converting numbers that are less than 1 expressed in scientific notation back to decimal numbers, move the decimal point to the left by the value of the exponent, adding zeroes. So 2.42 × 10−3 moves the decimal point by three places to the left, requiring two zeroes after the decimal point.
Worked Example 0.2
Write the following numbers expressed in non‐exponential form using scientific notation:
1 675
2 1 000 000
3 0.98
4 0.00355
Solution
1 To write 675 in scientific notation, we must move the decimal point (currently after the 5) two spaces to the left (i.e. divide by 100) and then multiply the remaining number by 100 or 102.675 becomes 6.75 × 102.
2 To write 1 000 000 in scientific notation, we must move the decimal point (currently after the last 0) six spaces to the left (i.e. divide by 1 000 000) and then multiply the remaining number by 1 000 000 or 106.1 000 000 becomes 1 × 106.
3 To write 0.98 in scientific notation, we must move the decimal point one space to the right (i.e. multiply by 10) and then multiply the remaining number by 0.1 or 10−1 (i.e. divide by 10).So 0.98 becomes 9.8 × 10−1.
4 To write 0.00355 in scientific notation, we must move the decimal point three spaces to the right (i.e. multiply by 1 000) and then multiply the remaining number by 0.001 or 10−3 (i.e. divide by 1 000).So 0.00355 becomes 3.55 × 10−3.
Worked Example 0.3
Convert the following numbers, expressed in scientific notation, to non‐exponential form.
1 7.01 × 102
2 6.912 × 105
3 8.05 × 10−1
4 2.310 × 10−4
Solution
1 Move the decimal point to the right by the number of spaces indicated by the exponent, i.e. by 2. 7.01 becomes 701.
2 Move the decimal point to the right by the number of spaces indicated by the exponent, i.e. by 5. 6.912 × 105 becomes 691 200.
3 Move the decimal point to the left by the number of spaces indicated by the exponent, i.e. by 1 space. 8.05 × 10−1 becomes 0.805.
4 Move the decimal point to the left by the number of spaces indicated by the exponent, i.e. by 4 spaces. 2.310 × 10−4 becomes 0.0002310.
0.4 Using metric prefixes
Scientists also have a shorthand for writing some large and small numbers using a prefix that represents a certain quantity. So the quantity 1 000 g, or 1 × 103 g in scientific notation, can be written as 1 kg. The letter ‘k’ represents the factor ‘kilo’, or 103. The quantity one‐thousandth of a gram or 0.001 g (1 × 10−3 g) can be written as 1 mg, where the prefix ‘m’ represents the factor ‘milli’, or 10−3. Table 0.3 gives some common prefixes, along with their symbols.
Note that the ‘k’ in kg is lowercase. The case of the unit can be very important, so make sure you pay attention to it!
Table 0.3 Some common prefixes and their values with quantities and symbols.
| Factor | Number | Name | Symbol | Factor | Number | Name | Symbol |
|---|---|---|---|---|---|---|---|
| 109 | 1 000 000 000 | giga | G | 10−9 | 0.000 000 001 | nano | n |
| 106 | 1 000 000 | mega | M | 10−6 | 0.000 001 | micro | μ |
| 103 | 1 000 | kilo | k | 10−3 | 0.001 | milli | m |
| 10−2 | 0.01 | centi | c | ||||
| 10−1 | 0.1 | deci | d |
0.4.1 Units of mass and volume used in chemistry
Mass
The mass of a substance is the amount we measure when we weigh the substance. Mass is a measure of the quantity of a substance. In science, we use the term mass as opposed to weight of a substance to describe the amount because the weight of a substance is related to the gravitational force acting on the substance. We often use the words ‘mass’ and ‘weight’ interchangeably, but they are not really the same thing scientifically.
The SI unit of mass is the kilogram. However, in the laboratory, chemicals and other objects are usually weighed in units of grams as the gram is a far more convenient amount to work with. Even though we use a balance to weigh a substance, we report its mass in grams, not its weight.
One kilogram is equivalent to one thousand grams, 1 000 g. 1 kg = 1 000 g. Therefore 1 g =
Another common measure of mass is the milligram, mg. One milligram is one‐thousandth
