Standard Form and Scientific Notation
SI Units, Prefixes and Engineering Notation
Orders of Magnitude

Standard From and Scientific Notation

In science and engineering we often deal with exceptionally large and exceptionally small quantities. Scientific notation therefore relies on 'standard form' to express these quantities in a way that is comprehensible and allows us to easily compare their relative scale.

For example, if I have a number like LaTeX: 325020456983 , writing it out can be unnecessarily long and precise. To make this more manageable, we can use standard form, as $LaTeX: 3.25\times10^{11}$ . We can determine the power of 10 by counting the number of digits between the first digit and the decimal point (in this case 11). We can also use standard form to represent very small values, using negative powers of 10. To determine the power, count the number of 0's before the first non-zero digit.

Note that in standard form, we use the following format: $LaTeX: a\times10^n$ , where $LaTeX: 1\le\vert a\vert<10$ .

Let's see some examples:

Value	Standard Form (2d.p)
	$LaTeX: 8.72\times10^9$
	$LaTeX: 4.82\times10^{-5}$
$LaTeX: 345\times10^6$	$LaTeX: 3.45\times10^8$
$LaTeX: 0.00823\times10^{-3}$	$LaTeX: 8.23\times10^{-6}$

On calculators and in scientific programming, we often find numbers written using LaTeX: E , where represents "10 to the power of".

Test Yourself

Standard form AQA Test Questions | BBC Bitesize

Resources

Standard Form Revision | BBC Bitesize

Scientific Notation | Wikipedia

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As these are conventions, there are other ways of writing numbers in the form \(a \times 1b^n\) with different constraints on \(a \) and \(n \) and different base \(b\) to suit different applications, e.g, see Engineering Notation and also use of base 2 for binary numbers in computing,

Scientific Notation Converter | calculatorsoup.com - converts numbers to scientific notation (and more).

Links to floating point notation, significant figures, etc.

SI Units, Prefixes and Engineering Notation

Other conventions for writing numbers very similar to standard scientific notation include engineering notation where the exponent \( n\) is restricted to 0, 3, 6, etc and -3, -6 and so on. These correspond to units, kilo-, Mega- and milli-, micro- prefixes in the SI (International System of Units).

For example, 2000 Newtons in engineering notation is 2.0 x 10³Newtons = 2 kN.

The full set of SI prefixes are given here (including the non-engineering prefixes of hecto- deca-, deci- and centi-).

Table of SI prefixes

When we refer to 'SI units', we are referring to the 'International System of Units' (Le Système International d'Unités). These are units from which the unit for any other measurable physical quantity can be derived. There are 7 SI 'base units' we can deal with:

Base Quantity	Base Unit	Base Symbol
Length	Meter	m
Mass	Kilogram	kg
Time	Second	s
Temperature	Kelvin	k
Electric Current	Ampere	A
Luminous Intensity	Candela	cd
Amount of Substance	Mole	mol

Let's take an example of how we can derive the unit for any other physical quantity from these units. For example, Joules (J):

We know Joules measure energy, and a possible formula for energy is: LaTeX: E=mc^2 (Energy = mass $LaTeX: \times$ [speed of light in a vacuum]²)

$LaTeX: \therefore J=kg\times(ms^{-1})^2=kg\cdot m^2s^{-2}$

Therefore, we get the unit for energy expressed in SI units: $LaTeX: \begin{equation*} kg\cdot m^2s^{-2} \end{equation*}$

Test yourself

Length Unit Conversion Problems | math-exercises.com

Resources

Metric prefix | Wikipedia

International System of Units | Wikipedia

Engineering Notation & Metric Prefixes | Ron Call

SUPER EASY way to remember SI Prefixes | Dr K

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Why were the SI-unit Prefixes Chosen to be a Multiple Power of 3? | Stack Exchange - Interesting discussion on StackExchange

SI units | NPL - The National Physics Laboratory, the official place to go for the UK's measurement conventions

Order of Magnitude

Order of magnitude simply refers to the closest power of 10 in which you are working. Let's look at the following cases

Let's say we have a value of $LaTeX: a\times10^b$ , where $LaTeX: 0.5<a\le5$ . In this case, our order of magnitude is equal to LaTeX: b .

If we have a value of $LaTeX: a\times10^b$ , where $LaTeX: 0.05<a\le0.5$ , in this case our order of magnitude is equal to LaTeX: b-1

If we have a value of $LaTeX: a\times10^b$ , where $LaTeX: 5<a\le50$ , in this case our order of magnitude is equal to LaTeX: b+1

Let's look at a table of examples:

Value	Order of Magnitude
$LaTeX: 2.10\times10^3$	3
$LaTeX: 3.62\times10^7$	7
$LaTeX: 5.11\times10^4$	5
$LaTeX: 0.41\times10^3$	2
$LaTeX: 8.94\times10^8$	9

Resources

Orders of Magnitude Explained Video 1 | Physics High

Order of Magnitude Explained Video 2 | Toppr JEE

Order of Magnitude Quiz

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Powers of Ten™ (1977) | Eames Office - classic video showing the relative size of things in the universe, or scale of the everyday, in powers of ten. Directed by design giants Ray and Charles Eames.

For future viewing: Orders of Magnitude VR – Information Website about the Educational Virtual Reality Experience - following the Powers of Ten film... new project in 2021.