So far you have learned about integers, decimals, fractions and powers. Now you will learn another important form numbers may take. This form is called the scientific notation. This notation is frequently used in scientific contexts. You have most certainly seen it on your calculator. There, it often looks something like this: $2.34$E7
. But what does it mean?
The number on the calculator $$\begin{array}{llll}\hfill 2.34E7& =2.34\cdot 1{0}^{7}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2.34\cdot 10\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =23\phantom{\rule{0.17em}{0ex}}400\phantom{\rule{0.17em}{0ex}}000.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
So, $E=\text{\u201c10raisedtothepowerof\u201d}$, or simply $E=\text{\u201c10tothepowerof\u201d}$, where the number after E
is the exponent of the base number 10.
Rule
$a\cdot 1{0}^{n}$, where $1\le a<10$ and $n\in \mathbb{Z}$.
$a\cdot 1{0}^{n}$, where $a$ is a number from 1 to 10, including 1, and $n$ can be any integer.
Why do you need this type of number? You need scientific notation when talking about very small or very large numbers. Scientific notation simplifies numbers with many digits. It is, as I have mentioned before, another mathematical method to make your life easier!
Example 1
The mass of an E. coli bacteria is
$$0.000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}95\phantom{\rule{0.17em}{0ex}}\text{kg}$$ |
This number is not easy to read, pronounce nor write. If you write it in scientific notation, it gets much easier to read. You will then get
$$0.000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}95\phantom{\rule{0.17em}{0ex}}\text{kg}=9.5\cdot 1{0}^{-16}\text{kg}$$ |
Example 2
The distance from the Earth to the Sun is $149\phantom{\rule{0.17em}{0ex}}600\phantom{\rule{0.17em}{0ex}}000$ km. This number can be written in a simpler form if you use scientific notation. Then you will get
$$149\phantom{\rule{0.17em}{0ex}}600\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}\text{km}=1.496\cdot 1{0}^{8}\text{km}.$$ |
How did we arrive at the numbers given in the examples? They follow the formula given in the rule above. Here is a simple guideline to write a number in the scientific notation:
Rule
With numbers larger than $10$, you move the decimal point to the left to write it in scientific notation. Then, the $n$ in $1{0}^{n}$ becomes positive.
With numbers smaller than $1$, but larger than $0$, you move the decimal point to the right to write it in scientific notation. Then, the $n$ in $1{0}^{n}$ becomes negative.
Example 3
Moving the decimal point to the left:
$$120\phantom{\rule{0.17em}{0ex}}000=120\phantom{\rule{0.17em}{0ex}}000.0=1.2\cdot 100\phantom{\rule{0.17em}{0ex}}000=1.2\cdot 1{0}^{5}$$ |
$$0.0024=2.4\cdot 0.001=2.4\cdot \frac{1}{1{0}^{3}}=2.4\cdot 1{0}^{-3}$$ |
You may skip the middle term, and write the answer directly by counting the number of times you move the decimal point in order to find the exponent. Be aware of the sign of the exponent!
Example 4
Write the numbers in scientific notation. $$\begin{array}{cc}34\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000=3.4\cdot 1{0}^{7}& \\ 0.000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}045=4.5\cdot 1{0}^{-11}& \end{array}$$