Read Online But How Do It Know? - the Basic Principles of Computers for Everyone by J Clark Scott - Free Book Online Page B
Authors: J Clark Scott
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ideas in this Arabic system was to have a symbol for zero. This is useful if you want to say that you have zero apples, but it is also a necessary thing to keep the positions of the digits straight. If you have 50 apples or 107 apples, you need the zeros in the numbers to know what position each digit is actually in, so you can multiply by ten the correct number of times.
Now these two ideas in the Arabic number system (the digits and the method) have one thing in common. They both have the number ten associated with them. There are ten different digits, and as you add digits to the left side of a number, each position is worth ten times more than the previous one.
In school, when they first teach children about numbers, they say something about our number system being based on the number ten, because we have ten fingers. So here’s an odd question: What if this number system had been invented by three-toed sloths? They only have three fingers on each hand, and no thumbs. They would have invented a number system with only six digits- 0, 1, 2, 3, 4 and 5. Could this work? If you had eight apples, how would you write it? There is no number ‘8’ in this system. The answer is, that since the first idea, the digits, was changed to only have six digits, then the second idea, the method, would also have to be changed so that as you add positions to the left, each one would have to be multiplied by sixes instead of tens. Then this system would work. As you count your apples, you would say “0, 1, 2, 3, 4, 5…” and then what? There’s no ‘6’ for the next number. Well, according to the method, when you want to go beyond the highest digit, you go back to ‘0’ and add a ‘1’ to the left. OK, “0, 1, 2, 3, 4, 5, 10, 11, 12.” Now you have counted all of your apples. What would this ‘12’ mean? It would be this many:. I guess you’d call it eight, but you’d write it ‘12’. Very odd, but it does work out - 1 times six plus two equals eight apples, it follows the Arabic method, but it is based on six instead of ten. If you continued counting up, when you got to ‘15,’ which is(one times six plus five,) the next number would be ‘20,’ but the ‘2’ would mean two sixes, or this many:. And 55 would be followed by 100. The ‘1’ in that third position would be how many ‘36’s there were (six times six)
This is a very odd number system, but guess what, you already use it in your everyday life. Yes, think of the way we write time, or the kind of clock that shows the numbers on its face. The right digit of the minutes and seconds follows our normal numbers, 0-9, 0-9, over and over. But the left digit of the minutes and seconds only goes 0-5. After 59 minutes, the clock goes to the next hour and 00 minutes. There are 60 minutes in an hour, numbered from 00 to 59. That left position never gets over 5. That position uses the number system based on six symbols (0-5). The hour part of the clock tells how many ‘60’s there are, though you will never see a 60 on the face of the clock. And you are so used to this that you don’t have to think about it. When the clock says 1:30, you know that this is halfway between 1:00 and 2:00, you don’t have to do any math in your head to figure it out. Have you ever had to add time? If you add 40 minutes and 40 minutes, you get 80 minutes, but to write that down in hours and minutes, you have to figure out how many 60s there are in 80, in this case 1, then figure out how many minutes there are beyond 60, in this case 20. So you write 1:20. The 1 represents 60 minutes, add 20 and you have your 80 again. So this is pretty complicated, two different number systems in the same number! But you have already been using it your whole life.
The hour positions are even stranger. On a 12 hour clock, it skips zero and goes 1-12 AM, then 1-12 PM. On a 24 hour clock, it goes from 00-23. We won’t try to analyze these. The point we wanted to make was that you are already familiar
Now these two ideas in the Arabic number system (the digits and the method) have one thing in common. They both have the number ten associated with them. There are ten different digits, and as you add digits to the left side of a number, each position is worth ten times more than the previous one.
In school, when they first teach children about numbers, they say something about our number system being based on the number ten, because we have ten fingers. So here’s an odd question: What if this number system had been invented by three-toed sloths? They only have three fingers on each hand, and no thumbs. They would have invented a number system with only six digits- 0, 1, 2, 3, 4 and 5. Could this work? If you had eight apples, how would you write it? There is no number ‘8’ in this system. The answer is, that since the first idea, the digits, was changed to only have six digits, then the second idea, the method, would also have to be changed so that as you add positions to the left, each one would have to be multiplied by sixes instead of tens. Then this system would work. As you count your apples, you would say “0, 1, 2, 3, 4, 5…” and then what? There’s no ‘6’ for the next number. Well, according to the method, when you want to go beyond the highest digit, you go back to ‘0’ and add a ‘1’ to the left. OK, “0, 1, 2, 3, 4, 5, 10, 11, 12.” Now you have counted all of your apples. What would this ‘12’ mean? It would be this many:. I guess you’d call it eight, but you’d write it ‘12’. Very odd, but it does work out - 1 times six plus two equals eight apples, it follows the Arabic method, but it is based on six instead of ten. If you continued counting up, when you got to ‘15,’ which is(one times six plus five,) the next number would be ‘20,’ but the ‘2’ would mean two sixes, or this many:. And 55 would be followed by 100. The ‘1’ in that third position would be how many ‘36’s there were (six times six)
This is a very odd number system, but guess what, you already use it in your everyday life. Yes, think of the way we write time, or the kind of clock that shows the numbers on its face. The right digit of the minutes and seconds follows our normal numbers, 0-9, 0-9, over and over. But the left digit of the minutes and seconds only goes 0-5. After 59 minutes, the clock goes to the next hour and 00 minutes. There are 60 minutes in an hour, numbered from 00 to 59. That left position never gets over 5. That position uses the number system based on six symbols (0-5). The hour part of the clock tells how many ‘60’s there are, though you will never see a 60 on the face of the clock. And you are so used to this that you don’t have to think about it. When the clock says 1:30, you know that this is halfway between 1:00 and 2:00, you don’t have to do any math in your head to figure it out. Have you ever had to add time? If you add 40 minutes and 40 minutes, you get 80 minutes, but to write that down in hours and minutes, you have to figure out how many 60s there are in 80, in this case 1, then figure out how many minutes there are beyond 60, in this case 20. So you write 1:20. The 1 represents 60 minutes, add 20 and you have your 80 again. So this is pretty complicated, two different number systems in the same number! But you have already been using it your whole life.
The hour positions are even stranger. On a 12 hour clock, it skips zero and goes 1-12 AM, then 1-12 PM. On a 24 hour clock, it goes from 00-23. We won’t try to analyze these. The point we wanted to make was that you are already familiar
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