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Computer Memory Definitions - What is the Binary Number System?

SUMMARY: Briefly explains the binary number system and its importance in computer technology.

As mentioned elsewhere in MalekTips, a bit is the smallest unit of memory a computer can hold. Bits can either hold a value of 0 or 1, on or off, yes or no, etc.

While a single bit holds very little data, bits are strung together to represent larger pieces of data. As these bits represent larger numbers, a system of numbering, binary, was adopted.

First, consider our standard "decimal" number system that has a base of 10. This means that after one counts ten numbers (0-9), to create the tenth number a second digit prefixes the value. When both digits reach 9 (the number 99), a third digit prefixes the value, making it 100. If all four digits reach 9 (the number 999), a third digit prefixes the value, making it 1000, and so forth.

If you add 5 + 5, there is no single digit that represents the result as each digit can only hold a value from 0-9. Two digits are needed to represent the number ten.

As opposed to the decimal number system, binary uses a base of 2. Each digit can only hold the value of 0 or 1, just like a bit can only hold the value of 0 or 1.

Thus in binary, if you add 1 + 1, a single digit cannot represent the result. Just as adding 5 + 5 in decimal creates a number containing a digit of 1 and a digit of 0, adding 1 + 1 in binary creates a number containing a digit of 1 and a digit of 0. This number in decimal would be 2.

If you add 1 to binary 10 (decimal 2), you would get binary 11 (decimal 3). Here is a chart:

Decimal  Binary
---------------
1        1      
2        10
3        11
4        100
5        101
6        110
7        111
8        1000 
...

If you understand exponentiation, note that every time a digit is added to a binary number the value goes up by a power of 2 (represented by 2^).

Decimal  Binary    Exponentiation
---------------------------------
1        1         2^0
2        10        2^1
4        100       2^2
8        1000      2^3
16       10000     2^4
32       100000    2^5
64       1000000   2^6
128      10000000  2^7
...

A byte is a series of 8 bits and can represent 256 (2^8) numbers, the smallest being binary 00000000 (decimal 0) and the largest is binary 11111111 (decimal 255, or 2^8 - 1).

Thus, each bit in a byte, when flipped from 0 to 1, raises the value of the number in decimal by 2^ and the bit's position (positions are read from right-to-left).

Decimal   Byte       Exponentiation
--------------------------------------------
1         00000001   2^0 (right-most bit 
                         = the 0th position)
2         00000010   2^1
4         00000100   2^2
8         00001000   2^3
16        00010000   2^4
32        00100000   2^5
64        01000000   2^6
128       10000000   2^7 (left-most bit 
                         = the 7th position)

 

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