numbers

Talking Basically About Bases

You may or may not know that we operate in a base-10 number system, which is a beautiful thing. It spares us from the agony of Roman numerals, where years looked something like this:

Instead of like this:

Remember all that talk about place value in elementary school? Ones, tens, hundreds, thousands, and so on? That’s the base-10 idea. Multiply 10 by itself successively, and you get the next place value.

Ten isn’t the only number to base a system on, though. Convenient with our ten-fingered anatomy, but it’s not even the only base we use on a regular basis. Time notoriously operates on non-ten bases. (Makes figuring elapsed time tricky for some students.) And since this particular country refuses to go metric, most of our measurements avoid the ease of base-10.

There are plenty of practical applications for other bases, but when I first learned about them in school, I remember just thinking it was cool to write a number that meant something other than what it looked like. Sort of a mathematical code.

For example, take base-8. We have to reassign all the place values. The ones place is still the ones place. The next place to the left is now the eights place. And the next is the eight-squareds (or sixty-fours) place, followed by the eight-cubeds (or five-hundred-twelves) place. So earlier I mentioned the year 1988. In base-8, that’d be

3 five-hundred-twelves
7 sixty-fours
0 eights
4 ones

So the year 1988 converts to 3704.

If someone gave me a worksheet of numbers to convert to different bases right now, I’d probably be a happy camper working through it.

And that concludes our Yes-I-Am-A-Geek moment for this week.

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Ooh, Look at the Pretty Numbers!

I’m a little OCD. Have I mentioned that before? Not to a degree that it interferes with my life, just noticeable in a few areas. Like when I leave my class with a substitute and everything’s out of place the next day—and that can be something as little as the books on a not-quite-full shelf being shoved to the right instead of the left. *shudder* Annoying.

As a math teacher, it’s only appropriate that one of my little quirks relates to numbers. Some are prettier than others. It’s not that I can’t function when “ugly” numbers come up. I just feel a little warm fuzzy when they’re pretty instead.

So, what are some of my “pretty” numbers? Palindromes are definitely way up there. Those are numbers that read the same forward and backward. When I look at a digital clock right when it’s 12:21 or 8:18? Love it. Catching when my odometer hits one? Love that, too.

Numbers that fall in order or in a pattern are nice, too. Speaking of my odometer, it recently passed 123,456 miles. (My car is well-loved.) That was awesome.

Then there’s my car stereo. The volume increments are pretty small, but anything much over 40 is usually permanent-damage-to-the-hearing range. Within the “safe” range, I get a little weird with settings that are and are not okay … and it has little to do with whether it’s loud or soft enough. In general, prime numbers = yuck. That means even numbers are mostly good, but something like 38 (a prime times two) isn’t as pretty as 35. Multiples of 5 are very pretty, as a rule. Multiples of 3 aren’t bad, either, which means 39 is slightly better than 38, but why not go the extra notch to 40, which is prettier than both combined?

I’m nuts. I know this.

Funny thing is, none of this matches what I mean by “pretty” and “ugly” numbers in my classroom. Rational numbers are pretty. Irrational numbers are ugly. Simple as that. If my students get a pretty answer, they know they should either write the exact decimal or the equivalent fraction. If they get an ugly answer, they should either round it appropriately, or leave it in square-root form (or as a multiple of pi, whatever applies).

I imagine that definition makes a lot more sense. But it doesn’t mean I’m not very much looking forward to twelve minutes after noon on December 12th of this year.

C’mon, guys, ‘fess up. What weird little quirks do you have that make you look just a little bit crazy?

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