Base numbers/Common bases

Common number bases are 2, 8, 10 and 16. The base doesn't necessarily have to be an integer. There are complex, irrational, negative, fractional, and many other kinds of bases. The best known one is phinary, which is base phi; others include "Fibonacci base" and base negative two.


Binary

Binary is base 2. It's a favorite among computer programmers. It has just two digits: $0$ and $1$.


Octal

Octal is base 8. It was also quite liked by programmers because the octal representation of numbers is 3 times shorter than the binary one and the conversion from octal to binary and back is very easy (can you guess why?). Besides, 8 is quite close to 10 and less than 10, so to learn doing addition and multiplication in base 8 is not very hard: you can basically count in base 10 with partial conversions to base 8 on the way. Let's multiply $12345_8$ by $7_8$. $5\cdot 7=35_{10}=43_8$ (to get the last result, just divide $35$ by $8$ with remainder). As usual, we write the last digit $3$ down and keep $4$ in mind. Now, $4\cdot 7+4=32_{10}=40_8$, so we write down $0$, getting $03$, and keeping $4$ in mind. And so on. The time needed to get the answer $111103_8$ only marginally exceeds the time of decimal multiplication (if you are good in division by 8 with remainder, of course).


Decimal

Decimal is base 10. It's the base that everyone knows and loves. Most numbers in the world are written without a specified radix and usually it can just be assumed that they are in base 10. The most commonly used explanation for the origin of base 10 for our number system is the number of fingers we have.


Hexadecimal

Hexadecimal is base 16. The digits in hexadecimal are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. One of its common uses is for color charts. Hexadecimal numbers are also used by programmers in the same way as octal numbers, but to learn to count in hexadecimal is harder than in octal.


See also

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