Tuesday, September 05, 2006

The important thing is to understand what you're doing rather than to get the right answer

It won't do you a bit of good to review math
It's so simple
So very simple
That only a child can do it!
--- Tom Lehrer, "New Math"

I had a hangover all day long at work. Ugh.


In Math Concepts, we reviewed last week's material (Venn diagrams and graphs), had the weekly quiz, and then turned our collective attention (such as it was) to numeration systems. Well, really just the one: our current Arabic-Indian-derived numeration system, with its emphasis on place value. That is, in our system, the value of numbers represented by digits is determined by the place in the number. So when you subtract, say,
you start at the right and say, "six minus two is four" --- and then where most people would go on to say, "five minus three is two," you should actually say "fifty minus thirty is twenty," because even though you're writing single digits, what you're really doing is subtracting three tens from five tens. Which is all very elementary, but it's good to be precise in your terms, mathematically speaking.

We also briefly talked about other number systems than base-ten. For example, what we write as 15 in our base-ten system is written, in base-five, as 30(f). Seems simple, because 30 is twice 15, just as 10 is twice 5, but it's not that easy. It's 30(f) because in base-five, fifteen is counted out:
1 2 3 4 10 11 12 13 14 20 21 22 23 24 30, or three groups of five.

So what is 38 in base-five? If 15 is 30(f), does 38 become 76(f)?


It's 123(f). 38 is 25 + 10 + 3. Twenty-five is base-five's 100, so you write that first 25 as 100(f). Ten is two groups of five, so it's 20(f). And three ones is still 3(f). Voila! 123!

Now you can understand Tom Lehrer's song, linked above.

Finally, we went over the properties of addition. The new one on me was the closure property. This says that for every set, if you do a certain operation (say, addition) and the result is a number not in that original set, the set is not closed for that operation. If the result of the operation is only numbers in the set, it's closed.

For example.

The set of {4,9} is an open set for addition, because 4 + 9 = 13 which is not in the set.
The set of {even numbers} is closed for addition, because any even number plus any other even number is always an even number.
The set of {odd numbers} is open for addition, because any odd number plus any odd number is an even number, which is not in the set.

Fascinating stuff, eh? No? ...Fine. Then here is an X-ray of a cell phone inside a prisoner's anal cavity.


Janet said...

Ok, I haven't had my coffee yet this morning, but I am just not grasping this base-5 stuff.

Chance said...

Non omnia possumus omnes.

It is rather confusing, made morte so by the use of base-ten logic to the base-five system (because, as we don't use it, we lack the terms to adequately describe it).

Base-ten has place values of one, ten, and hundred (and so on). Base-five has one, five, and twenty-five (five squared). So our 50 (two 25s) is 200(f) --- or two 100s in base-five. And 55 would be written 210, because 5 is base-five's 10.

Clear as glass, now! :)

A far more perplexing question is why I have to write a word verification to comment on my own blog.