Floating-point arithmetic is more complicated than it seems, and it may bring very surprising results!

1. Can you count on your computer?
Hamed Zakerzadeh
Hamed Zakerzadeh Can you count on your computer? 1 / 4

2. Fibonacci numbers 1,1,2,3,5,8,⋯
describe rabbit’s population:
They can be written as a recursive sequence:
for k ≥ 2 fk = fk−1 + fk−2, f0 = f1 = 0
Johannes Kepler (17th
century) showed that the ratio of consecutive
Fibonacci numbers converges to the golden ratio
rk =
fk+1
fk
= {1,2,1.5,1.67,1.63,1.62,⋯} or lim
k→∞
rk = ϕ ≈ 1.618
Hamed Zakerzadeh Can you count on your computer? 2 / 4

3. Can your computer confirm that rk → ϕ?
import numpy as np
N = 50
f = np.zeros(N,dtype=np.int32)
r = np.zeros(N-1)
f[0] = 1
f[1] = 1
r[0] = f[1] / f[0]
for k in range(2,N):
f[k] = f[k-1] + f[k-2]
r[k-1] = f[k] / f[k-1]
The division of positive numbers gets negative!
Your computer allocates 32 bits of memory to each Fibonacci number. So
they cannot be saved properly if they get bigger than 231
.
Around k = 40, they pass this limit. Ô⇒ everything collapses! /
Hamed Zakerzadeh Can you count on your computer? 3 / 4

4. Lesson of the day
Computers don’t perceive the numbers like us. They save them with
a limited precision (like floating-point format).
You may count on your computer but check possible sources of error.
Look here to see more practical examples:
Hamed Zakerzadeh Can you count on your computer? 4 / 4