The notability of $latex (82,000)_{10}$

The title is meant to read (82,000)_{10}, but WordPress’s Jetpack Plugin’s Beautiful Math component doesn’t execute latex markup in the Title/Subject. I’m still keeping it in the subject, though.

This post is about bases and numbers and elitism and of course, math and meta discussion. If you are interested, please read on!

Okay, first, brace yourself for what I am next about to tell you. I have been known to be… snooty.

There’s a certain kind of numerology that folks I know do, especially skeptics and those familiar with them, that I abhor. And I get a little shirty about it and, to be honest, I probably shouldn’t.  The numerology that I dislike is when folks take arbitrary numbers, dates,  times, and celebrate or play tricks with them to make other numbers and then find joy and solace in these random relationship. The kind I especially don’t like is when we celebrate 3/14/15 as a special \pi day or when we celebrate 11/11/11 11:11 am as a special moment.

I think things like the “There are 10 kinds of people in the world: those who understand binary, and those who don’t.” or the joke relationship of 10/31 to 12/25 or in radix notation, (31)_{8} == (25)_{10}, or in the joke, Oct 31 == Dec 25, or to explain it in mathic English, 31, in base 8, is equal to 25 in base 10, are less gross, but who am I to complain? Because I think 82,000 is a fabulous number and to be honest, it’s not that different from the numerical relationships I’m snobby about.

Almost all the Western world takes base 10 for granted. It’s how we count, it’s how we use calendars and figure time (albeit with limited subsets of the base), it’s how we do arithmetic. It’s so ubiquitous that we don’t even know how to signify we’re using it. We just use it and it and its associated numbers mean that we are using it.

There is a notation for it: It’s called radix. And we can notate it with subscripts and parentheses (as above), so it’s the number itself in the proper radix or base notation, and then the base.  So the radix method of marking out 10 in base 10 is (10)_{10}.

(Note that the assumption for the subscript for the radix notation is also base 10. Because if it were base 2, for instance, then “10” would actually equal 2, in base 10. Or if it were base 4, “10” would equal 4 in base 10.)

The world of bases for numerical values can be so weird and unnerving and disorienting that many folks take a look at it and run like hell back to the familiar world of base/radix 10. But at least for computer geeks like me, we have to know about and be able to count in and use base 2 and base 16 to go about daily business. And to be honest, even web designers need to know base 16, otherwise known as hexadecimal notation, to do colors in HTML And CSS.

Before we go into why 82,000 is such a lovely number, let’s first get the basics down in Base 2, Base 10, and Base 16. I just want to talk about notation and counting. And to ease discussion and not wind my brain around a post while writing about this, let’s just stick to base 10 while writing and thinking about this to start.

So for Base 10, with which you’re already familiar, each place in a whole number has a special meaning. You may not have thought about this explicitly since grade school, but the first digit on the right is the number of ones, the second from the right is the number of 10s, the third is the number of 100s. To put it another way, the 1s are actually 10^0, the 10s are 10^1 and the 100s are 10^2, and so on.

When you write 428, for instance, that’s the same number as 4\times(10^2)+2\times(10^1)+8\times(10^0). The other thing to remember from grade school is that when you run out of numbers in one place, you replace the highest number with a 0 and you carry that 1 over to the next place to the left. So in base 10, when you add 1 to 9, you put 0 in the 9’s place and carry the 1 to the next place up, the 10s place.

For this next part, I’m going to switch between the radix notation and the computational notation for base 2 and base 16, because the computation notation is a little easier and it’s good to know both. For reference, 10 base 2 is equal to radix: (10)_{2} is equal to computational: 0b10, and 10 base 16 is equal to radix: (10)_{16} is equal to computational 0x10. And in review, the decimal (or base 10) value equivalents are: 0b10 = 2 and 0x10 = 16. If you’re stuck here, go back and read or research or figure it out again.

(As an aside, think about what might be done with a negative base. Could we count, perhaps, the same numbers, but occasionally have them be negative? Or does it not work? Base 1 is trivial – we just string 1s together. Base 0 only counts 0.)

Now to base 2. Base 2 is interesting in that it has only 2 digits, 0 and 1. With base 2, you can count any number you can with any other base, it just takes longer and is more fiddly.

Since your places are powers of 2 (2^0 = 1s, 2^1 = 2s, 2^2 = 4s, and so on), breaking down and representing numbers works just like for base 10, but with smaller quantities. How do you represent 10 in base 2? (10)_{10} == (1010)_{2}. or 0b1010. To break it down, it’s 1\times(2^3) + 0\times(2^2) + 1\times(2^1) + 0\times(2^0), or (1\times8) + (0\times4) + (1\times2) + (0\times1) or finally, 8 + 0 + 2 + 0 = 10.

As an aside, base 2, or binary, is the way computers generally count. All those microscopic circuits in memory and CPUs these days are mostly just shuffling around values, stored in binary, to and from various places and comparing them to others.

Good? Okay, let’s move on to base or radix 16.

For base 16, aka hex, or hexadecimal, or of course, radix 16, we need more symbols than just 0 – 9 for each place. We need 6 more, so we take a, b, c, d, e, & f. To count from 0 to 15 in hex, we count… 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, a, b, c, d, e, \& f. And just like in base 2 and base 10, we run out of numbers for the 1s place so we put 0 in the 1s and carry the 1 to the 10s place, and 16, in hex, is 0x10.

Just for review, the places in hex are: 1s (which are 16^0), 16s (which are 16^1), 256s (16^2), 4096s (16^3) and so on. What’s 500 in hex? 0x1F4. To break that down again, for review that’s:

(1\times(16^{2})) + (15\times(16^{1})) + (4\times(16^{0}))

or (1\times256) + (15\times16) + (4\times1)

or, finally, 256 + 240 + 4 = 500.

I hope this is enough introduction or review on bases/radices.

So let’s move on to (82000)_{10}. It’s the 4th in a series of numbers: \{2, 3, 4, 82000, ...?\} The reason there’s a question mark in there is that we don’t yet know whether there’s another number in the series. If there is, it’s bigger than 82,000. And we’ve already checked all numbers up to 2,000 digits long. So we’ve checked up to (10)^{2000}.

How are these numbers a series? Well, 2 can be represented as a mix of 1s and 0s in base 2 – it’s kind of a cheat, but (10)_{2}. And 3 can be represented as a mix of 1s and 0s in base 2 and in base 3 ((11)_{2} and (10)_{3}). And 4 can be represented as a mix of 1s and 0s in base 2, base 3, and base 4 ((100)_{2}, (11)_{3}, and (10)_{4}), and 82000 can be represented as a mix of 1s and 0s in bases 2 through 5:

  • 82000 in base 2: 10100000001010000
  • 82000 in base 3: 11011111001
  • 82000 in base 4: 110001100
  • 82000 in base 5: 10111000

So even in bases with other numerals available for the different places, it’s just zeroes and ones for 82000.

There’s a great deal of this kind of work going on in mathematics and other universities. This kind of almost metamathematics where we check for and find various patterns in meta-properties of numbers, numerals, equations and functions. It’s certainly not all that mathematicians do, but looking for and isolating these patterns has done great things for both theoretical and practical mathematics. After all, without prime numbers, zeta functions, complex numbers, and other mathematical technologies/techniques, we wouldn’t have cryptography or optical physics.

So I think I will try, for now, not to be so snooty about coincidental meta-relationships.

And for your delectation and as a reward for having made it this far, here’s Numberphile’s YouTube coverage of the same topic.

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