Most real numbers are unknown—even to mathematicians
What is the most bizarre real number that you can imagine? Probably many people think of an irrational number such as pi or Euler’s number. And indeed, such values can be considered “wild.” After all, their decimal representation is infinite, with no digits ever repeating. Even such bonkers-looking numbers, however, together with all the rational numbers, make up only a tiny fraction of the real numbers, or numbers that can appear along a number line.
What Numbers Are Irrationals? All real numbers that cannot be represented by a fraction of two integers are irrational. Irrational numbers include, for example, the square root of 2, whose decimal representation is infinite without ever repeating. In fact, √2 is among the simplest irrational numbers because it is constructible—that is, it can be generated with a compass and ruler by drawing a right triangle with two sides that have a length of one unit.
Noncomputable Numbers Are Even Stranger Until the early 20th century, people assumed that transcendental numbers were the wildest thing that real numbers had to offer. But that was wrong. In 1937 British mathematician Alan Turing published a paper on computable numbers. He used this term to describe all those values for which there is a calculation rule that a computer can perform to calculate the numerical value with any degree of accuracy.
Cantor’s reasoning went as follows: Suppose one has a list of all the real numbers. Then one can imagine this list as a table. In each row, there is a number, with each column offering a position for a decimal place. Cantor demonstrated that if you draw a circle around a set of numbers forming a diagonal across this table , you can create a new real number by adding 1 to each diagonal entry. This new number cannot be contained in the list.
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