## Saturday, August 29, 2009

### Carnival of Mathematics #56

It's up at Reasonable Deviations. But much of it is above my head. The entries that looked interesting to me were:
• Rick Regan on decimal (base ten) numbers like 999 having binary representations that end in 111.
• Math Alive (Look at the lecture notes for pdfs.)
• Computer Science Unplugged (I downloaded a 111 page pdf of what look like great activities for young kids.)
• I'm not fond of war studies, but the German Tank problem was a very interesting application of statistics, which I'm likely to use in my stat classes.
• Not part of the carnival, but up at the top of the blog, was the previous post title, on a binary marble adding machine, made out of wood.
(It seemed the polite thing to do was to leave out the links, so you'd go visit RD to see these, and give them more traffic.)

If binary numbers are considered a part of computer science, then 60% of what I could really follow was the computer science material. Hmm...

1. Sue -- just one quibble with your wording "on decimals like 999...". I talk about integers written in decimal (AKA base 10), not decimals as in decimal fractions.

(And BTW, thanks for mentioning my article.)

2. Good point. I meant numbers in the decimal system, but it's not clear. I wonder what decimal fractions (like .99) would look like as binary fractions.

3. (edited to fix the confusing wording)

4. Most decimal fractions are infinite repeating in binary. 0.99 is 0.11(11110101110000101000), where the part in parentheses is repeating.

You can play with these yourself. I've written a converter that can be found at: http://www.exploringbinary.com/binary-converter/ (it doesn't mark the repeating part -- you'll have to infer that yourself).

5. Hmm, I'm wondering if there's any easy way to do the conversion process. I tried working on 1/10, and it seemed hideous.

I said (to myself) 1/10 is smaller than 1/2, 1/4, and 1/8, so there are 3 leading 0s. It's bigger than 1/16, so we have a 1 now, and I have to find 1/10-1/16. That equals 3/80, which is bigger than 1/32, so we get another 1, and need to find 3/80-1/32. That comes out more nicely than I expected, to 1/160.

I wasn't seeing any pattern and stopped that. Went to Wolphram alpha. Yep, I see .99, as you say, has 20 digits in its repeating pattern. But .9 only has 4. Hmm...

6. Good line of thought, but as you found out, the "subtract largest power of two" method is a lot uglier for fractions than whole numbers.

The simple method is "repeated multiplication by 2":

0.1 * 2 = 0.2
0.2 * 2 = 0.4
0.4 * 2 = 0.8
0.8 * 2 = 1.6
0.6 * 2 = 1.2
0.2 * 2 ... repeats

The answer is the integral parts of the products, which are subtracted out along the way: 0.00011..., or to use my repeating notation, 0.0(0011).

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