Types of Number Representations

1. Signed Integer Representations (n-bit)

For $n=8$ example range and representation of $-13$:

a) Sign-Magnitude (SM): MSB = sign (0=+ ,1=-), rest = magnitude.
Range $-(2^{n-1}-1)$ to $+(2^{n-1}-1)$; two zeros $\pm 0$.
$+13 = 00001101$, $-13 = 10001101$; 8-bit range -127 to +127.

b) 1's Complement: Negative = bitwise NOT of positive.
Range $-(2^{n-1}-1)$ to $+(2^{n-1}-1)$; two zeros.
$-13 = \overline{00001101}=11110010$.

c) 2's Complement (used in modern ALUs): Negative = 1's complement +1.
Range $-2^{n-1}$ to $+(2^{n-1}-1)$; single zero. Addition/subtraction unified.
$-13 = 11110010+1=11110011$.
Range 8-bit: -128 to +127.
Property: $A-B = A + (-B)_{2's}$.

Other: Excess/Biased $2^{n-1}$ or 127, base $-2$ not asked separately.

2. Fixed Point Representation

Binary point is implied fixed position.
Format: Sign | Integer part | Fractional part
E.g. $Q4.4$ format: 1 sign +3 integer +4 fractional.
Number $= -3.75 = 1|011|1100$ (SM). Value $= \sum b_i 2^i$.
Used for integer arithmetic and DSP fractional $[-1,+1)$.
Range depends on point position. E.g. 8-bit pure fraction $0.xxx$: $-1$ to $+0.992$.

3. Floating Point Representation

For very large/small reals: $N = \pm M \times B^{\pm E}$ where $M$ = mantissa/significand, $E$ = exponent, $B$ = base (2).

IEEE 754 Standard:

• Single Precision (32-bit): 1 sign | 8 biased exponent (bias 127) | 23 mantissa with hidden 1.
  Value $= (-1)^S \times 1.M \times 2^{E-127}$. $E=0$ and $255$ reserved for 0/subnormals/inf/NaN.
  E.g. $13.25 =1101.01_2 =1.10101\times2^3$: $S=0,E=130=10000010,M=1010100...0$ => 0x41540000.

• Double Precision (64-bit): 1|11|52, bias 1023.
  Special values: $E=0,M=0\Rightarrow\pm0$; $E=0,M
  eq0\Rightarrow$subnormal;$E=255,M=0\Rightarrow\pm\infty$;$E=255,M
  eq0\Rightarrow$ NaN.

Normalized form keeps $1.xxx$ before point to maximize precision.

4. Character Codes

To represent text/symbols.

• ASCII (7-bit, extended 8-bit): $2^7=128$ chars: 0-31 control, 32-127 printable. E.g. 'A'=65=0x41=1000001, 'a'=97, '0'=48. Extended ASCII 256 chars.
• EBCDIC (8-bit IBM): 256 chars, used in mainframes.
• Unicode: Universal. UTF-8 variable 1-4 bytes backward compatible with ASCII, UTF-16 2/4 bytes, UTF-32 fixed 4 bytes. Can represent >1M code points, covers all world scripts. E.g. 'A' = U+0041, '₹' = U+20B9.

Conversion example: $25.5_{10}=11001.1_2$ fixed point $00011001.1000$, floating $1.10011\times2^4$ IEEE single 0x41CC0000.