I've wondered how math would be different if we used a different base for counting (pi? e? most equations would be different). Attached are 2 images that I created to illustrate the concept. The first one is a base 4 system, and uses square grids to count and easily visualize quantities.
Then, I wondered if taking the same idea to the third dimension would help. Check out the second image, base 8.
I'm not sure if this is the correct usage of "base 4" and "base 8," it just seemed to fit.
As for the "proper question format," that is the bane of stack exchange:
Is there a developed system of counting and arithmetic that uses alternate bases, and a more visual representation?
I would be interested in knowing if this has been tried. While this may be hard for someone used to base 10 to transition to, if it were used from childhood, it would be like second nature.
Best Answer
the usage of various bases in is nothing new. Different bases have different benefits, but normally you use an integer as the base.
If you want familiarity to humans, stick to base-10. If you want easier computation, binary and balanced ternary have their benefits. In specific situations , stick to whatever suits that situation best.
You are right that bases that are powers of an integer have a special benefit in visualisation, but base-8 is rarely used to represent values, and base-27 is mostly used in recreational cryptography.
Drawing cubes of cubes of cubes is still a nice way to visualise large quantities, but it can be somewhat misleading. Here are 1000 cubes:
and here are 1024 cubes, only slightly more:
Base 10: its main benefit is familiarity to humans, and a pre-existing set of symbols to express numbers in base-10. Other than that, this base is not really special.
Base 12, Base 20, Base 60 have historically been used by different cultures (Mayas - base 20), (Babylonians - base 60)... base-12 can be seen in some marketing terminology: dozen = 12; gross = $12^2$ = 144; great gross = 12^3. Base-20 can still be seen in french numerals: quatre-vingt-dix-neuf = 4*20 + 10 + 9 = 99. The main benefit of these bases is that they are highly divisible. 12 = LCM(3, 4); 60 = LCM (3, 4, 5)
Mixed base 2/2/2.5: is the sequence 1-2-5-10 that has benefits in real life: it is "almost" regular, it converts well to decimal base, and has benefits over plain denary in situations where a value is represented by a set of tokens from a predefined set (think: money).
mixed base 6/10: is normally used to represent time. There are 60 seconds in a minute and 60 minutes in an hour. Thus, the sequence "1:00:00" can be understood as a single five-digit number where the digits have weights 3600, 600, 60, 10 and 1.
Base 6: this has special benefit with respect to the Collatz conjencture. In the Collatz conjencture, each step either divides a number by two or multiplies by three. In base-6, this translates to either dropping or not dropping the last digit. Each digit of each step - up to a shift by one place - depends only on two consecutive digits of the previous step.
Base 2: its main benefit is the ease of realisation in electronics. Either a wire voltage is zero volts and the wire opens PMOS transistors for which it is the base, or the wire voltage is a set positive value, and it opens the NMOS transistors in CMOS logic. Other digital systems of logic naturally converge to a two-valued system as well: resistor-transistor logic, transistor-transistor logic, emmiter-coupled logic. Binary signals are especially easy to rectify: just chain two inverters together. Its main drawback is the large amount of symbols needed to express any reasonably-sized value. One benefit in terms of computation is the simplicity of its multiplication table.
Base 8: This combines the information density close to the density of base-10 with the ease of conversion from base-2. If you have a large set of bits that you need to write down, just split them in groups of three, and write down each group as a single digit. This is the last power-of-two base before you run out of digits. Nevertheless, Base-16 is much more often used.
Base 16: the reason is that there are eight bits in a (modern) byte. Thus, you need three base-8 digits to express a byte, but only two base-16 digits. The disadvantage is that you need to find six extra symbols. However, A-F seemed to be the natural choice and it stuck. There are enough letters in the alphabet to get to the next power-of-two base, but if we add lowercase/uppercase distinction and two non-alphanumeric symbols, we get
Base 64: this is the largest power-of-two base before you run out of printable ASCII characters. Thus, Base64 (= base 64 with a particular choice for the set of symbols) is the de-facto standard for transmission of binary data. Once again, no particular benefit in terms of ease of computation.
Base 85: the main benefit is that 85^5 > 2^32. Thus, you only need five symbols to represent a four-byte number. However, it provides only little more information density (5/4 over 4/3) than base 64, at a cost of more expensive conversion. Also, there is less support in terms of standard libraries.
Various other bases are used for encoding data. If you have n symbols, use base-n. An example would be a comma code, where one symbol is a separator between numbers and the other symbols are digits: base-7, base-15, base-255...
Ternary (base-three) and balanced ternary: there has been some experiments in making computers in ternary. The main benefit is higher information density (1 trit ~ 1.6 bits) while keeping the amount of voltage levels relatively low. Balanced ternary uses the values -1, 0, 1 for digits. Balanced ternary is the largest base where multiplication of single digits produces a single digit. Base-9 and Base-27 then serve as a means to pack two digits into one. Another advantage is that balanced bases provide a unified means of representing negative integers along with positive integers.
base -2: while negative bases are rarely used, the most common representation of integers on computer, two's complement, is somewhat similar to negabinary: instead of alternating between negative and positive weights, standard binary is used for all but the top bit, and only the top bit has a negative weight.
base 2i: we can extend the idea of negative bases, and use imaginary numbers for the base. Then, we can encode complex numbers in a single digit. In base 2i, the four solutions to $x^4 = 1$ are 1, 10.2 (= 2i-i), 103 (= -4+3), 0.2(= -i)
irrational bases have a serious disadvantage: there is no way to exactly represent integers above the base. But, if we add a fractional part, algebraic irrational bases can still be used to represent integers. Meet phinary: 1, 10.01, 100.01, 101.01, 1000.1001...
but, closely related to phinary is the Fibonacci code. Fibonacci code is a numeric system that has no fixed ratio between the weights of successive digits, but the ratio does converge to phi. The weights are successive fibonacci numbers. The benefit is that every integer can be uniquely represented as a sequence of zeroes and ones where no two ones are consecutive. Stuff a one between two numbers and you know exactly where one begins and one ends - just look for a pair of ones. Thus, you've got a way to send a stream of integers.
Very large bases (~100 digits) come into play in cryptography, but then you are normally interested in the last digit only. Computation where you are only concerned about the last digit (rough description) is called modular arithmetic. If your base of choice is a product of two primes, then you can easily find pairs of numbers - digits in that base - so that if you use them as exponents, the last digit of the power is the same as the number being exponentiated. But, if you don't know the primes, you cannot easily determine one exponent from the other. So, you encrypt a message by exponentiating with one number and decrypt by exponentiating with the other. This is the basis of the RSA algorithm.