Irrational Bases and the Phi-cimal Number System // Math Minute [#43] [NUMBER THEORY] [ALGEBRA]
In Math Minute #42 ( • The One about 0.999… Repeating Equals 1 //... ), we discussed how rational numbers are, among other things, numbers that will terminate as a decimal in some rational base. But discussions of rational bases bring up the very reasonable question: what on earth would we mean by an irrational base? How would we ever represent numbers in an irrational base number system? And finally, we explore one particularly interesting irrational base: the base phi number system.
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Typically, a number system does two things for you: it uniquely describes any particular value uses powers of the base of the number system, and using symbols that are all the non-negative integers less than that base. So in the decimal number system, for example, we describe values using powers of 10, and we use the symbols (called digits) 0–9. In a binary number system, we describe values using powers of 2, and we use the bits 0–1.
Although the values we're used to representing (integer and rational numbers) are most easily represented in rational base number systems, it is also possible to describe them with other algebraic numbers. Today's video describes a "phicimal" or "phinary" number system, in which the base we use is the mathematical constant phi (related to the golden ratio).
Vi Hart released a series of videos on phi, the Fibonacci sequence, and other related questions several years ago. The one I quoted from in this video can be found here: • Video .
Finally, if you were intrigued by some of the questions left for the viewer, Jim Propp has an excellent discussion of possible imaginary or complex base number systems on his website here: https://mathenchant.wordpress.com/202....
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