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I once heard someone claim that if we had four digits on each hand our mathematics would be well in advance of what it is.

Presumably this is because 4 is a power of 2 and those clever computers calculate in powers of two. Personally, I think that they do so for electrical reasons, but am not enough of a mathematician to tell if math based on powers of 2 is easier, more intuitive, or in any way superior.

So, if we had four digits on each hand, would we be be mathematically more advanced?

Would there be any other major impacts? (and would cartoon characters only have three digits? ;-)


[Update] I doubt it makes any difference at all to mathematics, where symbols are involved. But to arithmetic? If multiplication & division were simpler, what then of pyramids? Or even just bridges, etc? Would earlier perfection of the arithmetic of engineering have had major impacts elsewhere?


[Upperdate] I am reminded of this 2,000+ year old computational device (complete with complex gearing). Could power of two based calculation have led to even greater devices?

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    $\begingroup$ Do you mean digits? By my count I already have four fingers on my hand, along with a thumb. I agree that the definition is somewhat fuzzy, but you might want to clarify that point before any other pedants arrive! $\endgroup$ – Joe Bloggs Dec 16 '15 at 12:48
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    $\begingroup$ That's one hell of a claim... I assume most mathematicians won't have problems working with octal numbers, or hexadecimal. People who don't use base 2, base 8, base 16 a lot find them complicated, but the mathemathical rules apply independent of the counting system you use. What did hinder mathematics for a very long time was the absence of a concept of zero. $\endgroup$ – Burki Dec 16 '15 at 13:36
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    $\begingroup$ @Burki That and the lack of symbolic notation for mathematics were likely both slowing down progress in mathematics. We are so used to see mathematical formulas expressed with symbols that we often forget that some of the best known formulas were originally expressed with words because the symbols hadn't been invented yet. $\endgroup$ – kasperd Dec 16 '15 at 15:12
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    $\begingroup$ Out of interest, the reason we use binary in computers is simplicity (and, as always, it's a trade off). Check out this episode of Computerphile for more details. $\endgroup$ – Tom Potts Dec 16 '15 at 15:15
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    $\begingroup$ @TomPotts I think this Numberphile video is also a good watch. Specifically because it counters, to some extent, the assumption that having four digits on your hand would naturally lead to base-four/base-eight systems. Even with standard five digitted humans, there's a variety of numbering/counting systems which aren't the standard decimal one. The 10 fingers -> decimal connection is something of a historical accident, rather than an foregone conclusion. $\endgroup$ – R.M. Dec 16 '15 at 18:45
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Not really. There's a similar argument that cultures with different languages are better at math due to how their number-words are constructed. For example, in English, the number of 11 is called "eleven" and the number 15 is "fifteen", but in Chinese, Japanese, Korean and Turkish that number is basically called "10 & 1" or "10 & 5". That doesn't sound difficult but for early language learners, adding the extra baggage takes up time. Fifteen plus twelve has to be translated to 15 + 12 then 10+5 + 10+2 and then you get the answer, 27, which is then 20+7 or twenty seven. It's actually worse in other languages... for example, 99 is ninety nine in English, in Korean it's "9 10's & 9", but in French it's quatre-vingt-dix-neuf... which is literally "4 20s & 10 & 9". Because there's less 'lag' in Chinese, Japanese, Korean and Turkish, they've found that children in those cultures as a whole understand math faster, which propagated the original stereotype that the entire culture is just seemingly better at math than Western cultures.

However, once those students get into more advanced mathematics including numbers in the abstract, all of that edge vanishes, so all cultures should be on the same level... however, there is a lingering psychological impact. Those who basically learned to hate math starting at those very early ages do even worse once math becomes abstract.

Having a base 8, or octal, math isn't going to be structured all that differently in the grand scheme of things. Anything that goes down to base 2 will be effectively the same for any large computations. Now if we had 4 fingers on one hand and three or five fingers on the other... THAT would most likely have an impact.

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On the one hand (pun intented), we might get a little bit better at the maths necessary for computer science, but on the other hand, we'd probably be worse at music, and maybe many other forms of art. Without the added precision that comes from our five-fingered hands, we may never have achieved some of the masterpieces we hold in such high esteem today. That said, everyone's standards would probably be lower, so I imagine this would not have been much of a problem, but there is a definite possibility that with fewer fingers, humans would be limited just enough to not invent certain things, and end up over a long time to not be nearly as advanced as we are today.

Plus, think about typing. You can always learn to think in octal, but it's another problem entirely to learn to type as quickly as someone with two extra fingers. It might only be a loss of a few seconds per person per day, but with millions of people typing for years this is going to add up. It may also become a bit more of a hassle to type out larger numbers, as they will require more characters. This could end up costing a lot of money for extra ink and paper.

