Proof without words
Encyclopedia
In mathematics
, a proof without words is a proof
of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than more formal and 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 able to be generalised.
—more specifically, the perfect square n2—can be demonstrated by a proof without words, as shown on the right. The first square is formed by 1 block; 1 is the first square. The next strip, made of white squares, shows how adding 3 more blocks makes another square: four. The next strip, made of black squares, shows how adding 5 more blocks makes the next square. This process can continue indefinitely.
can be proven without words as shown in the second diagram on right. The two different methods for determining the area of the large square give the relation
between the sides. This proof is more subtle than the above, but still can be considered a proof without words.
can also be proven graphically, as illustrated on the third diagram. The dashed curve along the X axis is the hypothetical distribution of X, while the dashed curve along the Y axis is the corresponding distribution of Y values. Note that the convex mapping Y(X) increasingly "stretches" the distribution for increasing values of X.
runs a regular feature entitled "Proof without words" containing, as the title suggests, proofs without words. The Art of Problem Solving and USAMTS websites run Java applet
s illustrating proofs without words.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a proof without words is a proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. 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...
of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than more formal and mathematically rigorous
Rigour
Rigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism...
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 able to be generalised.
Sum of odd numbers
The statement that the sum of all positive odd numbers up to 2n − 1 is a perfect squareSquare number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
—more specifically, the perfect square n2—can be demonstrated by a proof without words, as shown on the right. The first square is formed by 1 block; 1 is the first square. The next strip, made of white squares, shows how adding 3 more blocks makes another square: four. The next strip, made of black squares, shows how adding 5 more blocks makes the next square. This process can continue indefinitely.
Pythagorean theorem
The Pythagorean theoremPythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
can be proven without words as shown in the second diagram on right. The two different methods for determining the area of the large square give the relation
between the sides. This proof is more subtle than the above, but still can be considered a proof without words.
Jensen's inequality
Jensen's inequalityJensen's inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906. Given its generality, the inequality appears in many forms depending on the context,...
can also be proven graphically, as illustrated on the third diagram. The dashed curve along the X axis is the hypothetical distribution of X, while the dashed curve along the Y axis is the corresponding distribution of Y values. Note that the convex mapping Y(X) increasingly "stretches" the distribution for increasing values of X.
Usage
The College Mathematics JournalCollege Mathematics Journal
The College Mathematics Journal, published by the Mathematical Association of America, is an expository journal aimed at teachers of college mathematics, particular those teaching the first two years. It is a continuation of Two-Year College Mathematics Journal. It covers all aspects of mathematics...
runs a regular feature entitled "Proof without words" containing, as the title suggests, proofs without words. The Art of Problem Solving and USAMTS websites run Java applet
Java applet
A Java applet is an applet delivered to users in the form of Java bytecode. Java applets can run in a Web browser using a Java Virtual Machine , or in Sun's AppletViewer, a stand-alone tool for testing applets...
s illustrating proofs without words.