Irrational Numbers In Non-Base 10 Systems

Are irrational numbers simply an artifact of the particular numbering system being used or are they common accross numbering systems. For example, is 1/3 (.33 repeating) also irrational in all other numbering systems?

If the base is divisible by 3, then the expansion terminates. For example, 1/3 is 0.4 base 12. If the base is not divisible by 3, then the expansion repeats indefinitely. For example, 1/3 is 0.0101010101010101010101… base 2.

(Note that fractions like 1/3 are called rational numbers. Irrational numbers are those which cannot be expressed as a fraction, and irrational numbers have non-terminating, non-repeating expansions in any integer base.)

1/3 would be rendered as 1/10 in base-3. The decimal form would be .1 .

Some irrational numbers aren’t tamed quite so easily. My understanding is that pi is irrational in any integer-based counting system.

Can’t answer your question, but a quick correction: 1/3 is by definition rational (as in “can be expressed as a ratio”). Any number that repeats can be factored to a ration.

Pi is an example of an irrational number.

I leave the complicated stuff for better minds than mine.


Again, 0.3333 (repeating) is NOT irrational. A rational number is any number that can expressed in the form p/q, p and q integers, q not 0.

1/3 is a rational number. Recall that a number r is rational if there are integers p and q such that r = p/q. 1/3 is rational (p = 1, q = 3).

Irrational numbers are things like sqrt(2), [symbol]p[/symbol], etc. These are irrational in any integer base and have non-terminating, non-repeating representations.

:smack: ***…ratio… ** *

Ah, I get it – ratio, rational, and the light bulb dings.

I learned two things from one question – certainly getting my money’s worth this morning.

1/3 is not an irrational number because it can be expressed as a fraction, albeit not a decimal one. Pi is the standard example of an irrational number.

But regarding your question:

Imagine a base-3 numbering system, where 1 (Decimal) = A, 2 = B, and zero = 0.
3 would be A0, 4 would be AA, 5 would be AB (13 + 21), 6 would be B0 (23 + 01), 7 would be BA (23 + 11), 8 would be BB (23 + 21), and 9 finally would be A00 (1*3²).

More general, every integer can be expressed as x*3[sup]1[/sup] + y 3² + z3³, ad so on, ad infinitum.

This can be done analogously for numbers smaller than one. You could express it as multiples of 3[sup]-1[/sup], 3[sup]-2[/sup] and so on. So in our system, 1/3 ought to be 0.A.

Just want to say how I envy you math people.
I checked this out out of curiosity and I may as well be trying to read hieroglyphics. :frowning:

Irrational numbers are the same across all bases.

I have a truly marvelous demonstration of this proposition which this post is too small to contain.:smiley:

It should be mentioned that, as was said in a previous [thread=322665]GQ thread on irrational bases[/thread], rationality and irrationality are properties of a number while termination is a property of a number’s representation. 1/3 is a rational number since it can be expressed as a ratio of integers. In base 10, its representation is nonterminating (but periodic), while in base 3, it is terminating. We could presumably find bases (irrational bases, I guess) in which the representation of 3 would be neither terminating nor periodic, but the number itself would still be rational.

This is the other side of the “could there be a base pi number system” so that pi would no longer be irrational.

In a base pi system, while pi would be just 10, all integers greater than 3 would have non-terminating representations, e.g. 4 would be something like 10.22012202… (Arithmetic would become extremely hard in such a system).

Are we going to have to wait 300+ years to see it?

Base pi would make for very difficult arithmetic, but some other bases would make arithmetic simpler.

Base-60 is rather convenient for doing hand calculations in because 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12 and 1/15 all have terminating decimal notations. Adding 1/3 + 1/5 is a hassle in base-10, but is really easy in base-60.

Also, for an irrational base numbering system, base-e is pretty useful for lots of things.

When I add 1/3 and 1/5 by hand, I get 8/15.

Maybe, but those multiplication tables in base 60 would be a bear! Every kid would have to rote memorize what (base 10) 37 times 49 is, etc.

[symbol]p[/symbol] is irrational in any representation system. See post #12 for details.

As a general rule, if p and q are different prime numbers, then:
1/p + 1/q = (p+q)/(pq)