Significant Figures

Worked example

1. 1

   $0.00407$ — The question: how many significant figures? Look for the first digit that is not zero.

2. 2

   $0.00407$ — Skip the zeros in front. They only show where the point is, so start counting at the 4.

3. 3

   $4,0,7$ — After the 4, every digit counts. The 0 sits between the 4 and the 7, so it counts too.

4. 4

   $3\text{ significant figures}$ — Count them: 4, 0 and 7 make 3.

Worked example

1. 1

   $36.4827$ — The question: keep 3 significant figures. Start at the 3, the first digit that is not zero.

2. 2

   $36.4827$ — Count 3 figures from the left: 3, 6, 4. These are the ones you keep.

3. 3

   $36.4827$ — Look at the next figure, the 4th. It decides whether the 4 changes.

4. 4

   $8\ge 5$ — 8 is 5 or more, so round the last kept figure up: the 4 becomes 5.

5. 5

   $36.5$ — Drop every figure after the 3rd. The point stays where it was, so the number keeps its size.

Worked example

1. 1

   $2.39962$ — The question: write it correct to 4 significant figures. Start at the 2, the first figure.

2. 2

   $2.39962$ — Count 4 figures: 2, 3, 9, 9. These are the ones you keep.

3. 3

   $2.39962$ — The 5th figure is 6. That is 5 or more, so the last 9 rounds up.

4. 4

   $2.399+0.001=2.400$ — A 9 that rounds up becomes 10, so it carries left: 2.399 becomes 2.400.

5. 5

   $2.400$ — Keep both zeros. They are the 3rd and 4th significant figures, so they show you rounded to 4.

Worked example

1. 1

   $5.962$ — Keep 2 significant figures. Start at the 5, the first digit, so keep 5 and 9.

2. 2

   $5.962$ — The 3rd figure decides. It is 6, which is 5 or more, so round the 9 up.

3. 3

   $5.9+0.1=6.0$ — 9 tenths plus 1 tenth makes 10 tenths, which carries into the units. So 5.9 becomes 6.0.

4. 4

   $6.0$ — Keep the 0. It is the 2nd significant figure, so it shows you rounded to 2.

Worked example

1. 1

   $0.03068$ — Skip the zeros in front. The first significant figure is the 3.

2. 2

   $0.03068$ — For 2 figures, keep 3 and 0. This 0 counts, because it comes after the 3.

3. 3

   $0.03068$ — The next figure is 6. That is 5 or more, so the 0 rounds up to 1.

4. 4

   $0.031$ — Drop the rest. The front zeros stay, so the number keeps its size. That is 2 significant figures.

5. 5

   $0.03068$ — For 3 figures instead, keep 3, 0 and 6.

6. 6

   $0.03068\to 0.0307$ — The next figure is 8, so the 6 rounds up to 7.

Worked example

1. 1

   $48599$ — Keep 2 significant figures: the 4 and the 8.

2. 2

   $48599$ — The 3rd figure decides. It is 5, which is 5 or more, so the 8 rounds up to 9.

3. 3

   $49000$ — Put zeros where the dropped figures were. 48 599 is about forty-eight thousand, so the answer must stay in the thousands.

4. 4

   $48599$ — For 3 figures instead, keep 4, 8 and 5.

5. 5

   $48599$ — The 4th figure is 9, so the 5 rounds up to 6.

6. 6

   $48600$ — Fill the last two places with zeros, again to keep the size.

Worked example

1. 1

   $\frac{0.8-2.7^3}{3.6^2}$ — The question: give the answer to 3 significant figures. Type it with brackets round the whole top and the whole bottom.

2. 2

   $=-1.457021605$ — Write down every figure on the display. This line can earn a mark even if your rounding goes wrong.

3. 3

   $-1.457021605$ — Ignore the minus for now. Count 3 figures from the 1: 1, 4, 5.

4. 4

   $-1.457021605$ — The 4th figure is 7. That is 5 or more, so the 5 rounds up to 6.

5. 5

   $-1.46$ — Put the minus back. Rounding never changes the sign, so the answer stays negative.