Another way to express it would be to use powers of 10. Assuming that we are going to represent the concept of "raised to the power of" with the "^" symbol (so "10 squared" is written as "10^2"), another way to express it is like this:

What you can see from this expression is that each digit is a placeholder for the next higher power of 10, starting in the first digit with 10 raised to the power of zero.

That should all feel pretty comfortable -- we work with decimal digits every day. The neat thing about number systems is that there is nothing that forces you to have 10 different values in a digit. Our base-10 number system likely grew up because we have 10 fingers, but if we happened to evolve to have eight fingers instead, we would probably have a base-8 number system. You can have base-anything number systems. In fact, there are lots of good reasons to use different bases in different situations.

Bits
Computers happen to operate using the base-2 number system, also known as the binary number system (just like the base-10 number system is known as the decimal number system). The reason computers use the base-2 system is because it makes it a lot easier to implement them with current electronic technology. You could wire up and build computers that operate in base-10, but they would be fiendishly expensive right now. On the other hand, base-2 computers are relatively cheap.

So computers use binary numbers, and therefore use binary digits in place of decimal digits. The word bit is a shortening of the words "Binary digIT." Whereas decimal digits have 10 possible values ranging from 0 to 9, bits have only two possible values: 0 and 1. Therefore, a binary number is composed of only 0s and 1s, like this: 1011. How do you figure out what the value of the binary number 1011 is? You do it in the same way we did it above for 6357, but you use a base of 2 instead of a base of 10. So:

You can see that in binary numbers, each bit holds the value of increasing powers of 2. That makes counting in binary pretty easy. Starting at zero and going through 20, counting in decimal and binary looks like this:

 0 =     0
 1 =     1
 2 =    10
 3 =    11
 4 =   100
 5 =   101
 6 =   110
 7 =   111
 8 =  1000
 9 =  1001
10 =  1010
11 =  1011
12 =  1100
13 =  1101
14 =  1110
15 =  1111
16 = 10000
17 = 10001
18 = 10010
19 = 10011
20 = 10100

When you look at this sequence, 0 and 1 are the same for decimal and binary number systems. At the number 2, you see carrying first take place in the binary system. If a bit is 1, and you add 1 to it, the bit becomes 0 and the next bit becomes 1. In the transition from 15 to 16 this effect roles over through 4 bits, turning 1111 into 10000.

Bytes
Bits are rarely seen alone in computers. They are almost always bundled together into 8-bit collections, and these collections are called bytes. Why are there 8 bits in a byte? A similar question is, "Why are there 12 eggs in a dozen?" The 8-bit byte is something that people settled on through trial and error over the past 50 years.

With 8 bits in a byte, you can represent 256 values ranging from 0 to 255, as shown here:

  0 = 00000000
  1 = 00000001
  2 = 00000010
   ...
254 = 11111110
255 = 11111111

    0 = 0000000000000000
    1 = 0000000000000001
    2 = 0000000000000010
     ...
65534 = 1111111111111110
65535 = 1111111111111111

Bytes are frequently used to hold individual characters in a text document. In the ASCII character set, each binary value between 0 and 127 is given a specific character. Most computers extend the ASCII character set to use the full range of 256 characters available in a byte. The upper 128 characters handle special things like accented characters from common foreign languages.

You can see the 127 standard ASCII codes below. Computers store text documents, both on disk and in memory, using these codes. For example, if you use Notepad in Windows 95/98 to create a text file containing the words, "Four score and seven years ago," Notepad would use 1 byte of memory per character (including 1 byte for each space character between the words -- ASCII character 32). When Notepad stores the sentence in a file on disk, the file will also contain 1 byte per character and per space.

Try this experiment: Open up a new file in Notepad and insert the sentence, "Four score and seven years ago" in it. Save the file to disk under the name getty.txt. Then use the explorer and look at the size of the file. You will find that the file has a size of 30 bytes on disk: 1 byte for each character. If you add another word to the end of the sentence and re-save it, the file size will jump to the appropriate number of bytes. Each character consumes a byte.

