Scientific Notation
Scientific notation writes very large and very small numbers as a number from \(1\) up to \(10\) times a power of \(10\), such as \(4.5 \times 10^{-8}\). This NSW Year 9 Advanced maths subtopic uses it for measurements of any size (NESA outcome MA5‑MAG‑C‑01).
You will convert between decimals and scientific notation, read a calculator display such as 2.7E-4, order measurements, and multiply, divide and square numbers in scientific notation using the index laws. You will estimate a product, compare a raindrop with a grain of sand, find how long sunlight takes to reach Earth, and work out how many photos fit on a hard drive. It is part of Year 9 Advanced Indices and Scientific Notation.
Theory
Scientific notation writes a number as \(a \times 10^n\), with \(1 \le a \lt 10\) and \(n\) an integer. This Year 9 Advanced maths page covers converting both ways, ordering, calculator displays, calculating with the index laws and real problems with very large and very small measurements.
A number is in scientific notation (also called standard form) when it is written as \(a \times 10^n\), where \(a\) is at least \(1\) and less than \(10\), and \(n\) is an integer. So there is exactly one non-zero digit in front of the decimal point.
The index says how many places the decimal point moves. A positive index is for numbers \(10\) or more; a negative index is for numbers less than \(1\); \(10^0 = 1\) is for numbers from \(1\) to \(10\).
Scientific notation makes very large and very small measurements readable: the distance to the Moon is about \(3.84 \times 10^8\) m and a dust grain about \(5 \times 10^{-5}\) m.
To compare two numbers in scientific notation, compare the powers of 10 first. Only when the indices are equal do you compare the numbers in front.
A calculator shows scientific notation with an E: 5.2E-7 means \(5.2 \times 10^{-7}\).
| Ordinary number | Scientific notation | Point moves |
|---|---|---|
| \(52\,900\) | \(5.29 \times 10^{4}\) | \(4\) left |
| \(6.1\) | \(6.1 \times 10^{0}\) | none |
| \(0.61\) | \(6.1 \times 10^{-1}\) | \(1\) right |
| \(0.000\,009\,3\) | \(9.3 \times 10^{-6}\) | \(6\) right |
Converting
- To scientific notation: place the point after the first non-zero digit. Count how many places it moved. Large number: positive index. Number less than \(1\): negative index.
- To an ordinary number: move the point right for a positive index and left for a negative one, filling gaps with zeros.
- Is it in scientific notation? Check that the front number is from \(1\) up to, but not including, \(10\). \(23.7 \times 10^{3}\) and \(0.4 \times 10^{2}\) are not.
Calculating and applying
- Multiply or divide: work on the front numbers and the powers of \(10\) separately, then re-normalise (Example 2).
- Estimate: round each front number to one significant figure first, then calculate.
- Units first: change to the same unit before dividing (Example 3). Speed, distance and time: \(t = \dfrac{d}{s}\).
- How many times larger: divide the larger by the smaller. A percentage is the fraction \(\times 100\).
- Whole objects: if the answer counts whole things that must fit, round down.
- Missing index: simplify, re-normalise, then match indices (Example 4).
(i) Write \(3.016 \times 10^{6}\) as an ordinary number.
(ii) Write \(0.000\,704\) in scientific notation.
(i) The index is \(6\), so move the point \(6\) places right, filling with zeros.
| \(3.016 \times 10^6\) | \(=\) | \(\mathbf{3\,016\,000}\) |
The zero between the \(3\) and the \(1\) is part of the number and stays.
(ii) Put the point after the first non-zero digit, \(7\), and count the moves: \(4\) places right.
| \(0.000\,704\) | \(=\) | \(\mathbf{7.04 \times 10^{-4}}\) |
Check by undoing it: \(7.04 \div 10\,000 = 0.000\,704\).
(i) \((5 \times 10^{-3}) \times (8 \times 10^{7})\)
(ii) \(\dfrac{1.2 \times 10^{5}}{4 \times 10^{-3}}\).
Work on the numbers in front and the powers of \(10\) separately, then fix the front number so it is from \(1\) up to \(10\).
(i)
| \(5 \times 8\) | \(=\) | \(40\) |
| \(10^{-3} \times 10^{7}\) | \(=\) | \(10^{4}\) |
| \(40 \times 10^{4}\) | \(=\) | \(4 \times 10^{1} \times 10^{4}\) |
| \(\) | \(=\) | \(\mathbf{4 \times 10^{5}}\) |
(ii)
| \(1.2 \div 4\) | \(=\) | \(0.3\) |
| \(10^{5} \div 10^{-3}\) | \(=\) | \(10^{5-(-3)} = 10^{8}\) |
| \(0.3 \times 10^{8}\) | \(=\) | \(3 \times 10^{-1} \times 10^{8}\) |
| \(\) | \(=\) | \(\mathbf{3 \times 10^{7}}\) |
\(40\) and \(0.3\) are not allowed in front. Shifting the point one place changes the index by one.
Use the same unit for both lengths: \(3\text{ cm}\) \(= 0.03\text{ m}\) \(= 3 \times 10^{-2}\) m. Then divide.
| \(\dfrac{3 \times 10^{-2}}{7.5 \times 10^{-6}}\) | \(=\) | \(\dfrac{3}{7.5} \times 10^{-2-(-6)}\) |
| \(\) | \(=\) | \(0.4 \times 10^{4}\) |
| \(\) | \(=\) | \(4 \times 10^{3}\) |
| \(\) | \(=\) | \(\mathbf{4000}\) |
About \(4000\) cells. "How many fit" or "how many times larger" always means divide.
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
What is scientific notation?
It writes a number as \(a \times 10^n\) with \(1 \le a \lt 10\). For example \(52\,900 = 5.29 \times 10^{4}\) and \(0.0061 = 6.1 \times 10^{-3}\).
How do you know if the index is positive or negative?
If the number is \(10\) or more, the index is positive. If it is less than \(1\), the index is negative. If it is from \(1\) up to \(10\), the index is \(0\).
How do you multiply numbers in scientific notation?
Multiply the front numbers, add the indices of the powers of \(10\), then adjust so the front number is between \(1\) and \(10\), changing the index to match.
What does E mean on a calculator?
E means \(\times 10\) to the power of the number after it: 5.2E-7 means \(5.2 \times 10^{-7}\).
How do you order numbers in scientific notation?
Compare the powers of \(10\) first: a bigger index means a bigger number for positive values. Only if the indices are equal, compare the front numbers.