Standard Form…

This entry is part 10 of 10 in the series How Science Works

This is especially relevant to those of you doing Radley Physics Papers.

Joking aside, knowing a little mathematical notation can be very helpful. Often a “back-of-the-envelope calculation” is enough to use as a guide to the sort of answer you might expect.

Standard form is a way of expressing numbers as multiples of ten. It is excellent shorthand and makes quick estimates, using very large or very small numbers, much easier. It is also quite popular with the setters of certain scholarship exam papers (esp. Radley.) It is a great way to roughly calculate an answer where a really accurate answer is not needed.

There are a couple of maths sites that deal with this topic linked at the bottom of the page.

A nice, short and unsophisticated YouTube video…

10³ (ten to the power of three) means 10 x 10 x 10 i.e. 1000

Another way to write 2000 would be as 2 x 10³.

For numbers less than one, negative indices can be used e.g. 0.001 is 1 x 10^-3

To multiply numbers in standard form you multiply the numbers at the start and add the indices.

e.g. (2,000 x 300)

2 x 10³ × 3 x 10² = 6 x 10⁵

To divide numbers in standard form you divide the numbers at the start and subtract the indices.

e.g. (400,000 ÷ 2000)

4 x 10⁵ ÷ 2 x 10³ = 2 x 10²

The first number in standard form is always written less than 10, for example, the speed of light is 299 792 458 m/s. In standard form this would be 2.99792458 x 10⁸ m/s. This is a bit cumbersome so it is easier to round it off to one decimal place: 3.0 x 10⁸ m/s.

e.g. How fast does an object travel if it is moving at ¼ the speed of light?

299,792,458 m/s × 0.25 = 74,948,115 m/s

Or in standard form

3 x 10⁸ m/s × 2.5 x 10^-1 = 7.5 x 10⁷ m/s

Adding and subtracting in standard form are pretty simple too although it appears less commonly in scholarship questions. It is sometimes necessary to make the indices the same before you start.

e.g. 3 x 10³ + 3 x 10³ = 6 x 10³

If the “to the power ofs” are not the same to start with, you need to match them up before adding.

e.g. 3 x 10³ + 3 x 10⁴

First make the indices the same then add the first numbers together.

0.3 x 10⁴ + 3 x 10⁴ = 3.3 x 10⁴

There is a helpful video here that takes you through a couple of examples of adding and subtracting using standard form.

Let’s do a quick, ‘back of an envelope’ calculation to estimate the volume of the observable Universe. First I will make two assumptions.

a) The ‘Big Bang’ happened 13.8 billion years ago

b) The observable Universe has been expanding in every direction at the speed of light ever since

The formula to calculate the volume of a sphere is…

13.8 billion years is about 4.4 x 10¹⁷ seconds. The speed of light is roughly 3.0 x 10⁸ m/s. So the radius of the Universe is distance (m) = speed (m/s) × time (s).

3.0 x 10⁸ m/s × 4.4 x 10¹⁷ s = 13.2 x 10²⁵ m (correctly written as 1.3 x 10²⁶ m)

V = 4/3 × π × (1.3 x 10²⁶ m)³

V = 4/3 × π × 2.2 x 10⁷⁸ m³ = 9.2 x 10⁷⁸ m³

When I Google, “What is the volume of the Universe?” I get a figure of about 3.0 x 10⁸⁰ m³which is based on rather different assumptions because it turns out that the Universe’s expansion is accelerating and not constant. Standard form keeps your working clean and, especially if you are not allowed a calculator, makes difficult sums much easier to handle.

Annoyingly the comments section does not allow use of superscript and subscript text so it can be fiddly to type in standard form. The format that your calculator uses would be, for the speed of light, 3.0e+8 m/s or, for the volume of the Universe, 3.0e+80 m³

Questions…

What is the approximate speed of sound in air at sea-level and 20°C?
What is the nearest star to Earth?
The Sun is approximately 150,000,000 km from Earth. How long does the light take to reach us from the Sun in seconds?
Express 245.67352 in standard form.
Proxima Centauri is about 4.2 light years away. What is that in metres?
Roughly what shape is a cow (refers to the video below – if you have watched it!)?

I will leave you with a ten minute clip from physicist Professor Lawrence Krauss. This is the opening piece of a much longer lecture given several years ago. He is always very keen to promote physics as the answer to almost everything and one of his themes is the importance of being able to make good approximations.