When you feed a result back into the same equation over and over, small differences can compound. In the logistic equation x(n+1) = rx(n)(1-x(n)), some constants produce sequences that settle on one number, while others create chaos with no pattern at all. Even a tiny change in precision can shift the outcome from a steady pattern to total unpredictability.

Tiny changes in a starting point can cause wildly different results — and precision itself counts as a starting condition. When you run the iterative equation x(n+1) = rx(n)(1-x(n)) at different rounding levels, something striking happens: just changing the decimal precision, without touching the math itself, can switch the outcome from alternating to steady. Across 26 rounds with constants r ranging from 2.70 to 3.70, some values settle on one number, others alternate between two, and still others spiral into total unpredictability. That shift from order to chaos driven by rounding alone — not by any change in the equation — is chaos theory made visible in a spreadsheet.

When you run the equation x(n+1) = rx(n)(1-x(n)) for 26 rounds at 2, 3, 4, and 15 decimal places, the number of digits you carry through each step determines which long-term behavior emerges. For certain constants, a sequence might settle on one number at 15 decimal places but alternate between two values at just 2. That means changing the rounding level alone can switch the outcome from alternating to steady — not because the math changed, but because the exact digits you kept did.