# Explanation of Alternate Counting Systems

April 18, 2023

When somebody mentions counting or numbers, most people automatically think about the decimal system. It is the most common system of counting, but ten is not the only base available. The base of a counting system is the number at which a new digit is required.

When discussing bases, people may be confused by their names. It seems like it should be easy to call the decimal system “base-10,” but the use of numerals in base names is redundant. Every base is base-10. It is not meant to be read as “ten,” but reading it as “one zero” is better for understanding. The symbol for the size of a base does not exist within that base. Instead, it is the number of numerals in the base.

If that does not make sense, remember that there is a zero symbol to represent nothing. The digits from right to left starting at the fractional point (calling it a decimal point in other bases would be confusing) are multiplied by each increasing power of the base starting at x0. The same applies to digits to the right of the fractional point from left to right, but they are multiplied by each decreasing power beginning with x-1.

While any natural number greater than one can act as a base, most do not have any practical applications. Besides decimal, the most commonly used number systems are binary (base two), hexadecimal (base sixteen), and a hybrid between seximal/senary (base six) and decimal.

The binary system is often considered the language of computers because of its use in analog computers. It is quite special in that its only symbols are 1, which represents something, and 0, which represents nothing. This can be restated as a system made up of true and false values, respectively.

A code has been created to write using binary, aptly named binary code. It is composed of eight digit segments called bytes for every character commonly used in the English language. In the code, digits are called bits to have similarity to bytes, and they are the smallest unit of information. An easy way to understand binary code is to separate each byte into a three bit chunk and a five bit chunk. The first chunk, with three bits, determines the type of character, and the second chunk determines the character of that type.

For example, “01001000 01100101 01101100 01101100 01101111 00100000 01010111 01101111 01110010 01101100 01100100 00100001” is “Hello World!” As some may have noticed, the first three bits of the capital letters, lowercase letters, and punctuation are all different. Most punctuation marks, including spaces and numbers, begin with 001. Capital letters begin with 010, along with punctuation marks like @, ^, _, and other less used punctuation. Lower case letters begin with 011, along with some uncommon punctuation marks. The remaining five bits distinguish which character a byte represents among its type of character. The five bits are a number between zero and thirty-one, and each number represents a different character. The numbers one (00001) through twenty-six (11010) represent the letters of the Latin alphabet in order.

One thing to note is that, despite popular belief, morse code is not binary. It is composed of long units and short units, but it also relies on the space between those units as a third type of unit. This makes morse code a sort of trinary code, but that is the only real practical use of the trinary system.

Hexadecimal is also used by computers, specifically with RGB (red, green, blue) color palettes. It is base sixteen, so it needs extra symbols to represent ten, eleven, twelve, thirteen, fourteen, and fifteen. This task has been given to the first six letters of the alphabet respectively to avoid the task of creating entirely new symbols. The hexadecimal color palette functions by determining the amount of each primary color. The amount of a color can range from zero to two hundred fifty-five (ff). These amounts are placed in order following an octothorpe (also called pound or hashtag). For example, #fa7c05 is a shade of orange composed of two hundred fifty (fa) red units, one hundred ninety-nine (7c) green units, and five (5) blue units. These values are added to black (#000000) and can sum up to white (#ffffff). The values are taken from two bit chunks, otherwise, the example would be sixteen million four hundred fifteen thousand seven hundred forty-nine.

Another base that is unknowingly used often is base six or seximal/senary. Almost everybody using this system is unaware because they are used to using a hybrid of it and decimal. This hybrid is the standard system of time notation. This can be easily explained with an example, such as 12:34:56. The one and two represent the current hour, three and four the current minute, and five and six the current second. The digits in the six, four, two, and one’s positions are in decimal. They begin at zero and go up to nine before the digits to their left increase by one and they reset to zero. The digits in the five and three’s positions are in seximal because they only go up to five before they spill over to the next position. While this is the only common use of the seximal system, it is not the only practical use.

Finger counting can be done more efficiently in seximal than in decimal. Thumbs will be considered fingers during the following explanation for the sake of simplicity, even though they are scientifically not fingers. Most humans have five fingers on each hand, therefore it makes sense that a person using their fingers to count in decimal would only be able to count from zero to ten before using all of their fingers. On the other hand (pun intended), when counting fingers in seximal, one hand goes from zero to five and resets to zero as one finger raises on the other hand. The seximal system of finger counting can reach thirty-five (55) before all of the fingers are used. Click here for a visual explanation of this method. This still does not cover all of the practical uses of the system. The base of the decimal system has two and five as its prime factors, and seximal’s base has two and three as prime factors. In decimal, any fraction whose denominator is a multiple of three repeats, and denominators divisible by seven repeat even more wildly. Denominators divisible by five are not forced to repeat. In seximal, the fractions with denominators divisible by five do repeat, but denominators that are multiples of three are not forced to repeat. Fractions with denominators divisible by seven are also significantly more manageable in seximal.

Below are a few examples of equations in each of the aforementioned number systems to help with understanding:

13+15=32 (seximal) (9+11=20 in decimal)

2f-19=16 (hexadecimal) (47-25=22 in decimal)

100011/111+11001=11110 (binary) (35/7+25=30 in decimal)

#f1089b (hexadecimal)

#32aef1 (hexadecimal)

#2ffb8c (hexadecimal)

1101+10110=100011 (binary) (13+22=35 in decimal)

21+41=102 (seximal) (13+25=38 in decimal)

d+16=23 (hexadecimal) (13+22=35 in decimal)

Although the decimal system is used most often, it is one of an infinite set of number systems. Whether it is for programming a computer, encoding a message, counting efficiently, or simply for trivial knowledge, there are always practical benefits to expanding one’s knowledge.