Numeral systems

Positional numeral system

The value of the number $\pm (a_{k+1}\dots a_2{a}_1{a}_0.a_{-1}{a}_{-2}\dots a_{-l}{)}_b$ (Note that represents a sequence of digits, not multiplication) is

$n=\pm (a_{k+1}\times b^{k+1}+\cdots +a_1\times b^1+a_0\times b^0)+(a_{-1}\times b^{-1}+a_{-2}\times b^{-2}+\cdots +a_{-l}\times b^{-l})$

• $b$ basis (or radix)

• $D=\{d_1,d_2,\dots ,d_b\}$ set of symbols ($\forall i:a_i\in D$)

• $k=\lceil {\log }_{b}n\rceil$ is the number of digits to the left of the radix point (integer part)

• $l$ is the number of digits to the right of the radix point (fractional part)

• Binary

• Octal

• Decimal

• Hexadecimal

Bit numbering

• bit (b) is the smallest unit of data in a computer

• Byte (B) is 8 bits

  • Nibble is 4 bits

• Word is the natural unit of data used by a particular processor design, (in our course, it is 32 bits=4 bytes)

  • Halfword is 16 bits

• Least significant bit (LSB) is the rightmost bit

• Most significant bit (MSB) is the leftmost bit

Signed numbers

• sign–magnitude:
  • if $0d_2{d}_3\dots d_n$ is a positive number, then $1d_2{d}_3\dots d_n$ is the negative of that number
  • the most significant bit is the sign bit
  • the range is $-2^{n-1}+1$ to $2^{n-1}-1$
• two’s complement:
  • if $0\dots 0d_i\dots d_n$ is a positive number (where $d_i$ is the leftmost 1 bit), then $1\dots 1d_i\dots d_n$ is the negative of that number
  • the range is $-2^{n-1}$ to $2^{n-1}-1$
  • inverse of number in two’s complement:

    • method 1

4-bit signed binary numbers

Binary	Unsigned	Sign–magnitude	Two’s complement
0000	0	0	0
0001	1	1	1
0010	2	2	2
0011	3	3	3
0100	4	4	4
0101	5	5	5
0110	6	6	6
0111	7	7	7
1000	8	-0	-8
1001	9	-1	-7
1010	10	-2	-6
1011	11	-3	-5
1100	12	-4	-4
1101	13	-5	-3
1110	14	-6	-2
1111	15	-7	-1

8-bit Two’s complement

Binary	Unsigned	Two’s complement
00000000	0	0
00000001	1	1
$⋮$	$⋮$	$⋮$
01111110	126	126
01111111	127	127
10000000	128	-128
10000001	129	-127
$⋮$	$⋮$	$⋮$
11111110	254	-2
11111111	255	-1

Conversion

Decimal to any base

Converting the integral part

// integer n in base 10 to base b
while n>0
	divide n by b to get quotient and remainder;
	append remainder to the left of the result;
	n=quotient;

Converting the fractional part

// fractional part n in base 10 to base b
while n>0
	multiply n by b to get integer part and fractional part;
	append integer part to the right of the result;
	n=fractional part;

// note: the fractional part may never become zero, stop when the desired precision is reached

Any base to decimal conversion

Binary–hexadecimal conversion

4 binary digits can be represented by 1 hexadecimal digit

Binary–octal conversion

3 binary digits can be represented by 1 octal digit

Octal–hexadecimal conversion

we can convert octal to binary and then binary to hexadecimal or vice versa

Nonpositional numeral system