1 2.32 Mg = 2.32 × 106 g
2 1 000 mg = 1 000 × 10−3 g = 1 g
3 400 μg = 400 × 10−6 g = 4.00 × 10−4 g
Worked Example 0.5
Express the following quantities in the units stated:
1 1 cm3 in m3
2 1 000 m3 in dm3
3 1 g m−3 in g dm−3
Solution
1 1 cm is the same as 0.01 m or 1 × 10−2 m.Therefore 1 cm3 = (0.01 m)3 = (1 × 10−2)3 m3 = 1 × 10−6 m3
2 1 m is 10 times bigger than 1 dm: 1 m = 10 dm.Therefore 1 m3 = 1 × 103 dm3.1 m3 is 1 × 103 (one thousand) times bigger than 1 dm3.Therefore 1 000 m3 = 1 000 × 103 dm3 = 1 × 106 dm3.
3 1 dm3 is one‐thousandth of a m3.If the concentration is equal to 1 g in 1 m3, there will be 1 000 times fewer grams in 1 dm3.So 1 g m−3 = 1 × 10−3 g dm−3.
0.5 Significant figures
When carrying out calculations in science, the answer must be given to the same level of accuracy as the values in the question. For example, if a balance weighs to two decimal places, the mass of substance weighed may be read as 5.02 g. The total number of significant figures in this number is three. This means we know the mass accurate to 3 significant figures.
The number can be rounded to a smaller number of significant figures as shown:
Three significant figures = 5.02 g
Two significant figures = 5.0 g
One significant figure = 5 g
An answer should be given to the same number of significant figures as the number with fewest significant figures in the calculation. For example, if the mass of solid is 5.02 g and the mass of water the solid is added to is 50 g (the same as 50 mL or 50 cm3), the total mass of the solid and water would be 5.02 g + 50 g = 55 g. The answer can only be given to two significant figures as the mass of water is only known to this level of accuracy. In fact, the mass of water could actually be any mass between 49.50 g and 50.49 g, which are both equivalent to 50 g when given to two significant figures.
There are some rules for determining how many significant figures are in a number:
1 Any zeroes before a digit are not significant. For example, 0.005 is only accurate to 1 significant figure.
2 Any zeroes after a digit are significant. For example, 0.00500 is accurate to 3 significant figures.
3 Digits below the number 5 are always rounded downwards, and digits equal to or above the number 5 are rounded upwards, as in the following examples:The number 0.544 becomes 0.54 when written to two significant figures.The number 0.545 becomes 0.55 when written to two significant figures.The number 0.546 becomes 0.55 when written to two significant figures.
4 When performing a calculation, determine the required number of significant figures and round up or down at the end of the calculation, not at the steps in between.
5 Always give your answer to the same accuracy as that of the value known to the least number of significant figures in the calculation.
Worked Example 0.6
Convert the following numbers to values with two significant figures, and write the answer in scientific notation:
1 9 495 g
2 0.00940 g
3 0.09056 g
4 19.005 g + 1.515 g
Solution
1 9 495 g has four significant figures and becomes 9 500 g to two significant figures. In scientific notation, this is written as: 9.5 × 103 g.
2 0.00940 g has three significant figures and becomes 0.0094 g to two significant figures. In scientific notation, this is written as: 9.4 × 10−3 g.
3 0.09056 g has four significant figures and becomes 0.091 g to two significant figures when rounded up. In scientific notation, this is written as: 9.1 × 10−2 g.
4 The sum of 19.005 g + 1.515 g is 20.520 g, but this value has five significant figures. When it is rounded to two significant figures, it becomes 21 g as 0.52 rounds up to 1.0.
0.6 Calculations using scientific notation
0.6.1 Adding and subtracting
When adding numbers expressed using scientific notation, it is often useful to write the numbers using non‐standard coefficients to simplify the mathematical process.
For example, the number 4 242 can be written in any of the following ways and retains its original numerical value:
Only the last figure is standard scientific notation. However, if we were required to add the number 4 242 to 5.00 × 102 and give the answer in standard scientific notation, the following procedure could be used:
The example shows that the numbers are converted such that the exponent is common, i.e. 102 in this case, and then the numbers can be added.
Similarly, with numbers smaller than 1, the values are converted such that the exponents are common and added or subtracted in the normal manner. The exponent remains unchanged.
For example:
In this example the numbers are exact, that means we know them precisely. For example the number of people in a room. We therefore don’t need to give the answer to a specific number of significant figures as, effectively, the number of significant figures is infinite.
This answer must be rounded, as we only know the original values to an accuracy of three significant figures.
Worked Example 0.7
Calculate the values of the following, giving your answers in scientific notation:
1 102 + 1.310 × 103 =
2 0.057 90 + 1.3 × 10−4 =
3 3.120 × 10−2 − 5.7 × 10−4 =
4 6.375 × 103 − 0.103 × 102 =
Solution
1 Express both numbers in a format that has the same exponent, and then add:
2 Express both numbers in a format that has the same exponent, and then subtract:
3 Express both numbers in a format that has the same exponent, and then subtract:
4 Express both numbers