Decimal
Need to review decimals? Use this resource to get yourself up to speed.
Notation
Imagine trying to write out the mass of the Earth in kilograms; that would be \(5\,972\,000\,000\,000\,000\,000\,000\,000\textrm{ kg}\). Expressing such big numbers this way is hard and can lead to mistakes. In science and engineering, notation plays a key role in simplifying calculations and conveying information efficiently. Use this resource to learn how to write numbers in scientific and engineering notation.
Scientific notation
Scientific notation is a way of writing numbers in a compact form. It uses a number from \(1\) to less than \(10\), multiplied by a power of \(10\).
For example, the mass of the Earth would be concisely written as \(5.972\times10^{24}\). The charge on one electron, \(0.000\,000\,000\,000\,000\,000\,160\,2\textrm{ C}\) is \(1.602\times10^{-19}\textrm{ C}\).
By using scientific notation:
- the size of the number is more easily seen
- errors are less likely to occur
- calculations can be simplified using index laws.
Expressing numbers in scientific notation
To write a number in scientific notation:
- Move the decimal point so that there is one non-zero digit before the decimal point.
- Multiply by a power of \(10\) equal to the number of places that the decimal point has been moved. If:
- the decimal point is moved to the left, the power of \(10\) is positive
- the decimal point is moved to the right, the power of \(10\) is negative.
If the number is less than \(1\), the power of \(10\) will be negative. If it is more than \(1\), the power of \(10\) will be positive.
Example 1 – expressing numbers in scientific notation
Write \(5630\) in scientific notation.
Locate the decimal point in the number. For whole numbers, there is an imaginary decimal point at the end.
\[5630.\]
We need to move the decimal point so that we have one non-zero digital before the decimal point. In this case, it needs to be between the \(5\) and \(6\).
\[5\underleftarrow{630} = 5.630\]
We have moved the decimal point \(3\) places to the left, so the power of \(10\) is \(+3\).
\[5630=5.6\times10^{3}\]
Write \(0.00725\) in scientific notation.
Move the decimal point to be between the \(7\) and \(2\).
\[0\underrightarrow{.007}25=7.25\]
We have moved the decimal point \(3\) placed to the right, so the power of \(10\) is \(-3\).
\[0.00725=7.25\times10^{-3}\]
Exercise – expressing numbers in scientific notation
- Write the following numbers in scientific notation.
- \(58000\)
- \(0.0026\)
- \(70.6\)
- \(0.3\)
- \(2\,400\,000\)
- \(0.000\,000\,684\)
- \(0.0704\)
- \(0.260\)
- \(17\,600\)
- \(0.04080\)
- \(0.000\,500\)
- \(357\,000\,000\,000\)
- Write the following quantities in scientific notation.
- \(6\,370\,000\textrm{ m}\) – the mean radius of the Earth
- \(86\,400\textrm{ s}\) – the time for the Earth to orbit the Sun
- \(0.0018\textrm{ s}\) – the half-time of a radioactive isotope
- \(380\,000\,000\textrm{ m}\) – the average distance between the Moon and the Earth
- \(0.000\,000\,000\,001\,1\textrm{ s}\) – the time for a ray of light to pass through a window pane
- \(637\,000\,000\,000\,000\,000\,000\,000\textrm{ kg}\) – the mass of the planet Mars.
- Write the following numers in decimal form or in whole numbers.
- \(7\times10^{2}\)
- \(4\times10^{0}\)
- \(3.46\times10^{3}\)
- \(5.96\times10^{-5}\)
- \(9\times10^{7}\)
- \(3.98\times10^{1}\)
- \(2.78\times10^{5}\)
- \(6.78\times10^{-1}\)
-
- \(5.8\times10^{4}\)
- \(2.6\times10^{-3}\)
- \(7.06\times10\)
- \(3\times10^{-1}\)
- \(2.4\times10^{6}\)
- \(6.84\times10^{-7}\)
- \(7.04\times10^{-2}\)
- \(2.60\times10^{-1}\)
- \(1.76\times10^{4}\)
- \(4.080\times10^{-2}\)
- \(5.00\times10^{-4}\)
- \(3.57\times10^{11}\)
-
- \(6.37\times10^{6}\textrm{ m}\)
- \(8.64\times10^{4}\textrm{ s}\)
- \(1.8\times10^{-3}\textrm{ s}\)
- \(3.8\times10^{8}\textrm{ m}\)
- \(1.1\times10^{-12}\textrm{ s}\)
- \(6.37\times10^{23}\textrm{ kg}\)
-
- \(700\)
- \(4\)
- \(3460\)
- \(0.000\,059\,6\)
- \(90\,000\,000\)
- \(39.8\)
- \(278\,000\)
- \(0.678\)
Engineering notation
In engineering, a similar notation is used to handle practical applications. This is called engineering notation and it uses powers of \(10\) in multiples of \(3\), i.e. \(10^{3}\), \(10^{6}\), \(10^{12}\), \(10^{-6}\), etc.
