Level 5 · Ages 15–16
Read it again any time, including while you work through the exercises.
Your calculator shows 12 digits, but real mathematical intelligence is knowing which ones matter. Announcing that a trip takes 47.3821 minutes is absurd: nobody measures a trip to the ten-thousandth of a minute. Estimating and rounding sensibly is a scientist's skill.
First tool: the order of magnitude. Before computing 512 × 198, mentally replace it with 500 × 200 = 100000. In one second, you know the result is around one hundred thousand. It is your error radar: if the calculator shows 10137.6, you instantly know a typo slipped in somewhere.
For very large and very small numbers, we express the order of magnitude with a power of 10: the Earth-Moon distance, 384400 km, has an order of magnitude of 4 × 10⁵ km. Mind the threshold rule: 9.7 × 10⁶ is closer to 10⁷ than to 10⁶, because 9.7 is almost 10.
Second tool: sensible rounding. The precision of a result cannot exceed the precision of the starting measurements. If you measure with a tape graduated in centimetres, announcing a result to the tenth of a millimetre is a lie of precision. We keep a number of digits consistent with the instrument and the purpose.
In this course you will practise estimating before computing, choosing the right power of 10, and cutting off decimals that mean nothing. Precise when needed, fast when enough!
Without a calculator, give an order of magnitude for 49 × 21, then compare with the exact result.
49 ≈ 50 et 21 ≈ 20
I replace each factor with the nearest round number.
50 × 20 = 1000
The computation becomes instant: the result should be around 1000.
49 × 21 = 1029
The exact result, 1029, is indeed close to my estimate. If I had gotten 10290 or 103, I would have known something was wrong.
Result: Order of magnitude: 1000. Exact result: 1029. The estimate validates the computation.
The Earth-Moon distance is 384400 km. Give its order of magnitude as a power of 10.
384400 ≈ 400000
I round to the first significant digit: 384400 is close to 400000.
400000 = 4 × 10⁵
I write this round number as one digit times a power of 10: 4 followed by 5 zeros.
384400 km ≈ 4 × 10⁵ km
The order of magnitude of the Earth-Moon distance is 4 × 10⁵ km, a few hundred thousand kilometres.
Result: 384400 km ≈ 4 × 10⁵ km.