Binary Numbers

Here we are, thinking together about math through the internet. Doesn’t it all seem like some kind of magic sometimes? I’ve studied computers quite a bit, and am still mystified by what they can do.

Understanding binary numbers helps you understand how computers (and electronic gadgets in general) work. The difference between the first and third editions of the text is especially amusing in this lesson. In the first edition, Jacobs talks about computers being made of tubes and transistors. I remember radio tubes from when I was a kid. Sometimes we would take a few to the hardware store to test them there, to see which one was bad. They were about 4 inches long, about as big around as a C or D size battery. They were used in the first computers as on-off switches to represent numbers (and everything else).

Each on-off switch is called a bit (short for BInary digIT), and 8 of those is called a byte, and each byte is used to represent a number, a character, or a command. So there are 2 to the 8th or 256 characters available. People who program computers used to work directly with binary code. It was awfully hard to look at long strings of 1s and 0s and make some kind of sense out of them.. But if each byte is broken into 2 pieces of 4 characters each (for example, 1010 1101), then those pieces can be easily (by machine) converted to base sixteen (2 to the 4th), called hexadecimal, and it’s a little easier on the eyes. How does hexadecimal get its 16 different digits, when we only have 10 for normal use? It uses letters as digits. After 0 through 9 there’s A (the digit for ten), B, C, D, E, and F (the digit for fifteen). Sixteen is written as 10. (Don’t worry if you didn’t follow this… I'm trying to add historical color to this lesson, but none of this is necessary to what follows.)

I’ve always thought learning to convert between number bases was a good way to truly understand place value. (I don’t know if I’m right on that…) If you’ve looked at Mayan or Babylonian number systems, they both use other bases. Instead of ten, Mayan uses twenty and Babylonian uses sixty.

Here’s another connection with binary numbers: Have you ever played 20 questions with numbers?
Me: “I’m thinking of a number between 0 and 1023.” (A ten digit binary number.)
Friend: “Is it less than 512?” (Is the first binary digit a 0?)
M: “Yes.”
F: “Is it less than 256?” (Is the second binary digit a 0?)
M: “No.” (It’s a 1.)

If you get 20 questions, how big a span of numbers could you guess between?

I hope I’m making sense…

- Notes by Sue VanHattum