As an example, the number \(74\,000\) is \(7.4\times10^{4}\) in scientific notation, but in engineering notation, we write it as \(74\times10^{3}\).
Expressing numbers in engineering notation
To write a number in engineering notation:
- Move the decimal point in groups of \(3\) to give a number between \(1\) and \(1\,000\).
- Multiply by a power of \(10\) equal to the number of places that the decimal point has been moved. This will be a multiple of \(10\). If:
- the decimal point is moved to the left, the power of \(10\) is positive
- the decimal point is moved to the right, the power of \(10\) is negative.
Example 1 – expressing numbers in engineering notation
Write \(1\,636\,000\,000\) in engineering notation.
We need to move the decimal point in groups of \(3\).
\[ 1\,\underleftarrow{636}\,\underleftarrow{000}\,\underleftarrow{000} = 1.636\]
We have \(3\) groups of \(3\) places, so \(9\) places in total, to the left. The power is \(+9\).
\[1\,636\,000\,000 = 1.636\times10^{9}\]
Write \(0.4\) in engineering notation.
\[00\underrightarrow{.400} = 400\]
\[00\underrightarrow{.400} = 400\]
We have \(1\) group of \(3\) places, so \(3\) places in total, to the right. The power is \(-3\).
\[0.4 = 400\times10^{-3}\]
Write \(0.000\,004\,5\) in engineering notation.
\[0\underrightarrow{.000}\,\underrightarrow{004}\,5 = 4.5\]
\[0\underrightarrow{.000}\,\underrightarrow{004}\,5 = 4.5\]
We have \(2\) groups of \(3\) places, so \(6\) places in total, to the right. The power is \(-6\).
\[0.000\,004\,5 = 4.5\times10^{-6}\]
Write \(5.75\times10^{4}\) in engineering notation.
This value is already written in scientific notation. We just need to make the power a multiple of \(3\). In the case, we can make it a \(3\) by shifting the decimal point \(1\) place to the right.
\[5\underrightarrow{.7}5\times10^{4} = 57.5\times10^{3}\]
Write \(3.175\times10^{-1}\) in engineering notation.
We need to make the power a multiple of \(3\). Here, we can make it a \(-3\) by shifting the decimal point \(2\) places to the right.
\[ 3\underrightarrow{.17}5\times10^{-1} = 317.5\times10^{-3}\]
Write \(57.5\times10^{2}\) in engineering notation.
We need to make the power a multiple of \(3\). Here, we can make it a \(3\) by shifting the decimal point \(1\) place to the right.
\[5\underleftarrow{7.}5\times10^{2} = 5.75\times10^{3}\]
Exercise – expressing numbers in engineering notation
- Write the following numbers in engineering notation.
- \(53\,800\)
- \(145\,000\,000\)
- \(0.761\)
- \(0.000\,534\)
- \(0.002\,8\)
- \(9\,620\)
- \(0.348\,20\)
- \(0.6\)
- \(5.6\times10^{-2}\)
- \(54.8\times10^{8}\).
