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

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

$b$ is the basis (or radix)
$D = {d_{1}, d_{2}, \dots, d_{b}}$ is the set of symbols ( $\forall i : a_{i} \in D$ )
$k = ⌈ lo g_{b} n ⌉$ 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 $0 d_{2} d_{3} \dots d_{n}$ is a positive number, then $1 d_{2} d_{3} \dots d_{n}$ is the negative of that number
the most significant bit is the sign bit
the range is $- 2^{n - 1} + 1$ to $2^{n - 1} - 1$

One’s complement

the range is $- 2^{n - 1} + 1$ to $2^{n - 1} - 1$
the negative of a number is obtained by inverting all bits
- example: $5 = 0101 ⟹ - 5 = 1010$

Two’s complement

the range is $- 2^{n - 1}$ to $2^{n - 1} - 1$
If $k = 0 d_{2} d_{3} \dots d_{n}$ is a positive number, then we can find its negative $- k$ by:
- invert (flip) each bit (One’s complement) and add 1 to the result:
  - example: $5 = 0101 ⟹ - 5 = 1010 + 1 = 1011$
- subtract $k$ from $2^{n}$ :
  - example: $5 = 0101, n = 4 ⟹ 2^{4} - 5 = 16 - 5 = 11 ⟹ - 5 = 1011$

signed binary numbers

4-bit

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

8-bit

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

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

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

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Numeral systems