Mathematical Skills

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​In physics, we make measurements.

In many cases, these measurements involve extremely large or extremely small values. To work with these numbers, we use various mathematical tools to make sense of them. The following videos examine four different tools that allow us to manipulate numbers more effectively.

Scientific and Engineering Notation

When measuring physical variables, values can be extraordinarily small—such as the diameter of a proton—or unimaginably vast, like the scale of our galaxy, the Milky Way.

If we recorded these measurements using standard metres alone, expressing them would quickly become impractical:
• Diameter of a proton: ≈ 0.000000000000001 metres
• Diameter of the Milky Way: ≈ 1,000,000,000,000,000,000,000 metres

To streamline these values, we use scientific notation:
• Diameter of a proton: ≈ 1.0 × 10-15 metres
• Diameter of the Milky Way: ≈ 1.0 × 1021 metres

Much cleaner! Watch the video below to learn the rules for writing numbers in scientific notation, as well as its closely related counterpart, engineering notation.

To simplify things, we use scientific notation. Expressed this way, the diameter of a proton is approximately 1.0 × 10-15 metres, whereas the diameter of the Milky Way is around 1.0 × 1021 metres.

Much simpler!

Watch the video to learn about the rules for scientific notation, as well as its closely related counterpart, engineering notation.

Worksheet
Have a go in practising the concept learned by doing this worksheet
Prefixes

Another useful tool for working with large and small quantities is the use of prefixes. You are likely already familiar with a few of them.

For example, a kilometre (km) is equivalent to 1,000 metres (m), meaning the prefix "kilo-" represents 1,000 or 103.
Similarly, a millilitre (mL) is equivalent to 1⁄1000 of a litre (L), meaning the prefix "milli-" represents 1⁄1000 or 10-3.

By attaching these prefixes to base units, we can easily express scale without writing long strings of zeroes.

Watch the video below for a deeper explanation.


Here are two worksheets (though one is a bit of fun) to help practice prefixes
Significant Figures

When a physical quantity is measured, it is only known to a certain level of precision.

For example, if you measure a piece of paper using a 30 cm ruler, you might determine its length to be 21 cm or perhaps 21.5 cm. However, you are unlikely to be confident in a measurement like 21.354 cm.

The more decimal places you can reliably record, the more precise the measurement is. Naturally, this precision depends heavily on the tool you use.

In physics, we use significant figures to communicate the degree of precision in our measurements.

Watch the video below to learn how significant figures work and how to count them.

Have a go at practising Significant figures

Orders of Magnitude

Orders of magnitude are a powerful tool for quick estimation and comparing relative scales.

By rounding numbers to the nearest power of ten, we can quickly grasp the scale of a quantity and make rapid mental calculations without getting bogged down in precise details.

Watch the video to learn how to determine an order of magnitude and how to use it for estimation in physics.

Test your understanding
1. What is 0.000045 metres expressed in scientific notation?
2. How many millimetres (mm) are equivalent to 1 metre (m)?
3. How many significant figures are in the measurement 0.003050 kg?
4. What is the approximate order of magnitude of a human height (~1.7 m)?
5. Which prefix corresponds to the multiplier 10⁶?

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