-
- \(53\times10^{3}\)
- \(145\times10^{6}\)
- \(761\times10^{-3}\)
- \(534\times10^{-6}\)
- \(2.8\times10^{-3}\)
- \(9.62\times10^{3}\)
- \(348.2\times10^{-3}\)
- \(600\times10^{-3}\)
- \(56\times10^{-3}\)
- \(5.48\times10^{9}\)
Using prefixes and SI units
Engineering notation is more compatible with metrix prefixes like kilo (\(10^{3}\)), mega (\(10^{6}\)) and milli (\(10^{-3}\)), which are common in engineering contexts. The preferred prefixes are shown in the table. They all have a multiple of \(3\).
| Prefix | Symbol | Value | Example |
|---|---|---|---|
| giga | G | \(10^{9}\) | gigahertz (GHz) |
| mega | M | \(10^{6}\) | megavolt (MV) |
| kilo | k | \(10^{3}\) | kilometre (km) |
| milli | m | \(10^{-3}\) | milligram (mg) |
| micro | μ | \(10^{-6}\) | micrometre (μm) |
| nano | n | \(10^{-9}\) | nanoseconds (ns) |
| pico | p | \(10^{-12}\) | picofarad (pf) |
This helps engineers easily relate numerical values to practical units, making calculations and communication clearer.
Example 1 – using prefixes and SI units
Write the following quantities in SI units, using a preferred prefix.
- \(6\,000\textrm{ m}\)
- \(0.005\textrm{ V}\)
- \(0.000\,3\textrm{ s}\).
First, write each quantity in engineering notation.
\[6,\underleftarrow{000}\textrm{ m} = 6\times10^{3}\textrm{ m}\] \[0\underrightarrow{.005}\textrm{ V} = 5\times10^{-3}\textrm{ V}\] \[0\underrightarrow{.000}\,\underrightarrow{300}\textrm{ s} = 300\times10^{-6}\textrm{ s}\]
Match the power to the correct prefix and use it in place of the power.
\[6\times10^{3}\textrm{ m} = 6\textrm{ km}\] \[5\times10^{-3}\textrm{ V}=5\textrm{ mV}\] \[300\times10^{-6}\textrm{ s}=300\textrm{ µs}\]
Write \(9.625\times10^{-5}\textrm{ A}\) using a preferred prefix and SI unit.
Write the quantity in engineering notation.
\[9\underrightarrow{.6}25\times10^{-5}\textrm{ A} = 96.25\times10^{-6}\textrm{ A}\]
Match the power to the correct prefix and use it in place of the power.
\[96.25\times10^{-6}\textrm{ A} = 96.25\textrm{ µA}\]
Exercise – using prefixes and SI units
- Write the following in scientific notation, engineering notation, and using the correct prefix and SI unit.
- \(450\textrm{ m}\)
- \(63\,200\textrm{ W}\)
- \(0.000\,007\textrm{ F}\)
- \(37\,808\,000\textrm{ m}\)
- \(0.000\,000\,083\textrm{ m}\)
- \(0.800\textrm{ s}\).
- Write the following quantities in engineering notation, using the correct preferred prefix.
- \(0.002\textrm{ s}\)
- \(0.075\textrm{ km}\)
- \(0.05\textrm{ mA}\)
- \(0.003\,3\textrm{ µF}\)
- \(4\,500\textrm{ mm}\)
- \(0.25\textrm{ kV}\).
-
- \(4.5\times10^{2}\textrm{ m}\), \(450\textrm{ m}\) and \(450\textrm{ m}\) (no prefix)
- \(6.32\times10^{4}\textrm{ W}\), \(63.2\times10^{3}\textrm{ W}\) and \(63.2\textrm{ kW}\)
- \(7\times10^{-6}\textrm{ F}\), \(7\times10^{-6}\textrm{ F}\) and \(7\textrm{ µF}\)
- \(3.7808\times10^{7}\textrm{ m}\), \(37.808\times10^{6}\textrm{ m}\) and \(37.808\textrm{ Mm}\)
- \(8.3\times10^{-8}\textrm{ m}\), \(83\times10^{-9}\textrm{ m}\) and \(83\textrm{ nm}\)
- \(8\times10^{-1}\textrm{ s}\), \(800\times10^{-3}\textrm{ s}\) and \(800\textrm{ ms}\)
-
- \(2\textrm{ ms}\)
- \(75\textrm{ m}\)
- \(50\textrm{ µA}\)
- \(3.3\textrm{ nF}\)
- \(4.5\textrm{ m}\)
- \(250\textrm{ V}\)
