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In June 2008 I announced I was writing a book. The working title is 'Living in the Cloud'. It is a book intended to help people understand computers, the internet and other technology they use every day. I plan to have the book finished by the end of 2008, and will post excerpts as I write them for feedback and criticism. The following is one such excerpt:
2 + 2 = 100
Maths these days has largely been outsourced to computers, simply because they are so much better and faster at it, though after waiting for a few minutes while the shop assistant finds a calculator so she can work out what 10% off $100 is you might wonder if that is such a good idea. Nevertheless, for better or worse, so it goes. However, as is the case when outsourcing important work to anyone, it helps to know just enough about how to do it yourself so you can tell if your subcontractor is doing a good job.
Despite the fact computers can only understand ones and zeros, counting in binary isn't as complicated as you might think, though it can take a while for many people to wrap their heads around it at first. While we can all agree that 0 = 0 and 1 = 1, what does 2 look like in binary? Actually, it's 10. Here's how it works:
Our everyday system of decimal numbers is called a Base Ten system. That's because we have ten different numerals from 0-9. When we count, we run through these numerals one at a time until we hit 9, then we go back to 0 and put a 1 on the front. If we keep counting, we'll get to 19 then go back to 0 again and change the 1 to a 2. When we get all the way to 99, we simply go back to 00 and put another 1 on the front to get 100.
Binary is a Base Two system and it works exactly the same way. First 0, then 1. At this point we have already exhausted all our possibilities for numerals, just like when we reach 9 in the decimal system. That means our next move is back to zero and a new column in front. Therefore, in binary, 1 = 1 but 2 = 10. After that, 3 = 11 and 4 = 100.
Another way to look at it is in columns. The column furthest to the right in the decimal system is for single numbers (also called units) from 1-9. The next column is for tens, then hundreds, thousands and so on. If you think about it, each column is 10x the column before it. 100 is 10 x 10 and 10,000 is 10 x 1,000. This is because decimal is a Base Ten system. When we write the number 16 for example, what we are really saying is there is 1 ten and 6 units.
Since binary is a Base Two system, instead of tens, hundreds and thousands, each column represents 2x the column before it, so we end up with units, twos, fours, eights, sixteens and so on. Using the same example, that means instead of saying there is 1 ten and 6 units, we would say there is 1 sixteen and nothing left after that.
In practice this means it takes more to say less in binary, but this is because computers are limited to understanding only the concept of on/off. In a punch card, each 1 would be represented by a hole and each 0 by no hole.
The following graphic shows how to count from 0 - 16 in binary. Notice how it is possible to work out what the binary number is in real terms by adding up the columns with a 1 in them.
Taking the concept further, we can see that the number 167 in binary looks like this:
Or If we were looking at a punch card, like this:
As you can see, counting in binary is not particularly difficult, but it takes a lot of effort to say very little and quickly becomes tedious. Still, next time you see one of those shirts that says ‘There are 10 kinds of people, those who understand binary and those who don’t’, at least you’ll be in on the joke.
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