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Getting To Know the Unknowable Number (More or Less)

Only an infinite mind could calculate each bit

In this week’s podcast, “The Chaitin Interview IV: Knowability and Unknowability,” Walter Bradley Center director Robert J. Marks interviewed mathematician Gregory Chaitin on his discovery of the “unknowable number.” How can a number that is unknowable exist? Some numbers go on indefinitely (.999999999… ) but we can describe them accurately even if they don’t seem to come to an end anywhere. Some numbers, like pi (π), are irrational — pi goes on and on but its digits form no pattern. However, what does it mean to say that a number exists if it is unknowable? How do we even know it exists? That’s the topic of this series, based on the fourth podcast between Dr. Marks and Gregory Chaitin. Note:…

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Most Real Numbers Are Not Real, or Not in the Way You Think

Typical real numbers contain an encoding of all of the books in the US Library of Congress

Pick a random real number between zero and one. The number you choose, with probability one, will contain an encoding of all of the books in the US Library of Congress. This sounds absurd, but real numbers require infinite precision and every time you deal with the infinite, things get absurd. Infinities, including the infinite number of digits to express almost every real number, don’t exist. Curiously then, real numbers are not real. How do we choose a random number between zero and one? The easiest way to explain is using binary decimals. The binary number 0.1000… with zeros forever denotes the number ½ or, in base 10 notation, 0.5. The binary decimal 0.01000… with zeros forever is the number…

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Can We Add New Numbers to Mathematics?

We can work with hyperreal numbers using conventional methods. It could start in high school

Sometimes mathematics is moved forward by the discovery of new formulas and solutions to problems. However, sometimes mathematics grows by adding new kinds of numbers to the number system. In the early days of mathematics, it was thought that whole numbers were the only kind that existed. Sure, there were fractions, but fractions are merely ratios of whole numbers. It was thought that every possible number could be written in terms of whole numbers. These numbers were called rational numbers because they could be written as a ratio. There is a story about a Greek philosopher, Hippasus who discovered, roughly 2500 years ago, that certain numbers (specifically the square root of two) could not be written in terms of ratios…