Back to Use rational approximations of irrational numbers — Problem 3 · Task Set 3

8.NS.A.2 Tutor Intake -- Rational Approximations of Irrational Numbers

This short check helps your tutor see where to start. Work each problem on
your own, without a calculator. If you are not sure, give your best try —
every answer helps your tutor plan your first few sessions.

Grade 8·10 problems·~11 min·Common Core Math - Grade 8·container·8-ns-a-2

Work through problems with immediate feedback

Concepts

A student tracks the error in approximating $\sqrt{2}$ : at 1
decimal place the error is 0.04 (from $1.4^2 = 1.96$ ); at 2
decimal places the error shrinks to 0.0119 (from
$1.41^2 = 1.9881$ ). What happens to the error as more decimal
places are added?

The error stays exactly the same after the first two decimal places.

The error eventually grows again, because rounding mistakes accumulate the more digits you add, like a copying error compounding.

The error keeps shrinking — each additional digit makes the approximation roughly 10 times more accurate, and the process always converges, never gets worse.

The error becomes 0 once enough digits are added, since $\sqrt{2}$ can eventually be written exactly.