In
mathematics, a proof without words (or visual proof) is an illustration of an
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
or mathematical statement which can be demonstrated as
self-evident
In epistemology (theory of knowledge), a self-evident proposition is a proposition that is known to be true by understanding its meaning without proof, and/or by ordinary human reason.
Some epistemologists deny that any proposition can be self-e ...
by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or
mathematically rigorous proofs due to their self-evident nature.
When the diagram demonstrates a particular case of a general statement, to be a proof, it must be generalisable.
A proof without words is not the same as a
mathematical proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...
, because it omits the details of the logical argument it illustrates. However, it can provide valuable intuitions to the viewer that can help them formulate or better understand a true proof.
Examples
Sum of odd numbers
The statement that the sum of all positive
odd numbers up to 2''n'' − 1 is a
perfect square—more specifically, the perfect square ''n''
2—can be demonstrated by a proof without words.
In one corner of a grid, a single block represents 1, the first square. That can be wrapped on two sides by a strip of three blocks (the next odd number) to make a 2 × 2 block: 4, the second square. Adding a further five blocks makes a 3 × 3 block: 9, the third square. This process can be continued indefinitely.
Pythagorean theorem
The
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
that
can be proven without words.
One method of doing so is to visualise a larger square of sides
, with four right-angled triangles of sides
,
and
in its corners, such that the space in the middle is a diagonal square with an area of
. The four triangles can be rearranged within the larger square to split its unused space into two squares of
and
.
Jensen's inequality
Jensen's inequality can also be proven graphically. A dashed curve along the ''X'' axis is the hypothetical distribution of ''X'', while a dashed curve along the ''Y'' axis is the corresponding distribution of ''Y'' values. The convex mapping ''Y''(''X'') increasingly "stretches" the distribution for increasing values of ''X''.
Usage
''
Mathematics Magazine
''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a j ...
'' and the ''
College Mathematics Journal
The ''College Mathematics Journal'' is an expository magazine aimed at teachers of college mathematics, particular those teaching the first two years. It is published by Taylor & Francis on behalf of the Mathematical Association of America and is ...
'' run a regular feature titled "Proof without words" containing, as the title suggests, proofs without words.
The Art of Problem Solving and
USAMTS websites run
Java applet
Java applets were small applications written in the Java programming language, or another programming language that compiles to Java bytecode, and delivered to users in the form of Java bytecode. The user launched the Java applet from a ...
s illustrating proofs without words.
Compared to formal proofs
For a proof to be accepted by the mathematical community, it must
logically show how the statement it aims to prove follows totally and inevitably from a set of
assumptions. A proof without words might imply such an argument, but it does not make one directly, so it cannot take the place of a formal proof where one is required. Rather, mathematicians use proofs without words as illustrations and teaching aids for ideas that have already been proven formally.
See also
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Notes
References
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{{Mathematical logic
Articles containing proofs
Mathematical proofs