Scientific notation

Simon James; Chris Rawson; illustrated by Erin Cheffers, Deakin University

34 Scientific notation

Some numbers that apply to real phenomena are huge.

For example,

Our solar system is approximately 11 800 000 000 km across.
The distance light travels in one year is approximately 9 500 000 000 000 km.

Others are really small. For example, the time taken by light to travel one metre is roughly 0.000000003 seconds.

To write these out, we’d either need very long numbers with lots of zeros, or remember a bunch of different units or prefixes. Most people who are familiar with using long (very big or very small) numbers use scientific notation instead.

What is scientific notation?

Scientific notation is a way of writing large or very small numbers efficiently. After a bit of time, it gets a little more intuitive than counting zeroes, too.

The idea is simple enough: you start with a number between one and ten and then record the number of times it would need to be multiplied or divided by 10 to get to the actual figure.

Some examples of large numbers:

[latex]1.3 \times 10^3=1,300[/latex]
[latex]9 \times 10^8=900,000,000[/latex] or 900 million.

For smaller numbers, we still use indices (‘to the power of’), but recall that raising something to a negative power is effectively dividing it. So [latex]2^{-1}=\frac{1}{2}[/latex] and [latex]2 \times 10^{-1}=0.2[/latex].

[latex]5.8 \times 10^{-2}=0.058[/latex]
[latex]1.8 \times 10^-5=0.000018[/latex]

Definition of scientific notation

A number is in the form of scientific notation if it is written in the form of [latex]a \times 10^b[/latex], where:

[latex]a[/latex] is a number between 1 and 10, and
[latex]b[/latex] is an integer.

Non-standard notation

If the number a is not in the range 1 to 10, then it is not standard scientific notation. For example, if we want to write [latex]0.5 \times 10^4[/latex] in standard scientific notation, it should be rewritten as [latex]5 \times 10^3[/latex] (but both equal 5000).

We can see here that we’re using index notation, so let’s just quickly refresh our index laws.

Transcript

A useful thing about scientific notation is that it allows us to easily compare the magnitude of numbers.

If you have 18942000000000000000000000 and 213940000000000000000 it’s not so easy to see that the former is almost 100 000 times larger. However in scientific notation, we have 1.8942 x 10²⁵ compared to 2.1394 x 10²⁰ and we can see the difference in the power of 10 is 5, so almost 105 times difference. Sometimes this is referred to as order of magnitude, i.e. the two numbers differ by 5 in order of magnitude.

Real world examples

Now that we have a good understanding of scientific notation, we can consider a few interesting quantities:

Approximately 10¹¹neurons in the human brain, 10¹⁴synapses
More than 1.241 x 10¹²digits of pi known
About 3 to 7 × 10²² stars in the observable universe, 10⁸⁰atoms
A DNA molecule’s width is about 2 x 10^-7cm, (hair is about 2×10^-2)

The ones below are from Bill Bryson’s “A short history of nearly everything”, which relate some large quantities to a few benchmarks to help you build intuition about some impossible-to-fathom-ly large numbers.

Scientific notation can make routine mathematics simpler

Scientific notation can also make it easier to perform some operations. To multiply and divide numbers in scientific notation, we just need to remember our order of operations and index rules.

Let’s start with two routine problems to show you what we mean.

Indices

Suppose that [latex]a=1.4 \times 10^3[/latex] and that [latex]b=3.4 \times 10^{-2}[/latex]

What is [latex]ab[/latex]?
What is [latex]\frac{a}{b}[/latex]?

Watch the video, then have a go for yourself, then look through the worked solution below (in that order).

Transcript

Multiplying in scientific notation (worked solution)

Let’s start with [latex]ab[/latex].

First of all, what is the question asking us? We need to recall that, by the conventions of algebra, writing two things next to each other means without any other symbols in between means that they’re multiplied. For example, we can represent cows as having four legs by saying [latex]legs=4 \times cow[/latex] or [latex]l=4c[/latex].

So, [latex]ab=a \times b[/latex] or [latex](1.4 \times 10^3) \times (3.4 \times 10^{-2})[/latex].

There are two ‘tricks’ we can use here to make this easier to calculate.

It doesn’t matter what order you multiply them in: [latex]5 \times 3 \times 2[/latex], [latex]3 \times 2 \times 5[/latex] and [latex]2 \times 5 \times 3[/latex] all equal 30.
Because of the index laws, [latex]a^b \times a^c=a^{b+c}[/latex]. In other words, [latex]10^3 \times 10^{-2}=10^{(3-2)}=10^1=10[/latex]. To add a negative number, you subtract it. Eg [latex]3+(-2)=3-2[/latex].

Let’s put these two little tricks into practice to solve the problem above:

[latex]\begin{align}ab &= (1.4 \times 10^3) \times (3.4 \times 10^{-2}) \\&= (1.4 \times 3.4) \times (10^3 \times 10^{-2}) \\&= 4.76 \times 10^{(3-2)} \\&= 4.76 \times 10^{1}\\&= 4.76 \times 10 \\&= 47.6\end{align}[/latex]

It’s a good idea to check by writing it into your calculator as it originally appears to ensure the final numbers agree.

If you think about what this means, multiplying in scientific notation is made easier by the fact that you are working with base 10 (10 to the power of something), and that the power is always a single number.

All you need to do is

multiple the leading numbers together (1.4 and 3.4 in the example above),
add or subtract the powers of 10 (10^(3-2) in the example above)
Multiply the results of steps one and two together ([latex]4.76 \times 10[/latex]).

Dividing in scientific notation

Now let’s try [latex]\frac{a}{b}[/latex] or [latex]\frac{1.4 \times 10^3}{3.4 \times 10^{-2}}[/latex]

Here are the two ‘tricks’ to keep in mind for this one, that are related to the tricks in the multiplication above:

Just as the laws of indices state that [latex]a^b \times a^c=a^{b+c}[/latex], it also states that [latex]a^b \div a^c=a^{b-c}[/latex]. This makes sense because, for example, [latex]10^3 \times 10^2 = 1,000 \times 100=100,000 = 10^5[/latex]. Similarly, [latex]\frac{10^5}{10^3}=\frac{100,000}{1,000}=100=10^2[/latex].
The equation can be split up into two fractions that are multiplied together, rather than one single equation.

Let’s see it in action:

[latex]\begin{align}\frac{a}{b} &= \frac{1.4 \times 10^3}{3.4 \times 10^{-2}} \\ &=\frac{1.4}{3.4} \times \frac{10^3}{10^{-2}} \\ &=\frac{1.4}{3.4} \times 10^{(3-(-2))} \\ &=\frac{1.4}{3.4} \times 10^{5} \\ & = 0.411764705 \times 100,000 \\ &≈ 41,176.5\end{align}[/latex]

If you’re not quite convinced about being able to separate the equation into the multiplication at the first step, remember that when multiplying fractions, we just multiply the top and the bottom, so if we were working in the other direction we would have

[latex]\frac{1.4}{3.4} \times \frac{10^3}{10^{-2}} = \frac{1.4 \times 10^3}{3.4 \times 10^{-2}}[/latex]

The Feynman Technique for consolidating your understanding

Give an explanation of what scientific notation is in simple English. If there are things you’re not sure about, do some additional research and rewrite your explanation. Include examples of numbers that are in scientific notation and not in scientific notation and give an example of where it might be useful to use it.

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