The word binary in general refers to having two choices; in computer science binary refers to the base 2 numeral system, i.e. a system of writing numbers with only two symbols, usually 1s and 0s. We can write any number in binary just as we can with our everyday decimal system, but binary is more convenient for computers because this system is easy to implement in electronics (a switch can be on or off, i.e. 1 or 0; systems with more digits were tried but unsuccessful, they failed miserably in reliability). The word binary is also by extension used for non-textual computer files such as native executable programs or asset files for games.
One binary digit can be used to store exactly 1 bit of information. So the number of places we have for writing a binary number (e.g. in computer memory) is called a number of bits or bit width. A bit width N allows for storing 2^N values (e.g. with 2 bits we can store 4 values: 0, 1, 2 and 3, in binary 00, 01, 10 and 11).
At the basic level binary works just like the decimal (base 10) system we're used to. While the decimal system uses powers of 10, binary uses powers of 2. Here is a table showing a few numbers in decimal and binary:
Conversion to decimal: let's see an example that utilizes the facts mentioned above. Let's have a number that's written as 10135 in decimal. The first digit from the right (5) says the number of 10^(0)s (1s) in the number, the second digit (3) says the number of 10^(1)s (10s), the third digit (1) says the number of 10^(2)s (100s) etc. Similarly if we now have a number 100101 in binary, the first digit from the right (1) says the number of 2^(0)s (1s), the second digit (0) says the number of 2^(1)s (2s), the third digit (1) says the number of 2^(2)s (4s) etc. Therefore this binary number can be converted to decimal by simply computing 1 * 2^0 + 0 * 2^1 + 1 * 2^2 + 0 * 2^3 + 0 * 2^4 + 1 * 2^5 = 1 + 4 + 32 = 37.
To convert from decimal to binary we can use a simple algorithm that's again derived from the above. Let's say we have a number X we want to write in binary. We will write digits from right to left. The first (rightmost) digit is the remainder after integer division of X by 2. Then we divide the number by 2. The second digit is again the remainder after division by 2. Then we divide the number by 2 again. This continues until the number is 0. For example let's convert the number 22 to binary: first digit = 22 % 2 = 0; 22 / 2 = 11, second digit = 11 % 2 = 1; 11 / 2 = 5; third digit = 5 % 2 = 1; 5 / 2 = 2; 2 % 2 = 0; 2 / 2 = 1; 1 % 2 = 1; 1 / 2 = 0. The result is 10110.
TODO: operations in binary
In binary it is very simple and fast to divide and multiply by powers of 2 (1, 2, 4, 8, 16, ...), just as it is simply to divide and multiple by powers of 10 (1, 10, 100, 1000, ...) in decimal (we just shift the radix point, e.g. the binary number 1011 multiplied by 4 is 101100, we just added two zeros at the end). This is why as a programmer you should prefer working with powers of two (your programs can be faster if the computer can perform basic operations faster).
Binary can be very easily converted to and from hexadecimal and octal because 1 hexadecimal (octal) digit always maps to exactly 4 (3) binary digits. E.g. the hexadeciaml number F0 is 11110000 in binary (1111 is always equaivalent to F, 0000 is always equivalent to 0). This doesn't hold for the decimal base, hence programmers often tend to avoid base 10.
We can work with the binary representation the same way as with decimal, i.e. we can e.g. write negative numbers such as -110101 or rational numbers (or even real numbers) such as 1011.001101. However in a computer memory there are no other symbols than 1 and 0, so we can't use extra symbols such as - or . to represent such values. So if we want to represent more numbers than non-negative integers, we literally have to only use 1s and 0s and choose a specific representation/format/encoding of numbers -- there are several formats for representing e.g. signed (potentially negative) or rational (fractional) numbers, each with pros and cons. The following are the most common number representations:
As anything can be represented with numbers, binary can be used to store any kind of information such as text, images, sounds and videos. See data structures and file formats.
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