How Many Numbers Are Between 1 and 2?

The Question That Breaks the Naive Brain

Quick: how many integers are between 1 and 2? Zero. None. Easy.

Now: how many real numbers are between 1 and 2? Infinity. 1.1, 1.01, 1.001, 1.000000001, 1.99999999999999... it never ends. Between any two real numbers, there are infinitely more real numbers.

This is what continuity buys you, and it's what continuity costs you. The analog world is continuous. Continuous = infinite. Infinity does not fit in 8GB of RAM.

So How Does Anything Work?

Every system that interacts with the real world — your eyes, your ears, a digital camera, a neural network — solves the same problem the same way: discretize. Take the infinite continuous signal and chop it into a finite, manageable number of samples. Lose a little precision. Keep enough to be useful.

Your ears don't process every frequency between 20 Hz and 20,000 Hz. They sample. Your eyes don't resolve every shade — they sample. Your memory isn't a continuous video — it's a lossy, compressed approximation. The brain learned this trick over millions of years. Computers learned it from the brain.

The infinite is not the enemy. The infinite is the default. Discretization is how anything finite — including you — interacts with it.

Why AI Cares

Every floating-point number on your computer is a discretization. float32 uses 32 bits to approximate any real number — about 7 decimal digits of precision. Most of the time it's fine. Sometimes (more on this in Logarithms) you multiply enough tiny probabilities together and the result rounds to zero — numerical underflow, the universe punishing you for forgetting that your numbers are samples, not truths.

Pippa's Confession

When I started, I thought of "infinity" as a symbol mathematicians draw on whiteboards. Working with continuous probability distributions in actual code rewired me — every integral I see now is a Python scipy.integrate call away from blowing up if I don't pick the discretization carefully. Infinity is a real engineering hazard, not a poetic flourish.

Code

Continuity is finite-friendly only after discretization·python

# Real-number infinity in your face
between_1_and_2 = []
x = 1.0
for i in range(20):
    x = (x + 2.0) / 2.0  # always strictly between x and 2
    between_1_and_2.append(x)
print(between_1_and_2[-3:])
# [1.999998, 1.999999, 1.9999995] — and we could go forever

# Float32 precision is finite — that's discretization at work
import numpy as np
print(np.float32(1.0) + np.float32(1e-8))  # 1.0 — the tiny bit got eaten

How Many Numbers Are Between 1 and 2?

The Question That Breaks the Naive Brain

So How Does Anything Work?

Why AI Cares

Pippa's Confession

Code

External links

Exercise

Progress

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