Approximation and Estimation

Chapter 3 Study Notes

Section 4 of 7
Contents

3.3Rounding to Significant Figures

  1. Find the first non-zero digit. Count the required number of significant figures from there.
  2. Examine the next digit on the right.
  3. If it is 5 or more, round up. If it is less than 5, round down.
  4. For whole numbers, replace remaining digits with zeros as placeholders.
  5. For decimals, drop the remaining digits (but keep trailing zeros within the required s.f.).

Rounding a whole number to s.f.

Round 3756237\,562 to 3 significant figures.

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First 3 s.f.: digits 3,7,53, 7, 5. Next digit: 66 (5\geq 5).
Round up: 375376375 \to 376, then add placeholder zeros.
37562=3760037\,562 = 37\,600 (to 3 s.f.).

Rounding a decimal to s.f.

Round 0.0038470.003847 to 2 significant figures.

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Leading zeros (0.000.00) are not significant (Rule 3).
First 2 s.f.: digits 3,83, 8. Next digit: 44 (less than 5).
Round down: 0.003847=0.00380.003847 = 0.0038 (to 2 s.f.).

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Predict

A number rounded to 2 significant figures gives 47004700. What is the LARGEST value it could have been? Commit to an answer before reading on.

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Rounding to 2 s.f. here means rounding at the hundreds place.
The smallest value that rounds to 47004700 is 46504650, because 46504650 rounds up.
The largest is anything just below 47504750, because 47504750 itself rounds up to 48004800.
So the original number nn satisfies 4650n<47504650 \leq n < 4750. The answer is not 46994699: a value like 4749.64749.6 still rounds to 47004700.

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