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If we had eight fingers, would we still count to ten?
If humans had evolved - or were created with - only eight fingers, what would be our counting system? Easy: ONE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, TEN.
If we wanted to count further, we would do it the same way we do it today - almost: eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, twenty . . . fifty six, fifty seven, sixty . . . seven hundred seventy-five, seven hundred seventy-six, seven hundred seventy-seven, eight hundred. All we would do is leave out the numbers eight and nine.
The main thing to remember is that in the Base Eight counting system, 16 would be 14 for us. This would be the only time you would be right in saying to your teacher that five and seven are fourteen.
What if we had 12 fingers? Then we’d go: One, two, three, four, five, six, seven, eight, nine, A, B, ten. (Or, you could substitute A and B with whatever you think we might have called them, like “deck and decun;" the twelfth could be "dozin.”)
Why am I talking about numbering systems with a base that isn’t 10? Because there are non-base-10 systems all around us. Take the computer, for instance. It has only two fingers. How do you figure that? you ask. Because the computer knows only two things: that something either has a current in it, or it does not. Sorry, folks, that’s about the extent of computer’s awareness: something is either on or it isn’t. It’s either charged or it isn’t. Thus, it uses the “Binary” counting system: It goes like this: one, ten, eleven, one hundred, one hundred and one, hundred and ten, one hundred and eleven, one thousand, one thousand and one, one thousand and ten, one thousand and and eleven, one thousand and one hundred, and so on.
So how does a computer know how to calculate or print out a numeral nine, if it doesn’t know anything above one? Because the programmers show it how to make it appear that it knows the difference. The computer recognizes the number nine thusly:
0 0 0 0 1 0 0 1
If you hold up a finger each time you follow my counting in the binary system above, when you get to 1001, you will have nine fingers in the air. A fast way to translate from binary numbers (ones and zeros) to the base ten system, is to make a row of numbers as follows:
128 64 32 16 8 4 2 1
Note that - while going from right to left, and starting with one, each number doubles. Each position in this line represents the same position in the binary system. If a binary number has a zero on one of the eight positions above, then it’s as if the number in the corresponding position on the second row is zero as well. Now, looking at the binary number again, we see that the numbers in the fifth and eighth position are labeled as “1.” Therefore we add together only those numbers in those positions in the second row. Those numbers are 8 and 1. Lo, the result is 9. Any number you wish to have that’s 255 or under can be had by designating all or some of the positions as “1.” Try it, and see if I’m right.
As far as getting the computer to translate from binary to the base 10 system, the computer is programmed (with ones and zeros, ultimately), to go to a certain place in memory, depending on the numbers it’s faced with. The place in memory for a “0” could be represented by the number 26931. This number in the binary system would be:
0 1 1 0 1 0 0 1 0 0 1 1 0 0 1 1
The numbers “1” through “9” would be in each slot above that “address.” So “9” would be nine slots above 26931. So if we wanted a certain arabic number, the programmer would name the address for “0” and tell the computer to go to that address plus the number in question (in this case, 9). Therefore, the computer adds 00001001 to 0110100100110011 (since it IS a computer) and therefore goes to address number 0110100100111100 (26940)
What would the computer find at that address? Well, of course, another bunch of ones and zeros, as it knows nothing else:
At this point, the computer is programmed to print a spot of ink (whether on the screen or on a piece of paper) for every place there is a one. The results:
* * * *
* * * * *
* * * *
These days, computers are capable of making those dots very small, very numerous, and in different colors. In order to get the mega-results we have today, another non-base-ten system was invented: the “Hexadecimal,” which means the inventor thereof has 16 fingers.
That’s how we can see the complex varieties we get nowadays. By the way, each one or zero seen in these examples is a bit, and each row of ones and/or zeros (that consist of eight digits) is a byte. Now you can put a knowing image to all those numbers that describe computers.