Add to all this the fact that computers have only recently been developed, and that technological advancement is exponential, and I think it's far more important to make early advancements than late ones. Things like extra-precise building techniques and more complex art may have had more of an impact on early human development than maths, so I'd put my money on having less fingers actually being more of a hindrance to humanity in the long term.

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  • $\begingroup$ If we had fewer fingers, chord-based keyboards would be more popular and we'd type much faster. $\endgroup$ – Stephane Dec 16 '15 at 17:49
  • $\begingroup$ @Stephane I doubt that, simply because we don't use chord-based keyboards now. I wish I could use a chord-based keyboard, I just can't seem to find any good ones. $\endgroup$ – DaaaahWhoosh Dec 16 '15 at 17:52
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    $\begingroup$ I disagree on the music aspect of this - Music is full of binary notation. Note length divisions are all halves and quarters and eights - powers of two (multiplets of other numbers exist, but are less common). Double or halve the frequency of a tone and you get a perfect octave. Sure you'd lose some dexterity to play modern instruments by being short a finger, but in such a world, we'd have invented instruments that are optimized to be played with one less finger anyhow. $\endgroup$ – Darrel Hoffman Dec 16 '15 at 17:53
  • $\begingroup$ My father had lost the little finger on his right (dominant) hand. He never complained it got in his way and I never saw it causing him any difficulty. I can't see a species with one fewer fingers than humanity being at any significant disadvantage. $\endgroup$ – nigel222 Dec 16 '15 at 18:53
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At first I was going to say I doubt it, all bases have good and bad things about them. Not only that other bases have existed in human history, such as base 12. However, with 4 digits our base counting system would likely be Octal. which everyone would be using by default.

Computers use binary, and programmers used octal and still often use Hexidecimal to help make reading binary easier. So by having Octal be the base standard that everyone understands vs. having to learn it like learning a second language, many more people could have a better understanding of how to interact with computers with a closer number system. By volume alone this could have an impact on our technology. It also would have likely spawned earlier and more lasting work into computing machines. Better machines earlier would of course imply more advanced mathematics as well.

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    $\begingroup$ I disagree. Computers are a pretty modern invention - they've been around for less than 70 years. Yet mathematics were developed in ancient times, and served humanity in various ways over that time. Calculus was invented hundreds of years ago, etc. Making all those mathematical discoveries using a base 2 / 8 / 16 system would have been significantly more challenging than our own base 10 one. $\endgroup$ – AndreiROM Dec 16 '15 at 14:14
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    $\begingroup$ why would it have been more challenging in, say, base 8 or 12? $\endgroup$ – Mawg Dec 16 '15 at 14:16
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    $\begingroup$ @AndreiROM actually computing devices are a couple hundred years old, and Babage had one that is sometime thought of as the first computer (mechanical) back in the 1840's, in 1879 we had a mechanical cash register. On top of that why would base ten be any easier than 8? Just because we currently understand it better? $\endgroup$ – bowlturner Dec 16 '15 at 14:25
  • $\begingroup$ I would hardly consider a cash register a computer. It's a fancy calculator, nothing more. And base 10 mathematical systems are more intuitive. $\endgroup$ – AndreiROM Dec 16 '15 at 14:31
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    $\begingroup$ @AndreiROM they are only intuitive to people who are used to them. There is nothing more or less intuitive about base2,8, 16, or about base 23, for that matter. $\endgroup$ – Burki Dec 16 '15 at 14:42
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Switching the base does not give you any advantage. Mathematics has developed independently in many places and the most common numeric bases have been 5, 10 & 20 which is related to the number of digits one has (1 hand, vs 2 hands, vs hands & feet), but the Sumerians and later the Babylonians used base 60 and their understanding of math was excellent. It's all a matter of habit. If you use octal everyday for a year you'll find it's no different than base 10.

I work as a software engineer and can assure you that binary, octal & hex are rarely used in modern programming. The only use I can think of is when you are manipulating bits directly, which back in the day people did often, but nowadays is mostly taken care by the programming language / libraries.

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  • $\begingroup$ Note that 60 s still a multiple of 10. No hex? Lucky you - web guy? I am in embedded & still have to think in hex every day (actually, like anything it gets easy with practice. For fun, I extended my mental multiplication tables to hex, so that I now I can calculate things 0x1C * 0x7A in my head :-) $\endgroup$ – Mawg Dec 17 '15 at 8:05
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    $\begingroup$ No, not a web guy, I'm in aerospace now, did industrial control systems (including a bit of embedded work) before that. My point is that for the vast majority of applications knowledge of hex/binary/octal is not required. $\endgroup$ – ventsyv Dec 17 '15 at 15:31
  • $\begingroup$ I would agree with that. As a protocol guy (telecom & satcom), I often have to pick a few bits out of something larger, but that's probably niche stuff. In my home coding on windows/web I will admit that I can't remember having to use it. Kids today don't know they're born ;-) $\endgroup$ – Mawg Dec 18 '15 at 8:38

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