Number Representation

Signed Numbers

To evaluate how we store negative numbers, we measure against four key requirements:

1. Sign bit: Clear indication of polarity ($0 = +, 1 = -$).
2. Consistency: Incrementing the bit pattern corresponds to a logical increase in value.
3. Single Zero: Avoids logical ambiguity (prevents $+0$ and $-0$ logic errors).
4. Simple Arithmetic: Subtraction can use the same hardware as addition.

Comparison Table

Method	Sign Bit?	Consistent?	Single Zero?	Simple Math?
Sign-Magnitude	Yes	No	No	No
One’s Complement	Yes	Yes	No	No
Two’s Complement	Yes	Yes	Yes	Yes

Two’s Complement

• The Rule: To negate a number, invert all bits (NOT) and add 1.
• Why it wins: The CPU uses the same adder circuit for signed and unsigned integers. Subtraction is simply $A + (-B)$.
• Example (4-bit): $-5 \text{ (1011)} + 3 \text{ (0011)} = -2 \text{ (1110)}$

Bias (Offset) Encoding

Store value as: $Value_{Stored} = Value_{Actual} + Bias$.

• Purpose: Shifts the range so all stored bit patterns are non-negative.
• Benefit: Allows for unsigned comparison of signed values. This is why it is used for exponents in IEEE 754—it makes sorting floating-point numbers faster.

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Floating Point (IEEE 754)

Scientific Notation

Standard base-2: $1.xxxx \times 2^{exp}$

• The leading 1 is implicit (not stored) to maximize precision.

Single Precision (32-bit) Format

• Sign (1 bit): $0 = +, 1 = -$
• Exponent (8 bits): Biased by $127$.
• Significand (23 bits): The fractional part (mantissa).

Normalized Formula: $Value = (-1)^{Sign} \times (1 + Significand) \times 2^{(Exponent - 127)}$

Special Cases

Category	Exponent	Significand	Value/Purpose
Zero	0000 0000	$0$	$\pm 0.0$
Denormal	0000 0000	Non-zero	Underflow protection; No implicit $1$
Infinity	1111 1111	$0$	$\pm \infty$
NaN	1111 1111	Non-zero	Not a Number (e.g., $0/0$)

Denormalized Formula: Used for values too small for the standard format. The exponent is fixed at $-126$. $Value = (-1)^{Sign} \times (0 + Significand) \times 2^{-126}$

Precision and Step Size

Step Size: The gap between consecutive floating-point numbers (ULP - Unit in the Last Place).

• Normalized Step: $2^{(Exponent - 127 - 23)}$
• Denormalized Step: $2^{(-126 - 23)}$ (Constant gap)

Key Implications

1. Relative Precision: Accuracy is high near zero and decreases as magnitude increases.
2. Inexact Representation: Most decimal numbers (like $0.1$) cannot be represented exactly in binary floating point.
3. Absorption: If a number is large enough, adding $1.0$ to it does nothing because the “step” is larger than $1.0$.