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All Your Base Are Belongs To Us
Tuesday July 25, 2006 by Noam Samuel

Numbers. You see them all the time. 3, 27, 35, 91. They seem to be embedded into modern life. We rarely, however, stop and think about numbers such as XXII, ?”? or 2A. Indeed, in today’s world, it is easy to forget that modern numerical notation is just as its name suggests, a notation. There are countless ways to represent one number. In fact, there is an infinite amount of numerical notations.

Today, I am going to speak of a countably infinite subset of the set of numerical notations: The numerical bases. Numerical bases are representations of numbers based on a defined set of symols and of the powers of a certain number.

For example, let’s start with decimal notation, or, in its lesser known name, base 10. The last digit of any number represented in any numerical base is always multiplied by 1, or , where B is the base number. The second to last digit is multiplied by 10, which is also, considering that B=10, . The third to last is multiplied by 100, or . Thus, on and on, obviously adding together the results in the end.

This principal can be generalized for any natural number. The general formula is (An addition of , , and so on), were N is the number represented in base B, and is the number represented by the mth digit, counting from the end of the number and strarting with 0 (for 125, , and ). It’s important to note that even though I am using infinite sum notation, since integers have a finite number of digits, you can say that , and thus it behaves like a finite sum).

Each base B must have exactly B possible digits, each representing numbers from 0 to B-1. For example, in base 2 (binary), the only possible digits are 0 and 1. These two rules, put together, guarantee that each number can be represented in every base, and that there is only one representation of that number per each base (the proof for that statement warrants an entirely different article).

So, for example, there are many ways to represent the number 36. In base 2, it’s 100100. In base 3, however, it would be 1100. In base 4, it would be 210. In base 5, it would be 121. In base 6, it would be 100. In base 7, it would be 51. In base 8, it would be 44. In base 9, it would be 40, and in decimal it would be 36.

Finally, I have a riddle for you: How would you go about converting from one base to another. There are many way, each with its own benefit. The person whose comment best answers my question will appear in the next post on the subject, “Fun With Bases.”

P.S.
Sorry about the lack of posts lately, we are now going to return to the normal posting rate.

  1. You chickened out on the proof? Shame…

    And is the class of numerical notations actually a set? (*ominous music*)

    But yes, I’m glad to see the underground revived.


    — Patrick    Jul 25, 03:11 PM    #

  2. 1. Yes, I just couldn’t write it in a way that would be good, understandable, through, AND fit into a reasonably short article. I do have a version of the proofs in an internal (hidden) article, but it’s crap.

    2. Yes. Either something is a number or isn’t. Thus, there must exist a property of ‘numberism’. Thus, there must exist a set of numbers. QED.

    (Now let’s have all the logicians kill me at once)

    3.So am I, friend, so am I.


    — Noam Samuel    Jul 25, 05:59 PM    #

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