Base 10 is the number system most of the world uses, but there is so much more potential than our banal base 10. The next most common base is binary (2), which pins the foundations of computer science. Septenary (7), Duodecimal (12), and Sexagesimal (60) bases arise when talking about time on the scale of minutes, hours, weeks, and months, which we rarely think about. How do these bases work in the first place?
When there is a number, such as 284, in base 10, we are really writing $2\times 10^2+ 8\times 10^1+4\times 10^0$. The pattern of increasing powers of ten would continue if there were more digits, and decrease accordingly, if there were digits following a ‘.’ or ‘,’ in certain countries. To represent negative numbers, we simply add a ‘-‘ in front of a number to negate it, however other number systems do not need negative signs at all.
Negative bases exist, such as negabinary (base -2), that have the negative sign implicitly designed into the system. 284 in negabinary is represented as 101101100 (or 256-0+64-32+0-8+4-0+0). See? No negative sign needed. Negative bases are mostly useless, since they do not have a nice symmetry that is reflected in the use of the minus sign in non-negative bases.
Going one step further, there is no requirement to have integer bases. Rational bases such as base $\frac32$ are valid, but again have little use. Irrational numbers can also be used, and even have some applications. For instance, base $\sqrt[12]2$ is used in musical scales to differentiate between notes. The base that requires the least amount of digits to represent real values numbers on average is base $e$ (proof can be found here).
One fascinating irrational base is base $\pi$. The circumference of a circle is represented by the equation $C=2\pi r$. In base $\pi$, this means a circle of radius 10, has circumference 100. Likewise, area of a circle follows from the equation $A=\pi r^2$. The same circle of radius 10, will have an area of 1000 in base $\pi$. However, anything not involving circles becomes extremely cumbersome, since as far as we know, $\pi$ has an infinite not-repeating decimal in base 10, which would then lead anything not being a multiple of $\pi$ to take on this infinite non-repeating decimal in base $\pi$.
The last category of wacky bases are complex bases, such as the quater-imaginary base $2i$, which I find most interesting. The whole 2-dimensional complex plane, can be represented by the numbers 0,1,2, and 3 using base $2i$, which in a roundabout way proves that the real numbers and complex numbers are the same size of infinity. I built a java class for those curious to play around with the quater-imaginary base linked here.
The possibility of bases is infinite, and this is just one example of a topic that can seem very boring at first, but has a world of complexity underneath it. Strange bases, such as base $-1\pm i$ have applications where you would not expect it. To find other interesting examples of strange bases, I recommend checking out this thorough list of number systems here.
