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Number · Standard form

Chapter 1 · 4

The idea

Standard form in context

How standard-form arithmetic solves real scale problems — distances, masses, populations — and why unit conversion has to happen before the powers of ten can do their work.

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Number · Standard form

Standard form in context

How standard-form arithmetic solves real scale problems — distances, masses, populations — and why unit conversion has to happen before the powers of ten can do their work.

Why it works

Context washes away

Standard form exists because of context problems: light speed, atoms, national populations. The physics and geography wash away — underneath, every question is one of the four operations from the calculating toolkit, plus two habits that carry the marks.

Habit 1: convert units first

Powers of ten only compare like with like. If a length is 1.41.4 km and a cell diameter is 7×10−67 \times 10^{-6} m, the kilometres must become metres first: 1.4 km=1.4×1031.4 \text{ km} = 1.4 \times 10^3 m. Then the division is honest:

1.4×1037×10−6=0.2×109=2×108\frac{1.4 \times 10^3}{7 \times 10^{-6}} = 0.2 \times 10^{9} = 2 \times 10^8

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The rest of the explanation, plus 2 worked examples you step through move by move.

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