Measurement and uncertainty

Very large and very small numbers

A teaspoon of water holds about $170\,000\,000\,000\,000\,000\,000\,000$ molecules, and one molecule has a mass of about $0.000\,000\,000\,000\,000\,000\,000\,03\ \text{g}$. Numbers like these are hard to read and easy to copy wrongly. So we write them in scientific notation:

Here $n$ is a whole number, positive or negative. Count how many places the decimal point moves.

• $45\,600 = 4.56 \times 10^{4}$: the point moves 4 places to the left.
• $0.000\,72 = 7.2 \times 10^{-4}$: the point moves 4 places to the right.

To multiply, multiply the first parts and add the powers: $(2 \times 10^{3}) \times (3 \times 10^{4}) = 6 \times 10^{7}$. To add, first write both numbers with the same power: $4.0 \times 10^{3} + 2.0 \times 10^{2} = 4.0 \times 10^{3} + 0.20 \times 10^{3} = 4.2 \times 10^{3}$.

Every measurement has an uncertainty

A ruler marked in millimetres can show that a pencil is between $11.2\ \text{cm}$ and $11.3\ \text{cm}$ long. Your eye can judge a little more, so you may write $11.24\ \text{cm}$. The last digit is an estimate. We say the length has 4 significant figures: the digits that are certain, and one estimated digit after them.

Writing $11.240\ \text{cm}$ would claim a precision that the ruler does not have. Writing $11\ \text{cm}$ throws good information away. The number of digits you write tells the reader how good your measurement is.

The rules of significant figures

1. Every non-zero digit is significant. $285$ has 3.
2. A zero between non-zero digits is significant. $2.005$ has 4.
3. Zeros at the left, before the first non-zero digit, are not significant. They only place the decimal point. $0.0032$ has 2.
4. Zeros at the right are significant when there is a decimal point. $0.200$ has 3, and $4.50$ has 3.
5. Zeros at the right of a whole number with no decimal point are not clear. Is $100$ good to 1, 2 or 3 figures? Scientific notation settles it: $1.00 \times 10^{2}$ has 3.
6. Exact numbers have no uncertainty. Twenty students in a room, or the $1000$ in $1\ \text{L} = 1000\ \text{mL}$, have as many significant figures as you need.

Significant figures in calculations

Adding and subtracting. The answer keeps as many decimal places as the number with the fewest.

Multiplying and dividing. The answer keeps as many significant figures as the number with the fewest.

Rounding off

To round to a given number of figures, look at the first digit you drop.

• If it is more than 5, add one to the digit before it: $1.386 \rightarrow 1.39$.
• If it is less than 5, leave the digit before it: $4.334 \rightarrow 4.33$.
• If it is exactly 5, the textbook rule looks at the digit before it. An even digit stays the same, and an odd digit goes up by one: $4.365 \rightarrow 4.36$, but $4.375 \rightarrow 4.38$.

Round only the final answer. If you round at every step, the small errors add up.

Practise with new numbers

Precision and accuracy

These two words do not mean the same thing.

• Precision is how close repeated measurements are to each other.
• Accuracy is how close the measurements are to the true value.

Suppose the true mass of a sample is $2.00\ \text{g}$. One student reads $1.95\ \text{g}$ and $1.93\ \text{g}$: the two readings agree well, so they are precise, but they are not accurate. Another reads $2.01\ \text{g}$ and $1.99\ \text{g}$: precise and accurate. A balance that is badly set gives precise readings that are all wrong by the same amount.

Try it in Python

The program counts the significant figures of a number written as text. It follows the rules above.

Common mistakes

• Counting the zeros at the left. In $0.0045$, only the $4$ and the $5$ are significant.
• Using the wrong rule. Sums keep decimal places; products keep significant figures.
• Rounding at every step. Keep the extra digits until the end, then round once.
• Thinking precise means correct. Readings can agree with each other and still all be wrong.

Remember these

• Scientific notation is $N \times 10^{n}$, with $1 \le N < 10$.
• Significant figures are the certain digits and one estimated digit.
• Zeros at the left never count; zeros at the right count after a decimal point.
• Sums keep the fewest decimal places. Products and quotients keep the fewest significant figures.
• Precision is agreement between readings. Accuracy is closeness to the true value.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. How many significant figures are in $0.0503$, in $2.30 \times 10^{-3}$ and in $5000$?

2. Write $0.000\,0782\ \text{m}$ in scientific notation.

3. Work out $25.68\ \text{mL} - 3.1\ \text{mL}$ with the right number of figures.

4. Mohan measures a length three times and gets $9.81\ \text{cm}$, $9.80\ \text{cm}$ and $9.82\ \text{cm}$. The true length is $9.50\ \text{cm}$. Describe his readings.

5. Round $4.365$ and $4.375$ to three significant figures by the textbook rule.