Standard form lets you write very large and very small numbers compactly and compare them at a glance. This note covers writing numbers in standard form, calculating with them, and giving answers to a stated accuracy. It is used all through physics, chemistry and data analysis.
Key ideas
A number is in standard form when it is written as \(a \times 10^{k}\), where \(1 \le a \lt 10\) and \(k\) is an integer. The number \(a\) is called the mantissa and \(k\) is the exponent (or power of ten).
- Large numbers have a positive \(k\). For example, \(48\,000\,000 = 4.8 \times 10^{7}\).
- Numbers between 0 and 1 have a negative \(k\). For example, \(0.00052 = 5.2 \times 10^{-4}\).
- Numbers from 1 up to (but not including) 10 have \(k = 0\), for example \(7.3 = 7.3 \times 10^{0}\).
To convert, count how many places the decimal point must move so that exactly one non-zero digit sits in front of it. Moving the point left gives a positive power. Moving it right gives a negative power.
Calculating with standard form uses the laws of exponents:
- Multiplying: multiply the mantissas, add the exponents.
- Dividing: divide the mantissas, subtract the exponents.
- Adding or subtracting: first write both numbers with the same power of ten, then add or subtract the mantissas.
After any calculation, check that the mantissa is still at least 1 and less than 10. If it is not, adjust the mantissa and the exponent together so the value does not change.
Calculators show standard form in different ways. Look for the "\(\times 10^{x}\)" key or the "EE" key on your GDC and use it to enter a number such as \(3.2\times10^{5}\). Never type the power of ten as a separate multiplication in brackets without checking the display, and always write your final answer as \(a \times 10^{k}\), not in calculator style such as 3.2E5.
Rounding works as normal. Three significant figures in standard form means three digits in the mantissa, so \(4.5678 \times 10^{6}\) rounds to \(4.57 \times 10^{6}\).
Standard form also makes comparison easy. Compare the exponents first: the number with the larger exponent is larger, provided both mantissas are in the correct range. If the exponents are equal, compare the mantissas. For example, \(9.1 \times 10^{-3}\) is bigger than \(2.7 \times 10^{-4}\), because \(-3 \gt -4\).
Units often need converting before you start. A millimetre is \(10^{-3}\) m, a kilometre is \(10^{3}\) m, and a nanometre is \(10^{-9}\) m. Converting into one unit first keeps the powers of ten easy to handle.
Standard form is closely linked to significant figures. The mantissa carries the precision: \(2.50 \times 10^{3}\) has three significant figures, while \(2.5 \times 10^{3}\) has two. When you are told to give an answer to 3 significant figures, keep any trailing zero in the mantissa, because it shows the precision you have claimed. For instance, a speed of exactly two thousand five hundred metres per second to 3 s.f. is \(2.50 \times 10^{3}\) m s\(^{-1}\).
It also helps to know sensible sizes. The mass of a car is of order \(10^{3}\) kg, the width of an atom is of order \(10^{-10}\) m, and the population of the world is of order \(10^{10}\). Using these as a check, you can spot an answer that is wildly too big or too small, often because of a slip in a sign or an exponent.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| \(a \times 10^{k}\), \(1 \le a \lt 10\), \(k \in \mathbb{Z}\) | Standard form of a number. |
| \((a \times 10^{m})(b \times 10^{n}) = ab \times 10^{m+n}\) | Multiplying two numbers in standard form. Renormalise if \(ab \ge 10\). |
| \(\dfrac{a \times 10^{m}}{b \times 10^{n}} = \dfrac{a}{b} \times 10^{m-n}\) | Dividing. Renormalise if \(a/b \lt 1\). |
| \(10^{-n} = \dfrac{1}{10^{n}}\) | Negative powers mean reciprocals, so they give small numbers. |
| Significant figures | Count from the first non-zero digit. Standard form makes the number of significant figures obvious. |
Worked examples
Example 1 A hard drive stores \(3.2 \times 10^{5}\) files, each of average size \(4.5 \times 10^{-3}\) gigabytes. Find the total size in gigabytes, in standard form.
