Scientific Notation
Scientific notation writes very large and very small numbers as a number from \(1\) up to (but not including) \(10\) multiplied by a power of ten, so \(384\,400\) km becomes \(3.844 \times 10^5\) km. This NSW Year 9 Core maths subtopic is Stage 5 Core content under NESA outcome MA5‑MAG‑C‑01.
You will write large and small numbers in scientific notation and convert them back to decimals, read a calculator display such as 6.1E-5, and recognise when a number like \(34 \times 10^5\) is not in scientific notation. You will order numbers in scientific notation, multiply and divide them with the index laws, and estimate answers by rounding first. Examples include the distance to the Sun, red blood cells, the thickness of a coin and the size of a virus. It is part of Year 9 Core Indices and Scientific Notation.
Theory
Scientific notation writes a number as a value from \(1\) up to (not including) \(10\), multiplied by a power of ten. This Year 9 maths page shows how to write very large and very small numbers this way, convert them back, read a calculator display, and compare, multiply and divide them.
Scientific notation is a short way to write very large or very small numbers. The Earth's mass, the size of a cell and the speed of light are all written this way.
A number in scientific notation looks like \(a \times 10^n\). The first part \(a\) is at least \(1\) and less than \(10\), so it has exactly one non-zero digit before the decimal point. The index \(n\) is an integer: positive for large numbers, negative for small ones.
Large numbers have a positive index. Numbers between \(0\) and \(1\) have a negative index.
\(45 \times 10^3\) has the right value for \(45\,000\), but it is not in scientific notation, because \(45\) is not less than \(10\).
The form of a number in scientific notation:
| Ordinary number | Point moves | Scientific notation |
|---|---|---|
| \(38\,000\) | \(4\) places left | \(3.8 \times 10^4\) |
| \(6\,010\,000\) | \(6\) places left | \(6.01 \times 10^6\) |
| \(0.0025\) | \(3\) places right | \(2.5 \times 10^{-3}\) |
| \(0.000\,000\,9\) | \(7\) places right | \(9 \times 10^{-7}\) |
Multiplying and dividing use the index laws on the powers of ten:
How to write a number in scientific notation
- Place the decimal point just after the first non-zero digit.
- Count how many places the point moved.
- Choose the sign: moved left (a large number) gives a positive index; moved right (a small number) gives a negative index.
- Write it as \(a \times 10^n\) and check that \(a\) is between \(1\) and \(10\).
- Compare numbers by their powers first, then by the front numbers.
(i) \(73\,200\)
(ii) \(0.000\,46\) in scientific notation.
(i) Put the point after the first digit: \(7.32\). It moved \(4\) places left, so the index is \(4\).
| \(73\,200\) | \(=\) | \(7.32 \times 10\,000\) |
| \(\) | \(=\) | \(\mathbf{7.32 \times 10^4}\) |
(ii) Put the point after the first non-zero digit: \(4.6\). It moved \(4\) places right, so the index is \(-4\).
| \(0.000\,46\) | \(=\) | \(\mathbf{4.6 \times 10^{-4}}\) |
A number less than \(1\) always has a negative index.
(i) \(2.8 \times 10^5\)
(ii) \(9.1 \times 10^{-3}\)
(iii) the calculator display \(\texttt{5.3E-6}\)
(i) A positive index moves the point right. Fill the gaps with zeros.
| \(2.8 \times 10^5\) | \(=\) | \(\mathbf{280\,000}\) |
(ii) A negative index moves the point left.
| \(9.1 \times 10^{-3}\) | \(=\) | \(\mathbf{0.0091}\) |
(iii) On a calculator, E means "times \(10\) to the power".
| \(\texttt{5.3E-6}\) | \(=\) | \(5.3 \times 10^{-6}\) |
| \(\) | \(=\) | \(\mathbf{0.000\,005\,3}\) |
(i) \((8 \times 10^5) \times (3 \times 10^{-2})\)
(ii) \(\dfrac{8.4 \times 10^3}{2 \times 10^{-5}}\)
(i) Multiply the front numbers. Add the indices.
| \(8 \times 3\) | \(=\) | \(24\) |
| \(10^5 \times 10^{-2}\) | \(=\) | \(10^{3}\) |
| \(\text{product}\) | \(=\) | \(24 \times 10^3\) |
\(24\) is \(10\) or more, so move the point one place left and add \(1\) to the index.
| \(24 \times 10^3\) | \(=\) | \(\mathbf{2.4 \times 10^4}\) |
(ii) Divide the front numbers. Subtract the indices.
| \(8.4 \div 2\) | \(=\) | \(4.2\) |
| \(10^3 \div 10^{-5}\) | \(=\) | \(10^{3-(-5)} = 10^8\) |
| \(\text{quotient}\) | \(=\) | \(\mathbf{4.2 \times 10^8}\) |
(i) Order them from smallest to largest.
(ii) How many times longer is the largest than the smallest?
(i) Compare the powers first. The most negative power is the smallest number. Only when the powers match do the front numbers decide.
| \(10^{-5}\) | \(<\) | \(10^{-4} < 10^{-3}\) |
| \(1.9 \times 10^{-3}\) | \(<\) | \(4.8 \times 10^{-3}\) |
Smallest to largest: \(\mathbf{6.0 \times 10^{-5}}\), \(\mathbf{8.7 \times 10^{-4}}\), \(\mathbf{1.9 \times 10^{-3}}\), \(\mathbf{4.8 \times 10^{-3}}\).
(ii) "How many times" means divide the larger by the smaller.
| \(\dfrac{4.8 \times 10^{-3}}{6.0 \times 10^{-5}}\) | \(=\) | \(0.8 \times 10^{-3-(-5)}\) |
| \(\) | \(=\) | \(0.8 \times 10^{2}\) |
| \(\) | \(=\) | \(\mathbf{80}\) |
The largest particle is \(80\) times longer.
Common pitfalls
Frequently asked questions
What is scientific notation?
Scientific notation writes a number as a value from \(1\) up to but not including \(10\), multiplied by a power of \(10\). For example \(5\,200\,000 = 5.2 \times 10^6\).
How do you write a small decimal in scientific notation?
Move the point to just after the first non-zero digit and count the places it moved right. That count, made negative, is the index: \(0.00071 = 7.1 \times 10^{-4}\).
What does E mean on a calculator?
E means "times \(10\) to the power" of the number after it. \(\texttt{5.3E-6}\) means \(5.3 \times 10^{-6} = 0.000\,005\,3\).
How do you multiply numbers in scientific notation?
Multiply the front numbers and add the indices. If the front number is \(10\) or more, move the point one place left and add \(1\) to the index.
How do you compare numbers in scientific notation?
Compare the powers of \(10\) first: the larger index is the larger number. Only when the indices are the same do you compare the front numbers.