What if Humans Only Had 8 Fingers?

Most cultures developed a base ten counting system because we have, well, ten fingers. Imaging a prehistoric tribe which needs to count its members for the annual census. In order to accomplish the task, each member of the tribe would file past the tribal elders, one by one, while they did the counting. Now, since most prehistoric peoples only had number words for one, two and many, it would be awefully hard to get an accurate count using these few words. So instead, as the first person files past, the Tribal Counter would hold up one finger. As each of the next nine people came by, he would hold up another finger until all 10 of his fingers were raised. At this point, one of two things could happen:

• A second tribal counter could begin holding up each of his fingers for the next ten members, enabling a count up to 20, or
• A second tribal counter could hold up one finger to indicate that the first tribal counter had reached a full count, and then the first tribal counter could lower all of his fingers to prepare to count the next ten members.

If the first method is used, then it would take ten men to count 100 tribe members. If the second method is used, then it would take only two men to count 100 tribe members. The second method is the one that allows us to so neatly express quantities today.

Now, imagine that the tribe has 228 members. After the first tribal counter has run through his fingers ten times, the second tribal counter has all of his fingers raised. If the first tribal counter continues on the the eleventh set of ten, then the second tribal counter will not have enough fingers to keep track of this next group of ten. So, we need a third tribal counter to keep track of how many times the second tribal counter has lifted all of his fingers. When the second tribal counter has reached a full count, the third tribal counter raises one finger, while the second tribal counter returns his fingers to the down position. If you imagine him with closed fists, you can easily see how the number 100 is generated: the first and second counter's have their fists closed while the third has one finger raised. In the end, after all of the members had been counted, the first man would be holding up eight fingers, the second would be holding up two and the third would be holding up two. It would then be easy for us to read and recognize this number as 228.

But what if we only had 8 fingers to count on? What would change about our counting systems? Imagining the same scenarioas above, when the first tribal counter gets to 8, he has run out of fingers and must ask the second tribal counter to hold up one finger, while he returns his to the closed fist position. Now, we would see 10, even though it only represents 8 men. How many times could the first counter raise all of his fingers before using up all of the fingers of the second counter? Eight times, of course. Since each raised finger on the second man represents a group of 8 from the first man, this means that a full count on the second man represents a total of 64 people; when the third man raises his finger and the second man returns his to the closed fist position, we see 100 represented on their hands, but it really only means that 64 people have been counted. Therefore, the first man represents the ones place, the second man represents the eights place and the third man represents the sixty-fours place.

If we think of these numbers using the symbols we currently use to count with, we would see the written numerals as follows:

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, ..., 100, 101, 102, 103, 104, 105, 106, 107, 110, 111, 112...

Now, 10 does not mean ten, but eight.

Of course, it's easy to get confused because we're using the symbols that we use for a base 10 system, so numbers in a base 8 system can't be spoken the same way. A new vocabulary of number words would have to be created to match the meaning of the symbols.

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Suggested Activities

1. The digits used in a base 10 numbering system are 0 through 9. What are the allowable digits in a base 8 system? How about in a base 3 system?
2. How would you create enough symbols to represent all of the allowable digits in a base 16 numbering system? Remember, a digit can only be a single symbol, not a combination of other symbols.
3. What would the place values of a base 3 system be? What would the place values of a base 16 system be?
4. Try counting in base 3. List the first 20 numbers on a sheet of paper. Now list the first 20 numbers in a base 8 system. And finally, list the first 20 numbers in a base 16 system.

Welcome

After years of studying math and math history, I find myself with a large store of information and a limited audience with which to share it. So, it is with a love of math and teaching that I bring to you Math Stuff -- Things You Never Learned in School.... I hope you'll find it to be enjoyable, useful and intriguing.