Number Systems

Human beings use decimal (base 10) and duodecimal (base 12) number systems for counting and measurements (probably because we have 10 fingers and two big toes). Computers use binary (base 2) number system, as they are made from binary digital components (known as transistors) operating in two states - on and off. In computing, we also use hexadecimal (base 16) or octal (base 8) number systems, as a compact form for represent binary numbers.

Decimal Number System (Base 10):

Decimal number system has ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, called digits. It is a positional notation, for example,

735D = 7×102 + 3×101 + 5×100      

We shall denote a decimal number with an optional suffix D if ambiguity arises.

Binary Number System (Base 2):

Binary number system has two symbols: 0 and 1, called bits. It is also a positional notation, for example,

10110B = 1×24 + 0×23 + 1×22 + 1×21 + 0×20      

We shall denote a binary number with a suffix B. Some programming languages denote binary numbers with prefix 0b (e.g., 0b1001000), or prefix b with the bits quoted (e.g., b'10001111').

A binary digit is called a bit. Eight bits is called a byte (why 8-bit unit? Probably because 8=23).

Hexadecimal Number System (Base 16):

Hexadecimal number system uses 16 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F, called hex digits. It is a positional notation, for example,

A3EH = 10×162 + 3×161 + 14×160      

We shall denote a hexadecimal number with a suffix H. Some computer languages denote hex numbers with prefix 0x (e.g., 0x1A3C5F), or prefix x with hex digit quoted (e.g., x'C3A4D98B').

Each hexadecimal digit is also called a hex digit. Most languages accept lowercase 'a' to 'f' as well as uppercase 'A' to 'F'.

Computers uses binary system in their internal operations, as it is built from binary digital electronic components. However, writing or reading a long sequence of binary bits is cumbersome and error-prone. Hexadecimal system is used as a compact form or shorthand for binary bits. Each hex digit is equivalent to 4 binary bits, i.e., shorthand for 4 bits, as follows:

0H (0000B) (0D)1H (0001B) (1D)2H (0010B) (2D)3H (0011B) (3D)
4H (0100B) (4D)5H (0101B) (5D)6H (0110B) (6D)7H (0111B) (7D)
8H (1000B) (8D)9H (1001B) (9D)AH (1010B) (10D)BH (1011B) (11D)
CH (1100B) (12D)DH (1101B) (13D)EH (1110B) (14D)FH (1111B) (15D)

Number Systems Conversion

Conversion from Hexadecimal to Binary:

Replace each hex digit by the 4 equivalent bits, for examples,

A3C5H = 1010 0011 1100 0101B
102AH = 0001 0000 0010 1010B      

Conversion from Binary to Hexadecimal:

Starting from the right (least significant bit), replace each group of 4 bits by the equivalent hex digit, for examples,

1001001010B = 0010 0100 1010B = 24AH
10001011001011B = 0010 0010 1100 1011B = 22CBH      

It is important to note that hexadecimal number provides a compact form or shorthand for representing binary bits.

Conversion from Base r to Decimal (Base 10):

Given a n-digit base r number: dn-1 dn-2 dn-3 ... d3 d2 d1 d0 (base r), the decimal equivalent is given by:

dn-1 × r(n-1) + dn-2 × r(n-2) + ... + d1 × r1 + d0 × r0      

Conversion from Decimal (Base 10) to Base r:

Use repeated division/remainder. For example,

To convert 261D to hexadecimal:
261/16 quotient=16 remainder=5
16/16  quotient=1  remainder=0
1/16   quotient=0  remainder=1 (quotient=0 stop)
Hence, 261D = 105H      

Hint: You could use the Windows' Calculator to carry out number conversion by choosing the scientific mode (Launch "Calculator" ⇒ Select "View" menu ⇒ Choose "Scientific" mode).

Exercises

  1. Convert the following decimal numbers into binary and hexadecimal numbers.
    108; 4848; 9000          
  2. Convert the following binary numbers into hexadecimal and decimal numbers.
    1000011000; 10000000; 101010101010          
  3. Convert the following hexadecimal numbers into binary and decimal numbers.
    ABCDE; 1234; 80F          

You could use the Windows' Calculator (calc.exe) to check your answer, by setting it to the scientific mode (Select "View" menu ⇒ choose "Scientific" mode).

Answers:

  1. 1101100B, 1001011110000B, 10001100101000B, 6CH, 12F0H, 2328H;
  2. 218H, 80H, AAAH, 536D, 128D, 2730D;
  3. 10101011110011011110B, 1001000110100B, 100000001111B, 703710D, 4660D, 2063D.

References & Resources

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