Chapter 2: Integers and Floats — The Secrets of Numbers

Vol 3: Computer Core Expedition · Chapter 2

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Metadata Card

Attribute	Value
Difficulty	(Advanced)
Prerequisites	Chapter 1 (Bits, Bytes, Number Bases)
Keywords	Two's Complement, Unsigned Overflow, Signed Overflow, IEEE 754, Floating Point Precision, Rounding Mode

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Your Progress

"Descending deeper, the Core is paved with numbers — integers and floats. But these aren't the numbers you learned in Variable Village — their underlying representation hides countless traps."

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Encounter 1: Unsigned Integers — 0 to 2ⁿ-1

Unsigned range: 0 to 2ⁿ - 1

Unsigned overflow is well-defined in C — modular arithmetic (wrapping).

uint8_t x = 255;
x = x + 1;  // wraps to 0
x = x - 1;  // wraps to 255

Encounter 2: Signed Integers — Two's Complement

Two's complement solves three problems:

1. No negative zero
2. Same adder for add/subtract
3. Natural comparison

For n-bit signed:
  Positive: 0 to 2ⁿ⁻¹ - 1
  Negative: -1 to -2ⁿ⁻¹

8-bit example:
  01111111 → 127 (max)
  00000001 → 1
  00000000 → 0
  11111111 → -1
  10000000 → -128 (min)

Key insight: x - y = x + (~y + 1) — one adder handles both.

Encounter 3: Signed Overflow is Undefined Behavior

In C, signed integer overflow is undefined behavior. Compilers may assume it never happens and optimize accordingly.

int foo(int x) {
    // Compiler can assume x+1 > x is always true
    if (x + 1 <= x) {
        handle_overflow();  // May be deleted!
    }
    return x + 1;
}

Encounter 4: IEEE 754 Floating Point

Formula: (-1)ˢ × 1.m × 2ⁱᵉ⁻ᵇⁱᵃˢ⁾

Format	Bits	Sign	Exponent	Mantissa	Precision
float	32	1	8	23	~7 digits
double	64	1	11	52	~15-16 digits

Encounter 5: 0.1 + 0.2 ≠ 0.3

0.1 in binary is an infinitely repeating fraction. No finite-precision float can represent it exactly.

Golden Rule: Never compare floats with ==. Always use an epsilon:

#define EPSILON 1e-9f
int feq(float a, float b) {
    return fabsf(a - b) < EPSILON;
}

Common Pitfalls

1. Accumulation error: 10⁷ iterations of += 0.1f can accumulate significant error
2. Catastrophic cancellation: 1e20 + 1.0f - 1e20 = 0 in single precision
3. Non-associativity: (a + b) + c ≠ a + (b + c)

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Verification Checklist

• [ ] Can manually compute two's complement representation
• [ ] Can explain why signed overflow is UB in C
• [ ] Can explain why 0.1 + 0.2 ≠ 0.3
• [ ] Can write a safe float comparison function

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Traveler's Notes

• Two's complement: one adder for everything, at the cost of asymmetric range
• Signed overflow = UB (C). Unsigned overflow = well-defined modular arithmetic
• 0.1 + 0.2 ≠ 0.3 is a math fact, not a computer bug

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Chapter 3: Digital Logic — From Logic Gates to Computers