Friday 25 March 2016

Base systems


A base system is a system in which we count. Most people are familiar with the normal base-10 system, also called the decimal system. Each base system has its own unique set of numbers, like the decimal system, which has exactly ten.

The good thing about using a base system is that there does not have to be a different symbol for each number, which would be very confusing and hard to keep track of. Instead, the numbers count up through all of the symbols in the whole system, and then the system adds an extra digit and starts over again.

Computers have to bring this to the minimum, because an electrical current is either off or on. It is hard to get more information with simple digital devices, such as transistors, and it is hard to vary the current in any other easily detectable way. This results in only two "symbols" that a computer can use, so in counting, computers use the base-2 system, or the binary system.

Binary
The two 'symbols' in the binary system are usually represented by 0 and 1, and they are referred to as bits. These can combine to make long sequences that are used in computers. For example, whenever you press a letter on a standard keyboard, an eight-bit sequence of ones and zeroes gets sent to your computer.

Even though binary is simple, it also gets very long. All of the numbers from 0 to 10 in the decimal system can be represented as 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001 and 1010 in binary. All of the even numbers end with a zero, and there is a twos place, a fours place, an eighths place, a sixteenths place and so on. A computer byte, or eight digits of binary, can range through all of the numbers from 0 to 255. 109 in the decimal system is the same as 1101101 in binary.

Octal
The next counting system I want to focus on in this post is octal. This is the base-8 numeral system. Octal has been used by some of the native Americans for counting, because they counted on the spaces in between their fingers, and not the fingers themselves. Over history, octal has been proposed for many things such as coinage and counting, but in the present day, it is not widely used.

Octal uses only the digits 0-7. One helpful thing about the system, is that eight is the cube of two, or 2x2x2. This makes multiplication and division easier. The number 109 is represented by 155 in octal.

Decimal
The next system is also the most commonly used; the base-10 system, or the decimal system. We use the system only because we have ten fingers, so we have the decimal system solely due to evolution. If we had evolved with four fingers on each hand, we would be using octal!

The decimal system is very old. It was used by the ancient civilisations of Greece, Rome, Egypt and China. The oldest decimal multiplication table was made out of bamboo slips and came from the Warring States period in China. Romans had an interesting way to make decimal numbers, only needing numerals for 1(I), 5(V), 10(X), 50(L), 100(C), 500(D) and 1000(M). The Ancient Greeks did not use numerals, and instead used the letters Alpha-Theta as the numbers 1-9, Iota-Koppa as 10-90, and Rho-Sampi as 100-900. The number 148, for example, is translated as ρμη(RUE) in Greek(Notice that the Greek counting system includes the three letters Digamma, Koppa and Sampi, which are now obsolete in language). 109 in decimal is the same as the number 109. This example is not needed.

Duodecimal
One numeral system that is commonly used in America is the base-12 system, also called the duodecimal system or sometimes dozenal. This system is used today in foot-inch and single-dozen-gross-great gross systems as well as most clocks. You might sometimes refer to the number six as 'half a dozen', or twenty-four as 'two dozen'. In the duodecimal system, 24 translates to 20. The duodecimal system can be helpful because it is divisible by 2, 3, 4 and 6.

You may notice that in all of the systems I listed before, only numbers from the 0-9 set are used, however duodecimal has to count through all of the numbers 0-11 without adding another numeral place. This means that the system needs extra symbols to represent 10 and 11. These can be an inverted 2 for 10 and an inverted 3 for 11. The number that is 139 in the decimal system is the same as 37 in the duodecimal system.

Hexadecimal
The next numeral system is based on 2 raised to the fourth power: the hexadecimal system, or the base-16 system. This system requires sixteen different symbols, so the letters A-F are used for the numbers 10-15.

The hexadecimal system is highly involved in computer screens. Each pixel on the screen of a computer is made of three different lights, coloured red, green and blue. The brightness of each of these lights can be adjusted from 0(off) to 255(maximum brightness). This can vary the colours of each pixel. For example, red=255, green=128, blue=0 can colour a pixel bright orange. Each number in between 0 and 255 can be expressed as an eight-digit number in binary, or a two-digit number in hexadecimal. Six-digit series composed of hexadecimal numbers are used in HTML and other programs, two digits to represent each colour. In this form, bright orange would be #FF8000. 139 translated into hexadecimal would be 8B.

Vigesimal
The vigesimal system is the base-20 numeral system. It consists of the numbers 0-9 and the letters A-J. The letter J represents 19. It is used in the Mayan and Aztec language with its own symbols. This system is not all too different from the decimal system, because it is based on two times ten. 139 is 6J in vigesimal.

Sexagesimal
Sexagesimal is the last numeral system in this post. The extremely helpful thing about sexagesimal is that its base, the number 60, can be divided by 12 different factors including 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. It was used by the Babylonians, but I would not say that their counting system was completely base-60, because the Babylonians used decimal as a sub-base.

Of course, sexagesimal is still used today, in geographic coordinates(degrees, minutes and seconds), and time(hours, minutes and seconds). On clocks, the system fits together well with the duodecimal system, because exactly five minutes fits in between each hour. It is no wonder that this system, which was used by the Babylonians, is still used today!


Resources: Wikipedia, "List of numeral systems"