Counting in Binary
We start each class with a small exercise. This week the students need to count in binary from 0 to 15, padded out to 4 digits. As they say the number, I write it on the board.
I start with '0000' and then we go around the room with each student giving the next number: '0001', '0010', '0011', '0100', '0101', ...
The reason why we start with this counting (padded to 4 digits) is that they are actually practicing the binary-hex conversion table. They haven't learned hexadecimal yet, but we'll be covering that today.
[Note: After a week or two of this, all the students know it pretty well, so they all just say the numbers in unison.Binary-Decimal Conversion
Once another teacher overheard this exercise and was somewhat surprised to hear the students intoning "zero-zero-one-zero-one-one-...". They were all saying the numbers in a monotone and it sounded like they were chanting. I'll have to remember to dim the lights and light a few candles and incense for next time.]
"As we discussed last week, binary is for computers, and decimal is for humans. People can have a difficult time dealing with binary numbers directly, so it's good to know how to convert back and forth between the two number systems."
"Thus, we continue with some worksheets showing how to convert between decimal and binary:"
Worksheet: Converting from Binary to Decimal
Worksheet: Converting from Decimal to Binary
"The process isn't difficult, but it does take time and is hard to do in your head for anything but the most trivial conversions."
Rembering Binary Numbers
"In addition, binary numbers can be rather long and hard to remember. For example, I'm going to write a number on the board and ask you to remember it."
"For comparison, let's start with decmial. I'm going to write a number on the board:"
I write the following decimal number on the board:
571,842After about 5 seconds, I cover it up and then ask if anyone remembers the number. Pretty much anyone in the class is able to remember the number.
"Now let's try the same this with a binary number of roughly the same magnitude."
I write the following binary number on the board:
10100000010011110101As before, I cover it up and then ask if anyone can remember the number. I like to be a bit mean here and ask things like "Hmmmm... does anyone remember... was it 001001000 or 101101110?", using random sequences of '0's and '1's.
Now, with the binary number still covered up, I write the number again on the board. When I uncover the original number, everyone can see that they're the same number.
"So, how did I remember that long binary number? Well, I cheated. And I'll be showing you in this class what I did."
[Note: Of course, the trick is that I didn't really remember the binary number. I just thought of a 5-digit hex number and remembered that instead. When I needed to write the number, I just converted from hex to binary as I was writing. In the above example, I remembered the "A04F5" instead of the long string of binary digits.]Hexadecimal
"OK, so binary numbers are tedious to use directly, and there's no trivial way to convert them into decimal. What are we going to do?"
"The answer is hexadecimal, which is base 16."
"Hexadecimal is just like all the other number systems we've discussed except that we need 16 symbols. We re-use 0-9 from decimal, but we need 6 more symbols. We can choose anything, but by convention we use the first 6 letters of the alphabet:"
| Hex number: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F |
| Decimal equivalent: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
[Sidenote: Sexadecimal?Worksheet: Counting in Hexadecimal
Positional number systems are typically named using the Latin prefixes, hence octal (oct- = 8), decimal (deci- = 10) and duodecimal (duodeci- = 12). Given this convention, base-16 is more correctly referred to as sexadecimal since sexa- is the Latin prefix for 6 (as in sexagesimal for base-60).
The term hexadecimal is an unnatural combination of Greek (hexa- = 6) and Latin (deci- = 10) prefixes, but has gained acceptance as the preferred name for base-16.
This is probably a good thing, considering how often I tend to abbreviate the number system as "hex".]
"So, how does hexadecimal help us with binary? Let's compare the different number systems."
Activity: Number cards
First lay out the binary cards from right to left:
| Binary: | 256 | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
Now line up the octal cards with the binary cards:
| Binary: | 256 | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| Octal: | 64 | 8 | 1 |
Then add the hexadecimal cards:
| Hex: | 256 | 16 | 1 | ||||||
| Binary: | 256 | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| Octal: | 64 | 8 | 1 |
Now try to line up the decimal cards:
| Decimal: | 100 | 10 | 1 | ||||||
| Hex: | 256 | 16 | 1 | ||||||
| Binary: | 256 | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| Octal: | 64 | 8 | 1 | ||||||
Notice the following things:
- Octal lines up with every 3rd binary card
- Hexadecimal lines up with every 4th binary card
- Decimal doesn't line up with any binary card (except 1)
Worksheet: Converting from binary to octal
Worksheet: Converting from binary to hexadecimal
"So this solves one of the problems with binary - that there were too many digits to remember. It's still a bit of work to convert to/from decimal, but we can avoid this by sticking with hexadecimal as much as possible."
"Note that we just as easily could have used octal instead of hexadecimal to solve this problem. In fact, many early computer systems made extensive use of octal and grouped bits in sets of 3 (and had byte sizes of 6 or 9 bits). While there are a few places where octal is still used in computer science, hexadecimal is by far the preferred base to convert binary. Besides, grouping 4 bits at a time seems more "power of two"-ish than grouping them by 3's."
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