Total size = number of files \(\times\) size of each file.
\((3.2 \times 10^{5}) \times (4.5 \times 10^{-3})\)
Multiply the mantissas: \(3.2 \times 4.5 = 14.4\). Add the exponents: \(5 + (-3) = 2\).
This gives \(14.4 \times 10^{2}\). The mantissa is bigger than 10, so write \(14.4 = 1.44 \times 10^{1}\).
\(14.4 \times 10^{2} = 1.44 \times 10^{1} \times 10^{2} = 1.44 \times 10^{3}\).
Answer: \(1.44 \times 10^{3}\) gigabytes.
Example 2 A town has \(4.2 \times 10^{5}\) inhabitants and a neighbouring village has \(3.5 \times 10^{4}\). Find the total population in standard form.
The powers of ten are different, so rewrite the smaller number with the larger power.
\(3.5 \times 10^{4} = 0.35 \times 10^{5}\)
Add the mantissas: \(4.2 + 0.35 = 4.55\).
Answer: \(4.55 \times 10^{5}\).
Example 3 A human hair has a width of \(7.0 \times 10^{-5}\) m. A silicon chip feature is 14 nm wide. How many times wider is the hair than the feature?
Convert to the same unit. \(14\) nm \(= 14 \times 10^{-9}\) m \(= 1.4 \times 10^{-8}\) m.
Divide: \(\dfrac{7.0 \times 10^{-5}}{1.4 \times 10^{-8}} = \dfrac{7.0}{1.4} \times 10^{-5-(-8)} = 5 \times 10^{3}\).
Answer: the hair is \(5.00 \times 10^{3}\) times wider.
Common errors
- Common error: leaving an answer such as \(14.4 \times 10^{2}\) or \(0.5 \times 10^{3}\). The mantissa must satisfy \(1 \le a \lt 10\).
- Common error: adding exponents when adding numbers. The rule "add the exponents" only applies to multiplication. For \(4.2\times10^{5}+3.5\times10^{4}\) the powers must be matched first.
- Common error: getting the sign of the exponent wrong for small numbers. \(0.003\) is \(3\times10^{-3}\), not \(3\times10^{3}\).
- Common error: copying the calculator display "3.2E5" into the answer line. Write it as \(3.2\times10^{5}\).
- Common error: rounding to the wrong number of significant figures by counting the power of ten as digits.
Exam tip: Do the mantissa and exponent parts separately and write both lines out. Then check that your final mantissa lies between 1 and 10 before you write the answer.
Check yourself
- Write \(0.000\,603\) in standard form.
Answer
Move the decimal point four places to the right to get 6.03. So \(0.000\,603 = 6.03 \times 10^{-4}\). - Calculate \((5 \times 10^{6})(8 \times 10^{-2})\), giving your answer in standard form.
Answer
\(5 \times 8 = 40\) and \(6 + (-2) = 4\), so the product is \(40 \times 10^{4}\). Renormalise: \(40 = 4 \times 10^{1}\), so the answer is \(4 \times 10^{5}\). - Find \(3.6 \times 10^{7} + 2.4 \times 10^{6}\).
Answer
Write \(2.4 \times 10^{6} = 0.24 \times 10^{7}\). Then \(3.6 + 0.24 = 3.84\), so the sum is \(3.84 \times 10^{7}\). - A virus particle is \(1.2 \times 10^{-7}\) m wide. How many of them placed side by side fit along a line of length 3 mm? Give your answer in standard form.
Answer
Convert 3 mm to \(3 \times 10^{-3}\) m. Number \(= \dfrac{3 \times 10^{-3}}{1.2 \times 10^{-7}} = 2.5 \times 10^{-3-(-7)} = 2.5 \times 10^{4}\).