How would math change if humans were born naturally with six fingers on each hand instead of the usual five?

My seventh-grade math teacher asked us what was essentially the same question. She held up three fingers on each hand and asked us how we would count if this were what our hands were like.

She let us puzzle over it for a while. Finally she explained that “this many” (the number we represent with the symbol “6”) would be “ten” in our counting system. We’d be operating in base 6. She taught us to count and do arithmetic in base 6, base 8, and base 16, and, most interesting of all, base 2 (binary). She taught us what zero means.

Earlier, she had taught us the difference between a number and a numeral and said that “this many” (holding up fingers) was a number, while a numeral was only a symbol that stood for that many. By the time she got to different bases, we were ready for the abstraction.

This was in 1958.

So yes, I think “this many” (the value we represent with the symbols we call “twelve”) would be equal to the base of our arithmetic system, which we call “ten.” A full column on the abacus would have “this many” beads:

o o o o o o o o o o o o

and when we filled the column, we would call it “ten,” it would end with a zero, and we would move over one row.

Math would be the same. Humans might however show a greater appreciation for Base 12 numbers.

Math was discovered, base ten was invented.

Well… It wouldn’t do to have us born with only one finger on each hand, because there are 10 types of people in the world: those who understand binary, and those who don’t. Also, you wouldn’t be able to gesture in any non-obscene way.

Maybe 8 fingers would be nice. We could count in base 2, octal, 10, or hex.

Who knows. It may be different, but It may also be the same.
The first number system used by humans was base60, used by the babylonians, who probably did not have 60 fingers.

@ETpro How would our fingers be useful to count in or to indicate numbers in Hexadecimal (apart from using writing instruments or typing?

@Dr_Lawrence If you had 8 fingers per hand, you could count on your fingers in base 16 or 8 just as easily as you can count in base to with our current 5-finger norm.

1, 2, 3, 4, 5, 6, 7, 8
9, A, B, C, D, E, F

We would probably have had a base twelve duodecimal instead of a decimal system. It would have the slight advantage that, since 12 is divisible by 3, ⅓ and 1/6 would be represented as .4 and .2 instead of repeating decimals, although 1/10 and 1/5 would now be repeating decimals. I remember being surprised, when learning about different bases, that there is nothing special about the number 10, other than that we have 10 fingers.

I’m thinking we just would fold our thumbs in and not count those. Base 12 sucks, but we certainly use it in America every day. 12 inches in a foot. Oh, and two 12 hours sessions make up a day.

Meanwhile, how many digits I have on my hand really has nothing to do with how I do math. I use as many fingers as I need to help me with a math problem, if it is even feasable to do it on my hands.

12 sucks? Why? Would 10 eggs be better than a dozen? Would John Holmes be more impressive with 10 inches than 12? What’s so special about 10, aside from Bo Derek once upon a time? If she was a 10, who would rate 12?

@JLeslie, that’s not a base 12 system.

I forgot about the eggs. Although, a baker’s dozen is 13.

12, 8, 16. America’s aversion to ten makes our lives more difficult mathematically. In Metric, base 10, you can easily just move a decimal point, rather than divide and multiply by 8 or 12. Figuring how many meters in a kilometer is way easier than figuring how many yards in a mile. Unless you have it memorized of course.

@Jeruba What isn’t?

@JLeslie Other base counting systems work exactly the same. They are equally easy to move decimal points in. You’re just used to base 10.

Math has a very specific logic, and it could not exist without said logic, so my guess is that it would be the same as it is now.

@ETpro We use base 10 for decimals, but we have to convert. For instance on a time card if we work one hour and 30 minutes, writing 1.3 hours isn’t accurate, unless the computer system the company uses is programmed to understand it correctly. Back in my teens I was getting insufficient hours because my manager was inputing our time incorrectly, so I have a history with that. There were about 15 employees working there, only I noticed it. Not even because of the size of my paycheck, but because of how she totalled the hours on the time sheet. Corporate hadn’t caught the mistake either.

Another example 1 foot 5 inches is not 1.5 feet. We use a base ten decimal system.

The number of inches in a foot and the number of eggs in a dozen are not examples of base 12. We still count them using base 10.

Here is how you count in base 2:

1
10
11
100
101
110
111
1000

In base 2 (binary), the number written as 100 equals this many:

o o o o

100 is always the square of the base (which is written “10”); that is, raised to the power of 2.

The number written as 1000 equals this many:

o o o o o o o o

1000 is always the cube of the base; that is, raised to the power of 2.

In base 12, the square of the base would amount to what we know in the decimal (base 10) system as “144” but it would be written “100.”

(If there is something amiss in my understanding of these concepts after all these years, I’m sure a math person will correct me.)

We would count in base 12 – the Duodecimal system.

