# Number bases in a different world

We have always worked in the seemingly intrinsic base 10 which I believe is due to the fact that we have 10 fingers. What if we had 6 fingers and were unavoidably lead to work in base 6 or weirder still, what if we had an odd number of digits like 7 or nine? What would these things mean for math and science as a whole? Would it have any effect at all provided everything else remained the same? Would we seamlessly invent and create all algebraic rules and functions just as we have now? Would it alter the rules of math? Would it be detrimental to our development?

Caveat: I am not a mathematician so pardon me if I sound dumb Note by Olawale Olayemi
6 years, 6 months ago

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If you ask any alien species what base do you use, they'd all answer, "That's obvious. Base 10". Is it not interesting?

- 6 years, 6 months ago

Very true. For those of you who don't get it, just think about it for a bit more...

- 6 years, 6 months ago

There are 10 types of people in the world, those who understand binary, those who don't , and those who didn't expect this joke to be in base 3.

- 6 years, 6 months ago

There is a popular saying. "There are 10 types of people. Those who get this and those who don't". The choice of the base of a number system poses no problems as long as one is consistent.

- 6 years, 6 months ago

Different bases have been used over the centuries by different civilizations. The Babylonians used a base of 60, (hence the 60 minutes in an hour, 60 degrees being the angles of an equilateral triangle), the Mayans used a base 20 system, the Romans used a fraction system using base 12, (duodecimal), and many African societies have used base 12 as well. (Base 12 is quite practical, actually, as 12 is a highly composite number). Computers use binary, (base 2), ternary, (base 3), octal, (base 8) and hexadecimal, (base 16). In modern mathematics the most "natural" base is $e$, but since we have a bias towards integers such a transcendental base would not go over well.

Base 10 is probably the most commonly used, both historically and presently, but not using it would by no means be a detriment to our development. It's the colonizers that write the history books and decide which numbering systems to use, that's all. And as far as mathematics is concerned, numbering systems are the medium, not the message, so what and how much we know about math and science would be no different if a base other than 10 were more prevalent. I've always used base 10, of course, so it's hard to imagine otherwise, but I'm also unilingual so it's hard for me to imagine thinking in another language, but I know others can, so I see no reason why the same can't be done for numbering systems.

- 6 years, 6 months ago

Now that you have shed light on this, it seems so sensible and really just a matter of the victor writing the history. It's almost like asking if we would have been less advanced if Latin was the official language of the world not to mention the great strides in science and thought that were made in that language.

It seems though, that a lot of currency has been placed on 'even' bases probably out of the duality that man observes around him. Could you please explain more why $e$ is the most natural base?

Thanks!

- 6 years, 6 months ago

Euler's constant $e$ has several properties that make it a natural base. In calculus, we have that

$\dfrac{d}{dx}e^{x} = e^{x}$, $\displaystyle\int e^{x} dx = e^{x} + C$ and $\dfrac{d}{dx}\ln(x) = \dfrac{1}{x}.$

The inverse of the exponential function $e^{x}$ is the natural logarithm $\log_{e}(x) = \ln(x)$, which as a result of the elegant calculus properties of $e^{x}$ makes it the most convenient base to work with in many situations. Much of the natural world is governed by exponential growth and decay, so as a result $e$ emerges as a factor in so many branches of science. Even in finance, compound interest can be accurately modeled as exponential, (exactly so in the case of continuously compounded interest). In statistics the standard normal distribution has $e$ as it's centerpiece, so it serves as a natural base in statistical analyses as well. The list goes on, but this should give you an idea of why this number is so significant.

Beyond being the most natural base, it is also, surprisingly, connected to all integers through the series

$e^{x} = \displaystyle\sum_{n=0}^{\infty} \dfrac{x^{n}}{n!}.$

Saving the best for last, we have arguably the most important formula in mathematics, Euler's formula,

$e^{ix} = \cos(x) + i \sin(x),$ and the related Euler's identity $e^{i\pi} + 1 = 0.$

- 6 years, 6 months ago

Great stuff as always. I appreciate you taking out the time!

- 6 years, 6 months ago

Haha , I would love to see some changes in Base 10 or a new Base getting equally popular ! In fact what I am more interested is to see how people react to it :)

- 6 years, 6 months ago

Yeah, I'd love to see how people would react if $e$ were declared the base of operation; mathematicians would become so popular. :P When we make 'first contact' with aliens I think our first question for them should be, "What base do you work in?" :)

- 6 years, 6 months ago

Yup! And the aliens will answer , $\pi$ ! It would be fun :)

- 6 years, 6 months ago

Haha. Right before they annihilate us. That would be scary!

- 6 years, 6 months ago

Exactly !

Imagine some aliens from Planet Zeta arrived on our Earth on their spaceship Dirichlet 4869 . Their mission is to capture our planet . They are more technically advanced than us (ofc) , so they have brought with them their Beta Launchers and what not !

Btw it's a nice idea that you've brought up , sir :)

- 6 years, 6 months ago

Thanks Roopesh!

- 6 years, 6 months ago