If you were to look at the file as a computer looks at it, you would find that each byte contains not a letter but a number -- the number is the ASCII code corresponding to the character (see below). So on disk, the numbers for the file look like this:

 F   o   u   r     a   n   d      s   e   v   e   n 
70 111 117 114 32 97 110 100 32 115 101 118 101 110 

Standard ASCII Character Set
The first 32 values (0 through 31) are codes for things like carriage return and line feed. The space character is the 33rd value, followed by punctuation, digits, uppercase characters and lowercase characters.

  0   NUL
  1   SOH
  2   STX
  3   ETX
  4   EOT
  5   ENQ
  6   ACK
  7   BEL
  8   BS
  9   TAB
 10   LF
 11   VT
 12   FF
 13   CR
 14   SO
 15   SI
 16   DLE
 17   DC1
 18   DC2
 19   DC3
 20   DC4
 21   NAK
 22   SYN
 23   ETB
 24   CAN
 25   EM
 26   SUB
 27   ESC
 28   FS
 29   GS
 30   RS
 31   US
 32
 33   !
 34   "
 35   #
 36   $
 37   %
 38   &
 39   '
 40   (
 41   )
 42   *
 43   +
 44   ,
 45   -
 46   .
 47   /
 48   0
 49   1
 50   2
 51   3
 52   4
 53   5
 54   6
 55   7
 56   8
 57   9
 58   :
 59   ;
 60   <
 61   =
 62   >
 63   ?
 64   @
 65   A
 66   B
 67   C
 68   D
 69   E
 70   F
 71   G
 72   H
 73   I
 74   J
 75   K
 76   L
 77   M
 78   N
 79   O
 80   P
 81   Q
 82   R
 83   S
 84   T
 85   U
 86   V
 87   W
 88   X
 89   Y
 90   Z
 91   [
 92   \
 93   ]
 94   ^
 95   _
 96   `
 97   a
 98   b
 99   c
100   d
101   e
102   f
103   g
104   h
105   i
106   j
107   k
108   l
109   m
110   n
111   o
112   p
113   q
114   r
115   s
116   t
117   u
118   v
119   w
120   x
121   y
122   z
123   {
124   |
125   }
126   ~
127   DEL

 

Name	Abbr.	Size
Kilo	K	2^10 = 1,024
Mega	M	2^20 = 1,048,576
Giga	G	2^30 = 1,073,741,824
Tera	T	2^40 = 1,099,511,627,776
Peta	P	2^50 = 1,125,899,906,842,624
Exa	E	2^60 = 1,152,921,504,606,846,976
Zetta	Z	2^70 = 1,180,591,620,717,411,303,424
Yotta	Y	2^80 = 1,208,925,819,614,629,174,706,176

You can see in this chart that kilo is about a thousand, mega is about a million, giga is about a billion, and so on. So when someone says, "This computer has a 2 gig hard drive," what he or she means is that the hard drive stores 2 gigabytes, or approximately 2 billion bytes, or exactly 2,147,483,648 bytes. How could you possibly need 2 gigabytes of space? When you consider that one CD holds 650 megabytes, you can see that just three CDs worth of data will fill the whole thing! Terabyte databases are fairly common these days, and there are probably a few petabyte databases floating around the Pentagon by now.

Binary Math
Binary math works just like decimal math, except that the value of each bit can be only 0 or 1. To get a feel for binary math, let's start with decimal addition and see how it works. Assume that we want to add 452 and 751:

  452
+ 751
  ---
 1203

Binary addition works exactly the same way:

  010
+ 111
  ---
 1001

To see how boolean addition is implemented using gates, see How Boolean Logic Works.

Quick Recap

• Bits are binary digits. A bit can hold the value 0 or 1.
• Bytes are made up of 8 bits each.
• Binary math works just like decimal math, but each bit can have a value of only 0 or 1.

For more information on bits, bytes and related topics, check out the links on the next page.