds for all , or even for all . This form of mathematical induction is actually a special case of the previous form, because if the statement to be proved is then proving it with these two rules is equivalent with proving for all natural numbers with an induction base case .
Example: forming dollar amounts by coins
Assume an infinite supply of 4- and 5-dollar coins. Induction can be used to prove that any whole amount of dollars greater than or equal to can be formed by a combination of such coins. Let denote the statement " dollars can be formed by a combination of 4- and 5-dollar coins". The proof that is true for all can then be achieved by induction on as follows:
''Base case:'' Showing that holds for is simple: take three 4-dollar coins.
''Induction step:'' Given that holds for some value of (''induction hypothesis''), prove that holds, too. Assume is true for some arbitrary . If there is a solution for dollars that includes at least one 4-dollar coin, replace it by a 5-dollar coin to make dollars. Otherwise, if only 5-dollar coins are used, must be a multiple of 5 and so at least 15; but then we can replace three 5-dollar coins by four 4-dollar coins to make dollars. In each case, is true.
Therefore, by the principle of induction, holds for all , and the proof is complete.
In this example, although also holds for
, the above proof cannot be modified to replace the minimum amount of dollar to any lower value . For , the base case is actually false; for , the second case in the induction step (replacing three 5- by four 4-dollar coins) will not work; let alone for even lower .
Induction on more than one counter
It is sometimes desirable to prove a statement involving two natural numbers, ''n'' and ''m'', by iterating the induction process. That is, one proves a base case and an induction step for ''n'', and in each of those proves a base case and an induction step for ''m''. See, for example, the
proof of commutativity accompanying ''
addition of natural numbers''. More complicated arguments involving three or more counters are also possible.
Infinite descent
The method of infinite descent is a variation of mathematical induction which was used by
Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
. It is used to show that some statement ''Q''(''n'') is false for all natural numbers ''n''. Its traditional form consists of showing that if ''Q''(''n'') is true for some natural number ''n'', it also holds for some strictly smaller natural number ''m''. Because there are no infinite decreasing sequences of natural numbers, this situation would be impossible, thereby showing (
by contradiction) that ''Q''(''n'') cannot be true for any ''n''.
The validity of this method can be verified from the usual principle of mathematical induction. Using mathematical induction on the statement ''P''(''n'') defined as "''Q''(''m'') is false for all natural numbers ''m'' less than or equal to ''n''", it follows that ''P''(''n'') holds for all ''n'', which means that ''Q''(''n'') is false for every natural number ''n''.
Prefix induction
The most common form of proof by mathematical induction requires proving in the induction step that
:
whereupon the induction principle "automates" ''n'' applications of this step in getting from ''P''(0) to ''P''(''n''). This could be called "predecessor induction" because each step proves something about a number from something about that number's predecessor.
A variant of interest in
computational complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) ...
is "prefix induction", in which one proves the following statement in the induction step:
:
or equivalently
:
The induction principle then "automates"
log2 ''n'' applications of this inference in getting from ''P''(0) to ''P''(''n''). In fact, it is called "prefix induction" because each step proves something about a number from something about the "prefix" of that number — as formed by truncating the low bit of its
binary representation
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one).
The base-2 numeral system is a positional notation ...
. It can also be viewed as an application of traditional induction on the length of that binary representation.
If traditional predecessor induction is interpreted computationally as an ''n''-step loop, then prefix induction would correspond to a log-''n''-step loop. Because of that, proofs using prefix induction are "more feasibly constructive" than proofs using predecessor induction.
Predecessor induction can trivially simulate prefix induction on the same statement. Prefix induction can simulate predecessor induction, but only at the cost of making the statement more syntactically complex (adding a
bounded universal quantifier), so the interesting results relating prefix induction to
polynomial-time computation depend on excluding unbounded quantifiers entirely, and limiting the alternation of bounded universal and
existential
Existentialism ( ) is a form of philosophical inquiry that explores the problem of human existence and centers on human thinking, feeling, and acting. Existentialist thinkers frequently explore issues related to the meaning, purpose, and valu ...
quantifiers allowed in the statement.
One can take the idea a step further: one must prove
:
whereupon the induction principle "automates" log log ''n'' applications of this inference in getting from ''P''(0) to ''P''(''n''). This form of induction has been used, analogously, to study log-time parallel computation.
Complete (strong) induction
Another variant, called complete induction, course of values induction or strong induction (in contrast to which the basic form of induction is sometimes known as weak induction), makes the induction step easier to prove by using a stronger hypothesis: one proves the statement
under the assumption that
holds for ''all'' natural numbers
less than
; by contrast, the basic form only assumes
. The name "strong induction" does not mean that this method can prove more than "weak induction", but merely refers to the stronger hypothesis used in the induction step.
In fact, it can be shown that the two methods are actually equivalent, as explained below. In this form of complete induction, one still has to prove the base case,
, and it may even be necessary to prove extra-base cases such as
before the general argument applies, as in the example below of the
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
.
Although the form just described requires one to prove the base case, this is unnecessary if one can prove
(assuming
for all lower
) for all
. This is a special case of
transfinite induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.
Induction by cases
Let P(\alpha) be a property defined for a ...
as described below, although it is no longer equivalent to ordinary induction. In this form the base case is subsumed by the case
, where
is proved with no other
assumed;
this case may need to be handled separately, but sometimes the same argument applies for
and
, making the proof simpler and more elegant.
In this method, however, it is vital to ensure that the proof of
does not implicitly assume that
, e.g. by saying "choose an arbitrary