Code_ The Hidden Language of Computer Hardware and Software - Charles Petzold [22]
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Find the two numbers you wish to add in the top row and the left column. Follow down and across to get the sum. For example, 4 plus 6 equals 10.
Similarly, when you need to multiply two decimal numbers, you follow a somewhat more complicated procedure but still one that breaks down the problem so that you need do nothing more complex than adding or multiplying single-digit decimal numbers. Your early schooling probably also entailed memorizing a multiplication table:
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81
What's best about the positional system of notation isn't how well it works, but how well it works for counting systems not based on ten. Our number system isn't necessarily appropriate for everyone. One big problem with our base-ten system of numbers is that it doesn't have any relevance for cartoon characters. Most cartoon characters have only four fingers on each hand (or paw), so they prefer a number system that's based on eight. Interestingly enough, much of what we know about decimal numbering can be applied to a numbering system more appropriate for our friends in cartoons.
Chapter 8. Alternatives to Ten
Ten is an exceptionally important number to us humans. Ten is the number of fingers and toes most of us have, and we certainly prefer to have all ten of each. Because our fingers are convenient for counting, we humans have adapted an entire number system that's based on the number 10.
As I mentioned in the previous chapter, the number system that we use is called base ten, or decimal. The number system seems so natural to us that it's difficult at first to conceive of alternatives. Indeed, when we see the number 10 we can't help but think that this number refers to this many ducks:
But the only reason that the number 10 refers to this many ducks is that this many ducks is the same as the number of fingers we have. If human beings had a different number of fingers, the way we counted would be different, and 10 would mean something else. That same number 10 could refer to this many ducks:
or this many ducks:
or even this many ducks:
When we get to the point where 10 means just two ducks, we'll be ready to examine how switches, wires, lightbulbs, and relays (and by extension, computers) can represent numbers.
What if human beings had only four fingers on each hand, like cartoon characters? We probably never would have thought to develop a number system based on ten. Instead, we would have considered it normal and natural and sensible and inevitable and incontrovertible and undeniably proper to base our number system on eight. We wouldn't call this a decimal number system. We'd call it an octal number system, or base eight.
If our number system were organized around eight rather than ten, we wouldn't need the symbol that looks like this:
Show this symbol to any cartoon character and you'll get the response, "What's that? What's it for?" And if you think about it a moment, we also wouldn't need the symbol that looks like this:
In the decimal number system, there's no special symbol for ten, so in the octal number system there's no special symbol for eight.
The way we count in the decimal number system is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and then 10. The way we count in the octal number system is 0, 1, 2, 3, 4, 5, 6, 7, and then what? We've run out of symbols. The only thing that makes sense is 10, and that's correct. In octal, the next number after 7 is 10. But this 10 doesn't mean the number of fingers