2.3 Practice problems
You may need to revise how to round numbers to complete some of these problems.
Why do we round numbers?
In general, there are two reasons:
- To make a number simpler but keep its value close to what it was.
- To round to the correct number of significant figures.
Rules for rounding
If the number to the right of the rounding digit is less than 5, then keep the rounding digit and change the rest of the digits right of the rounding digit to 0.
If the number to the right of the rounding digit is greater than or equal to 5, then add one to the rounding digit and change the rest of the digits right of the rounding digit to 0.
When rounding significant figures, the standard rules of rounding numbers apply, except those non-significant digits to the left of the decimal are replaced with zeros.
Example: 356 rounded to 2 significant digits is 360.
Note
These are ‘rules of thumb’. There are a number of methods for rounding. For example, some statisticians will round to the nearest even number if the final digit is 5 to avoid bias. For example, 12.5 is rounded to 12 and so is 11.5. Ultimately, in practice you need to think about why you are rounding in the first place and apply a consistent method.
- How many significant figures are in each of these measurements?
- 95.0°C
- 120°C
- 501 g
- 15 patients
- 0.450 ml
- 250 mM
- 2.38 × 10–3 g
- 0.00238 g
- 2,050 mmol
- 2.050 mmol
- Rewrite the following answers so they are rounded to the correct precision (correct number of significant figures).
-
- 1.22 M × 1.3 l = 1.586 mol
- 500.00 g ÷ 125 ml = 4 g ml–1
- 516.15 g − 0.005 g = 516.145 g
- 1.023 moles / 2.1 l = 0.4871 M
- 9.000 s − 0.5 s = 8.50 s
- The average height of 3 students with heights 170.3 cm, 167.23 cm and 171 cm
Solution to Practice Problem 2.2a.
Solution to Practice Problem 2.2c
Significant figures and logarithms
Logarithmic values[1] when expressed as a decimal have two parts. The integer part is called the characteristic while the part to the right of the decimal is called the mantissa. So, for example log1015.0 = 1.176. In this case 1 is the characteristic and 0.176 is the mantissa.
In terms of significant figures in logarithmic values only the numbers to the right of the decimal place (mantissa) count as significant. That is, the number of significant figures in the mantissa of a value expressed in scientific notation equals the number of significant figures to the right of the decimal in the logged value.
This is important for logarithmic quantities such as pH.
pH = –log[H+]
For example, pH = 8.85 has only 2 significant figures. It corresponds to a [H+] of 1.4 × 10–9 M (2 significant figures).
A pH of 7.6 corresponds to a concentration known to 1 significant figure.
Therefore, a pH given in a whole number such as pH = 4 corresponds to a concentration expressed only as a power of 10, in this case 10–4 M.
Significant figures in pH readings
In the case below, the pH reading of 7.29 involves only 2 significant figures. The leading digit in front of the decimal is not significant, only the numbers after the decimal are significant figures.
This is because the 7 indicates the magnitude of the number while the .29 indicates the number itself. This pH corresponds to a [H+] of 5.1 × 10–8 M, as pH = –log[H+]
To see why this is so, take the log of 5.1 × 10–8
That is, log (5.1 × 10–8)
The log of a product is equal to the sum of the logs of each multiplier.
Therefore, log (5.1 × 10–8) = log (5.1) log (10–8)
log (5.1) = 0.71 (2 significant figures, reflecting the uncertainty in the last digit of 5.1)
log (10–8) = –8.000000 … an infinite number of significant figures, as 10–8 is an exact number
Therefore, 0.71 (–8.000000 …) = –7.29
Remember, pH = –log[H+] so pH = 7.29
Here you are following the rules for addition of significant figures. That is, the number of places after the decimal point in the answer is less than or equal to the number of decimal places in every term in the sum.
- For more information on logarithms see Chapter 12. ↵
