Take care!
3 × 102 = 300 (i.e. two zeroes after the number before the decimal point) 3 × 10−2 = 0.03 (i.e. just one zero after the decimal point before the number)
When a number is written in exponential form, it is called scientific notation. The standard format is that the number is written with one digit before the decimal point and the multiplier 10n is added after the number. The following examples show how numbers can be rewritten using scientific notation:
4 is written as 4 × 100.Any number raised to the power of zero is equal to 1, so 100 = 1.
42 is written as 4.2 × 101.It isn't usual to use the factors 100 and 101 unless specifically required to give an answer in scientific notation.
242 is written as 2.42 × 102.
2424 is written as 2.424 × 103.
To express numbers smaller than 1 using scientific notation, i.e. a decimal number, the number is converted to a number between 1 and 10 and the exponential factor 10–n used to indicate the position of the decimal point, as in these examples:
0.4 is written as 4 × 10−1.
0.042 is written as 4.2 × 10−2.
0.00242 is written as 2.42 × 10−3.
Box 0.1
When 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