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Mathematics is the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations. It evolved through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the systematic study of positions, shapes and motions of physical objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

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The second Borel-Cantelli lemma implies that a chimpanzee like this one typing at random will almost surely produce the complete works of Shakespeare, given enough time.

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type or create a particular chosen text, such as the complete works of William Shakespeare. Note that "almost surely" in this context is a mathematical term with a specific meaning, and that the "monkey" is not an actual monkey; rather, it is a vivid metaphor for an abstract device that produces a large, random sequence of letters.

The theorem graphically illustrates the perils of reasoning about infinity by imagining a vast but finite number. If every atom in the visible universe were a monkey producing a billion keystrokes a second from the Big Bang until today, it is still very unlikely that any monkey would get as far as "slings and arrows" in Hamlet's most famous soliloquy. The infinite monkey theorem is straightforward to prove, even without appealing to more advanced results.

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Image credit: User:Chris 73

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Credit: Jtico

In projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:

In a projective space, two triangles are in perspective axially if and only if they are in perspective centrally.

The above picture illustrates Desargues' theorem. Another important feature of projective geometry noticable in the picture is all lines meet at exactly one point (e.g. there are no parallel lines).

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...that a sphere can be cut up and reassembled into two spheres the same size as the original (Banach-Tarski paradox)?
...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitrary quintic equation?
...that Euler found 59 more amicable numbers while for 2000 years, only 3 pairs had been found before him?
...that you cannot knot strings in 4-dimensions? You can, however, knot 2-dimensional surfaces like spheres.
...that there are 6 unsolved mathematics problems whose solutions will earn you one million US dollars each?
...that there are different sizes of infinite sets in set theory? More precisely, not all infinite cardinal numbers are equal?
...that every natural number can be written as the sum of four squares?
...that the largest known prime is over 9 million digits long?
...that the set of rational numbers is equal in size to the subset of integers; that is, they can be put in one-to-one correspondence?

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