SI prefixes, conversion of units and orders of magnitude

Too many zeros

A red blood cell is about $0.000\,007\ \text{m}$ across. The Sun is about $150\,000\,000\,000\ \text{m}$ away. Count the zeros and you will make a mistake sooner or later.

Physics has two tools for this: prefixes and scientific notation.

Prefixes

A prefix in front of a unit multiplies it by a power of ten. Kilo means a thousand, so $1\ \text{km} = 1000\ \text{m}$.

So a $2.4\ \text{GHz}$ Wi-Fi signal has a frequency of $2.4 \times 10^{9}\ \text{Hz}$, and a $350\ \text{nm}$ light wave is $350 \times 10^{-9}\ \text{m}$ long.

Watch out. A prefix belongs to the unit with its power. $1\ \text{cm}^2$ means $(1\ \text{cm})^2 = (10^{-2}\ \text{m})^2 = 10^{-4}\ \text{m}^2$, not $10^{-2}\ \text{m}^2$. And $1\ \text{cm}^3 = 10^{-6}\ \text{m}^3$.

Scientific notation

We write a number as $a \times 10^{n}$, where $1 \le a < 10$ and $n$ is a whole number.

• $0.000\,007 = 7 \times 10^{-6}$: the point moved $6$ places to the right, so $n = -6$.
• $150\,000\,000\,000 = 1.5 \times 10^{11}$: the point moved $11$ places to the left, so $n = +11$.

To multiply, multiply the fronts and add the powers: $(2 \times 10^{3})(4 \times 10^{5}) = 8 \times 10^{8}$. To divide, divide the fronts and subtract the powers.

Changing units: multiply by one

Here is the one idea you need. Since $1\ \text{km} = 1000\ \text{m}$, the fraction $\dfrac{1000\ \text{m}}{1\ \text{km}}$ equals one. Multiplying by one does not change a quantity. It only changes how it is written.

Change $54\ \text{km/h}$ to $\text{m/s}$:

The km cancel, the h cancel, and m and s are left. If a unit does not cancel, the fraction is upside down.

A quick rule. $\dfrac{1000}{3600} = \dfrac{1}{3.6}$. So km/h to m/s: divide by 3.6. And m/s to km/h: multiply by $3.6$.

A worked example

Mercury has a density of $13.6\ \text{g/cm}^3$. What is that in $\text{kg/m}^3$?

1. Top. $1\ \text{g} = 10^{-3}\ \text{kg}$.
2. Bottom. $1\ \text{cm}^3 = 10^{-6}\ \text{m}^3$.
3. Together. $13.6\ \text{g/cm}^3 = 13.6 \times \dfrac{10^{-3}\ \text{kg}}{10^{-6}\ \text{m}^3} = 13\,600\ \text{kg/m}^3$.

Think first. Is a bigger number sensible? A cubic metre is a million cubic centimetres, so it holds $13.6 \times 10^{6}\ \text{g}$ of mercury. That is $13\,600\ \text{kg}$. Yes.

Order of magnitude

Often we only need the size of a number: about ten, a thousand, a million? The order of magnitude is the power of ten nearest to the number.

Write the number as $a \times 10^{n}$. Then:

• if $a \le 5$, the order of magnitude is $10^{n}$;
• if $a > 5$, it is $10^{n+1}$.

The mass of a car, $1.2 \times 10^{3}\ \text{kg}$, has the order $10^{3}\ \text{kg}$. The radius of the Earth, $6.4 \times 10^{6}\ \text{m}$, has the order $10^{7}\ \text{m}$.

Some books use $3.16$ instead of $5$, because $3.16 \approx \sqrt{10}$ is the true halfway point on a scale of powers of ten. See it here.

Both rules agree except for a front number between $3.16$ and $5$. Use the rule your book uses.

Estimating with powers of ten

How many breaths do you take in a year? About $15$ a minute. A year has about $60 \times 24 \times 365 \approx 5 \times 10^{5}$ minutes. So about $15 \times 5 \times 10^{5} \approx 7.5 \times 10^{6}$ breaths: of the order of $10^{7}$.

An estimate like this, called a Fermi estimate, uses round numbers on purpose. It tells you quickly whether an answer is sensible.

Practise with new numbers

Try it in Python

The program turns prefixed values into plain units, and finds an order of magnitude with the rule of $5$.

Remember these

• A prefix is a power of ten: k $10^3$, M $10^6$, G $10^9$, c $10^{-2}$, m $10^{-3}$, $\mu$ $10^{-6}$, n $10^{-9}$.
• The prefix goes with the power: $1\ \text{cm}^2 = 10^{-4}\ \text{m}^2$, $1\ \text{cm}^3 = 10^{-6}\ \text{m}^3$.
• Change a unit by multiplying by a fraction equal to one; the old unit must cancel. km/h to m/s: divide by $3.6$.
• Order of magnitude: $a \times 10^{n}$ goes to $10^{n}$ if $a \le 5$, else to $10^{n+1}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A sheet of paper is about $0.09\ \text{mm}$ thick. Write this in metres, in scientific notation.

2. A car moves at $25\ \text{m/s}$. What is its speed in km/h?

3. A floor tile is $30\ \text{cm} \times 30\ \text{cm}$. What is its area in square metres?

4. Estimate the order of magnitude of the number of seconds in a human life of $70$ years.