Bits, Bytes and Binary

If you have used a computer for more than five minutes, then you have heard the words bits and bytes. Both RAM and hard disk capacities are measured in bytes, as are file sizes when you examine them in a file viewer.

You might hear an advertisement that says, "This computer has a 32-bit Pentium processor with 64 megabytes of RAM and 2.1 gigabytes of hard disk space."

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.

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: 01101011. How do you figure out what the value of the binary number 01101011 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:

OK Now you try one.  Convert this binary digit into a decimal - 00101100.  

Help, I need a clue - click here

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.

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.

DEC  ASCII  BINARY 0 NUL 0000 0000 1 SOH 0000 0001 2 STX 0000 0010 3 ETX 0000 0011 4 EOT 0000 0100 5 ENQ 0000 0101 6 ACK 0000 0110 7 BEL 0000 0111 8 BS 0000 1000 9 TAB 0000 1001 10 LF 0000 1010 11 VT 0000 1011 12 FF 0000 1100 13 CR 0000 1101 14 SO 0000 1110 15 SI 0000 1111 16 DLE 0001 0000 17 DC1 0001 0001 18 DC2 0001 0010 19 DC3 0001 0011 20 DC4 0001 0100 21 NAK 0001 0101 22 SYN 0001 0110 23 ETB 0001 0111 24 CAN 0001 1000 25 EM 0001 1001 26 SUB 0001 1010 27 ESC 0001 1011 28 FS 0001 1100 29 GS 0001 1101 30 RS 0001 1110 31 US 0001 1111 32 Space  0010 0000

DEC  ASCII  BINARY 33 ! 0010 0001 34 " 0010 0010 35 # 0010 0011 36 $ 0010 0100 37 % 0010 0101 38 & 0010 0110 39 ' 0010 0111 40 ( 0010 1000 41 ) 0010 1001 42 * 0010 1010 43 + 0010 1011 44 , 0010 1100 45 - 0010 1101 46 . 0010 1110 47 / 0010 1111 48 0 0011 0000 49 1 0011 0001 50 2 0011 0010 51 3 0011 0011 52 4 0011 0100 53 5 0011 0101 54 6 0011 0110 55 7 0011 0111 56 8 0011 1000 57 9 0011 1001 58 : 0011 1010 59 ; 0011 1011 60 < 0011 1100 61 = 0011 1101 62 > 0011 1110 63 ? 0011 1111 64 @ 0100 0000

DEC  ASCII  BINARY 65 A 0100 0001 66 B 0100 0010 67 C 0100 0011 68 D 0100 0100 69 E 0100 0101 70 F 0100 0110 71 G 0100 0111 72 H 0100 1000 73 I 0100 1001 74 J 0100 1010 75 K 0100 1011 76 L 0100 1100 77 M 0100 1101 78 N 0100 1110 79 O 0100 1111 80 P 0101 0000 81 Q 0101 0001 82 R 0101 0010 83 S 0101 0011 84 T 0101 0100 85 U 0101 0101 86 V 0101 0110 87 W 0101 0111 88 X 0101 1000 89 Y 0101 1001 90 Z 0101 1010 91 [ 0101 1011 92 \ 0101 1100 93 ] 0101 1101 94 ^ 0101 1110 95 _ 0101 1111 96 ` 0110 0000

DEC  ASCII  BINARY 97 a 0110 0001 98 b 0110 0010 99 c 0110 0011 100 d 0110 0100 101 e 0110 0101 102 f 0110 0110 103 g 0110 0111 104 h 0110 1000 105 i 0110 1001 106 j 0110 1010 107 k 0110 1011 108 l 0110 1100 109 m 0110 1101 110 n 0110 1110 111 o 0110 1111 112 p 0111 0000 113 q 0111 0001 114 r 0111 0010 115 s 0111 0011 116 t 0111 0100 117 u 0111 0101 118 v 0111 0110 119 w 0111 0111 120 x 0111 1000 121 y 0111 1001 122 z 0111 1010 123 { 0111 1011 124 | 0111 1100 125 } 0111 1101 126 ~ 0111 1110 127 DEL 0111 1111

Lots of Bytes
When you start talking about lots of bytes, you get into prefixes like kilo, mega and giga, as in kilobyte, megabyte and gigabyte (also shortened to K, M and G, as in Kbytes, Mbytes and Gbytes or KB, MB and GB). The following table shows the multipliers:

　

Name	Abbr.	Size in Bytes
Kilobyte	KB	210 = 1,024
Megabyte	MB	220 = 1,048,576
Gigabyte	GB	230 = 1,073,741,824
Terabyte	TB	240 = 1,099,511,627,776
Petabyte	PB	250 = 1,125,899,906,842,624
Exabyte	EB	260 = 1,152,921,504,606,846,976
Zettabyte	ZB	270 = 1,180,591,620,717,411,303,424
Yottabyte	YB	280 = 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.

  452
+ 751
  ---
 1203

Binary addition works exactly the same way:

  010
+ 111
  ---
 1001

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.

Try This

Can you convert this binary code into normal ASCII text?

01100010 01101001 01101110 01100001 01110010 01111001 00100000 01100011 01101111 01100100 01100101 00100000 01101001 01110011 00100000 01100110 01110101 01101110 

What is the message?