Significant figures: the digits you can trust, plus one

Certain digits and one doubtful digit

In Fig. 1.14 a rod ends between 4.6 cm and 4.7 cm. One student writes 4.6 cm, another 4.7 cm. Both agree on the 4. The next digit is only an estimate, so it is the doubtful digit.

The accurately known digits plus the first doubtful digit are the significant figures. They show at a glance how fine the measurement was.

All digits 1 to 9 count. Zeros need rules, because some zeros are measured and some only place the decimal point.

Significant figures = accurately known digits + the first doubtful digit. Leading zeros never count; a zero after the decimal at the end always does.

Key terms

Significant figures

The digits of a measurement that are reliably known.

Doubtful digit

The first digit found only by estimation.

Leading zeros

Zeros on the left that only place the decimal point.

Trailing zero after a decimal

A zero written at the end after the point, because it was measured.

Short questions with model answers

1. Q1. Problem 1.6 (a), (b): how many significant figures in 0.0045 m and 2.047 m?

   • 0.0045 m: the three zeros are on the left, so only 4 and 5 count.
   • 2.047 m: the zero sits between 2 and 4, so all four count.

   2 and 4, as in the book's answers.

2. Q2. Quick Quiz (b)–(d): 12.5 cm, 0.125 m and 0.000125 km. How many significant figures in each?

   • 12.5 cm = 0.125 m = 0.000125 km
   • Ignore the leading zeros: 1, 2, 5 remain each time.

   3 in every case.

Common mistakes

• ✗ Counting 0.0034 m as 5 significant figures.

  ✓ Zeros on the left only place the point: 2.

• ✗ Treating 2.40 mm as 2.4 mm with 2 significant figures.

  ✓ The final zero after the decimal was measured: 3.

• ✗ Counting the 10 and the 4 in 3.50 × 10⁴ m.

  ✓ Only digits before the power of ten count: 3.

MCQs

1. 1. Problem 1.6 (c): significant figures in 3.40 m:

   1. (a) 2
   2. (b) 3
   3. (c) 4
   4. (d) 1
   Show answer

   (b) The final zero after the decimal counts (rule c).

2. 2. Problem 1.6 (d): significant figures in 3.420 × 10⁴ m:

   1. (a) 3
   2. (b) 2
   3. (c) 5
   4. (d) 4
   Show answer

   (d) 3, 4, 2, 0 before the power of ten (rules c, d).

3. 3. Significant figures in 0.0034 m:

   1. (a) 2
   2. (b) 4
   3. (c) 5
   4. (d) 3
   Show answer

   (a) Leading zeros do not count (rule b).

4. 4. Significant figures are:

   1. (a) All the digits written
   2. (b) Only the certain digits
   3. (c) The accurately known digits and the first doubtful digit
   4. (d) The digits after the decimal point
   Show answer

   (c) Your book's definition, p. 21.

5. 5. Significant figures in 5.06 m:

   1. (a) 2
   2. (b) 1
   3. (c) 4
   4. (d) 3
   Show answer

   (d) A zero between two digits counts (rule a).

Quick revision

• Significant figures are the known digits plus the first doubtful one.
• Zeros between digits and trailing zeros after a decimal count; leading zeros do not.
• In scientific notation, count only the digits before the power of ten.