The Duodecimal Society of Ameraca , now called the Dozxenal Society of America believes Base 12 still offers many advantages and should be the standard.

Base 12 would make fractional numbers so much better. It would be easy to divide a whole into: half, third, quarter and sixth parts of a whole There would be fewer repeating decimals.
Say the base 12 numbering system was 0,1,2,3,4,5,6,7,8,9,A,B,
@jeruba spoke about whole numbers 10 would be 12base10, 100 would be 144 base 10
(For the rest of this answer I will define unlabelled numbers as Base 12 and use whisper mode for Base 10, e.g. 10 .
But let’s look at fractions and the Duodecimal representation. They are great.
½ would be 0.6 exactly
⅓ would be 0.4 exactly
¼ would be 0.3 exactly
1/6 would be 0.2 exactly
Sure there would be some repeating decimals but the most common ones would be simple

For units of time we might modify the second so there are 100 seconds 144 seconds in a minute and 144 minutes in an hour. There would be 10 hours in a day. How many seconds in a day? 1000. Easy. How many in half a day? 600. Easy. We’d learn to think that way quickly.

@LuckyGuy @Jeruba I understand all that. The math you present all makes sense to me.

I also remember finding answers in various bases in my 7th grade math class. The would give a number like 18, and ask for the answer in base 12 for instance. The answer would be 6, unless I am remembering incorrectly. 7 in base 6 was 1. To me the 12 hour clock is base 12. When we convert from feet to inches that is in base 12. When I convert pounds to ounces base 16. Am I wrong?

@JLeslie , You are mixing up change of base with modular arithmetic. What you are describing is modular arithmetic, which is to take the remainder of a number after dividing by the modulus. 18 mod 12 = 6, 7 mod 6 = 1 and so on.

Changing bases is a whole different matter. Let me jog your memory on this. Consider counting in the decimal system. You start with the numbers 1 to 9. When we get to ten, we write a 1 in the tens column followed by 0. The various columns stand for powers of ten.

If we write numbers in base 8, we start with the numbers 1 to 7. We write 8 as 10. The various columns stand for powers of 8.

The most widely used base other than ten is base 2 binary numbers, consisting of ones and zeros, because that is how computers represent numbers.

Base 12 is the same idea, but we would need to create symbols for ten and eleven so that we could count to eleven with single digits before switching to 10 for 12.

@JLeslie. In Base 12 there would be 10 inches in a foot. It would be like the decimal system. I wonder if we would still call it the metric system.

I’m a bit late to this math party, but I can still contribute.

Everything would indeed most likely be in base twelve. Fractions of the form x/(12^y) would all have terminating representations. In decimal, ⅓ is 0.33333333… but in base 12, ⅓ is 0.4.

One downside is that the fraction 1/5 would have an infinitely long representation. Instead of 0.2, it would be 0.249724972497….

Since a lot more numbers are divisible by three than by five, the advantage with thirds will probably outweigh the disadvantage with fifths.

@LuckyGuy We will probably still call it the metric system. Metric comes from the word for meter, which comes from the word for measure. Nothing about it implies base 10.

@JLeslie That is modular arithmetic, not a base change
eighteen mod twelve = six, but eighteen in base twelve = 16 (not sixteen!).

Ok, got it. My memory must be wrong, but it is so clear in my memory. I googled a little and found things like this. I think it is saying what I said? Not that Yahoo is where I really go for answers.

Thanks everybody!

@JLeslie I don’t consider time to be base 12. When you write out the time, there are only ten number-symbols that we use (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). That’s practically the definition of base ten. If it were true base twelve, there would be twelve symbols involved.

@PhiNotPi It’s seems like maybe at one moment and time the vocabulary was used differently. But, I doubt that is the case, I think I just confused the terms. I am not fighting for my position; I fully accept I was using it wrong. At minimum I would not be affectively communicating what I mean by using the term base to explain what I was getting at. If my memory serves I remember the being taught in each “base” the answer cannot exceed the base I was given as a question. For instance we might have been given the number 20 and asked how to express it in base 4 base 12, etc. But, from the answers above I think I must have confused it with modular arithmetic. Confused the terms. It is absolutely possible and I completely believe I did. Memory is a funny thing. I remember learning the concept I am talking about, it seemed so obvious and easy to me. The 24 hour vs 12 hour clock had always been easy for me to understand and figure out, and this concept I was being taught was the same. I couldn’t understand and still don’t why people have trouble converting the 24 hour clock to 12 and vice a versa. It is barely math to me. It’s like 2+2. But, it seems it is not related to base. I’m glad @Jeruba corrected me.

@JLeslie I’m not so much criticizing you as I am criticizing the author of the article.

@PhiNotPi I didn’t take it as a criticism. My only point was there is more than one person out there, others besides me, who use the term base the same. I find that odd. Odd because it is wrong from what all the math jellies wrote here.