THE DEFINITION
e := lim(k→∞) (1 + 2⁻ᵏ)^(2ᵏ)
Compound interest interpretation:
• Rate per period: r = 2⁻ᵏ
• Number of periods: n = 2ᵏ
• Final amount: (1 + r)ⁿ → e
Compound interest interpretation:
• Rate per period: r = 2⁻ᵏ
• Number of periods: n = 2ᵏ
• Final amount: (1 + r)ⁿ → e
CURRENT COMPUTATION
STEPS (2ᵏ)
1024
STEP SIZE (2⁻ᵏ)
0.000977
MULTIPLICATIONS
10
ERROR
~10⁻³
CONVERGENCE TABLE
| k | Estimate | Error |
|---|
WHY DYADIC IS PERFECT
• Step size Δ = 2⁻ᵏ (exact in binary)
• Number of steps = 2ᵏ (power of 2)
• Repeated squaring: only k multiplies!
• (1 + Δ)^n via: square k times
• Each k adds ~1 bit of precision
• Number of steps = 2ᵏ (power of 2)
• Repeated squaring: only k multiplies!
• (1 + Δ)^n via: square k times
• Each k adds ~1 bit of precision
DIFFERENTIAL EQUATION VIEW
e solves: dy/dt = y, y(0) = 1
Euler's method with step Δ = 2⁻ᵏ:
y(t+Δ) = y(t) · (1 + Δ)
After 2ᵏ steps (t = 1):
y(1) = (1 + 2⁻ᵏ)^(2ᵏ) → e
Euler's method with step Δ = 2⁻ᵏ:
y(t+Δ) = y(t) · (1 + Δ)
After 2ᵏ steps (t = 1):
y(1) = (1 + 2⁻ᵏ)^(2ᵏ) → e