Scientific Notation
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Theory
Scientific notation writes a number as a number from \(1\) up to (but not including) \(10\), times a power of \(10\), such as \(4.7 \times 10^{8}\). This Year 10A maths page revises it from Year 9: converting to and from decimals, metric prefixes, calculating with numbers in scientific notation (including powers), finding a missing index, and comparing very large and very small measurements.
A number is in scientific notation when it is written as \(a \times 10^{n}\), where \(1 \le a < 10\) and \(n\) is an integer. It is used for very large and very small measurements, such as distances in space or the size of a cell.
For a large number, \(n\) is positive: it counts how many places the decimal point moves to the left. For a number less than 1, \(n\) is negative: the point moves to the right.
To write a number as a decimal, reverse the process: a positive index moves the point right (the number gets bigger); a negative index moves it left (the number gets smaller).
Metric prefixes stand for powers of \(10\): kilo \(10^3\), mega \(10^6\), giga \(10^9\), milli \(10^{-3}\), micro \(10^{-6}\), nano \(10^{-9}\).
Significant figures are the digits that carry meaning, counted from the first non-zero digit. \(0.004\,056\,3\) to two significant figures is \(0.0041\). In scientific notation, every digit of \(a\) is significant.
The form, and the index laws for powers of \(10\):
| Prefix | Symbol | Power of 10 | Example |
|---|---|---|---|
| giga | G | \(10^{9}\) | \(3\) GW \(= 3 \times 10^{9}\) W |
| mega | M | \(10^{6}\) | \(5\) MB \(= 5 \times 10^{6}\) bytes |
| kilo | k | \(10^{3}\) | \(7\) km \(= 7 \times 10^{3}\) m |
| milli | m | \(10^{-3}\) | \(4\) mm \(= 4 \times 10^{-3}\) m |
| micro | μ | \(10^{-6}\) | \(9\) μm \(= 9 \times 10^{-6}\) m |
| nano | n | \(10^{-9}\) | \(6\) ns \(= 6 \times 10^{-9}\) s |
How to write a number in scientific notation
- Place the decimal point after the first non-zero digit to get \(a\).
- Count the places the point moved. Left gives a positive index; right gives a negative index.
- Write \(a \times 10^{n}\), dropping zeros at the end of \(a\).
How to calculate
- Multiply or divide the numbers in front; add or subtract the indices.
- Add or subtract only after rewriting both numbers with the same power of \(10\).
- Fix the form if \(a\) is \(10\) or more, or less than \(1\).
- Convert units first if needed, such as cm to m or seconds to minutes, then round to the significant figures asked for.
- A missing index: do the arithmetic on the side you know, re-normalise it, then match the powers of \(10\) on both sides.
(ii) Write \(0.000\,507\) in scientific notation.
(iii) Write \(6.3 \times 10^{-5}\) as a decimal.
Place the decimal point after the first non-zero digit, then count how far it moved.
| \(3\,060\,000\) | \(=\) | \(\mathbf{3.06 \times 10^{6}}\) |
| \(0.000\,507\) | \(=\) | \(\mathbf{5.07 \times 10^{-4}}\) |
| \(6.3 \times 10^{-5}\) | \(=\) | \(\mathbf{0.000\,063}\) |
(iii) A negative index makes the number small: move the point \(5\) places left, filling the gaps with zeros.
(i) \((4 \times 10^{6}) \times (7.5 \times 10^{-2})\)
(ii) \((8.4 \times 10^{5}) \div (2.1 \times 10^{-3})\)
Work with the numbers in front and the powers of \(10\) separately, using the index laws.
| \((4 \times 7.5) \times 10^{6 + (-2)}\) | \(=\) | \(30 \times 10^{4}\) |
| \(\) | \(=\) | \(3 \times 10^{1} \times 10^{4}\) |
| \(\) | \(=\) | \(\mathbf{3 \times 10^{5}}\) |
| \((8.4 \div 2.1) \times 10^{5 - (-3)}\) | \(=\) | \(\mathbf{4 \times 10^{8}}\) |
In (i), \(30\) is not between \(1\) and \(10\), so rewrite it as \(3 \times 10^1\) and add \(1\) to the index.
(ii) A dust grain is \(35\) micrometres wide. Write this in metres, in scientific notation.
(i) To add, both numbers need the same power of \(10\).
| \(8 \times 10^{5}\) | \(=\) | \(0.8 \times 10^{6}\) |
| \(5.2 \times 10^{6} + 0.8 \times 10^{6}\) | \(=\) | \(6.0 \times 10^{6}\) |
| \(\) | \(=\) | \(\mathbf{6 \times 10^{6}}\) |
(ii) Micro means \(10^{-6}\).
| \(35 \times 10^{-6}\) | \(=\) | \(3.5 \times 10^{1} \times 10^{-6}\) |
| \(\) | \(=\) | \(\mathbf{3.5 \times 10^{-5}}\text{ m}\) |
Simplify the left side in terms of \(n\), then put it in scientific notation before matching.
| \(6.4 \div 8\) | \(=\) | \(0.8\) |
| \(10^{n} \div 10^{-3}\) | \(=\) | \(10^{n+3}\) |
| \(0.8 \times 10^{n+3}\) | \(=\) | \(8 \times 10^{n+2}\) |
| \(n + 2\) | \(=\) | \(9\) |
| \(n\) | \(=\) | \(\mathbf{7}\) |
Matching \(n + 3 = 9\) before fixing \(0.8\) gives the wrong answer \(n = 6\).
Common pitfalls
Frequently asked questions
How do you write a number in scientific notation?
Put the decimal point after the first non-zero digit, then multiply by \(10\) to the number of places the point moved. Moving left gives a positive index, moving right a negative one: \(3\,060\,000 = 3.06 \times 10^{6}\).
What does a negative power of 10 mean in scientific notation?
It means the number is less than \(1\). \(6.3 \times 10^{-5} = 0.000\,063\): move the decimal point \(5\) places to the left.
How do you multiply numbers in scientific notation?
Multiply the numbers in front and add the indices of \(10\). If the number in front ends up \(10\) or more, adjust it: \(30 \times 10^{4} = 3 \times 10^{5}\).
What are significant figures?
They are the digits in a number that carry meaning, counted from the first non-zero digit. \(0.004\,056\,3\) to two significant figures is \(0.0041\).
How do you enter scientific notation on a calculator?
Use the \(\times 10^x\) key, often marked EXP, instead of typing \(\times 10\) and a power. Put brackets around each number when you divide.