 Mathematics

# Different Bases Shawn Schafer's image for:
"Different Bases"
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Image by: The most common base of course decimal, also known as base-ten. This includes numbers zero through nine represented with one digit, 0 through 99 with two, and so on and so forth. The word Decimal comes from the root deca which means ten. This stands for ten numbers being able to be represented with one digit. Some may get confused as you can only represent numbers up to nine with one digit in base-ten, but remember that zero is counted.

Some other common bases include Binary(base-two), Octal(Base-eight), and hexidecimal(base-sixteen). A large reason why these other bases are fairly common is their use in computer systems, but thats a whole other series of articles. Binary consists of one's and zero's. Do you know how when you go beyond 9 in the decimal system, the number goes to 10, a 1 in the ten's place and a 0 in the ones place? Well the same basic system works for numbers in base-two and all other bases for that matter. So 2 in decimal would be '10' in binary. 0 is 0, 1 is 1, 2 is 10,3 is 11, 4 is 100, and so on and so forth. The place beyond the ones place is seldom called the "two's" place as you might expect but this would be an accurrate way to describe it.

Octal's work similarly, with numbers up to 8 being able to be displayed with one digit, 63 with two, and up to 511 with three. Hexidecimal also works similarly, but one more piece of information is needed to understand it. Since there aren't one digit numbers beyond 9, 10 to 15 are represented with letters A through F in hexidecimal, which will also be known as hex from here on out. So 10 is A, 11 is B, 12 is C, 13 is D, 14 is E, and 15 is F. When a number goes beyond 15(or F) it is displayed as 10 in hex, which has a value of 16. 1A is how you would represent 26, 1B is how you would represent 27 and so on and so forth. Basically, just treat letters A through F as if they are normal, fundamental numbers. Numbers up to 15 can be represented with 1 digit, up to 255 with two, up to 4095 with three, and 65535 with four.

Now, some might be mildly impressed that I memorized how large of numbers a certain number of digits can represent in different bases. Don't be because I didn't. There is a very simple formula to figure it out. To find out what the lowest number of a certain number of digits is in a certain base, you take what base it is and put it to the power of the number of digits minus one. In case that is confusing to some, I'll clarify. 10 in Octal is 8, which is 8^1 or 8^(2-1), 100 is 64 which is 8^2 or 8^(3-1), 1000 is 512 which is 8^3 or 8^(4-1), and so on and so forth. In hex, 10 is 16 which is 16^1, 100 is 256 which is 16^2, and 1000 is 4096 which is 16^3. The same works for binary, 10, 100, 1000 and so forth all increasing from one to the next by powers of 2( 4,8,16,etc.)

As far as applications of these go, I'll admit my knowledge is limited. The first time I used hexidecimal was in using gameshark codes for a pokemon video game. Gamesharks and other similar video game cheating devices manipulate hexidecimal variables within hexidecimal codes. As far as octal goes I really have no idea what it is used for. Binary is the core mechanic of all computers. At the end of the day, computers are just a series of transistors that are either activated or not( 1s or 0s), although it isn't really used in programming very often anymore because languages have been developed that translate more intuitive code into 0s and 1s.

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