Uses of Surds and Indices in the Real World

How is GCSE maths used in everyday life?

Here are 10 examples of where people use Surds and Indices in everyday life or in their jobs. From understanding earthquakes to following a career in engineering, Surds and Indices are super useful!

Please note: The maths careers website is a resource which is aimed at the whole of the United Kingdom, and any mention of GCSEs and A-levels can also be applied to equivalent qualifications in Scotland.

Indices

Compound Interest

Compound interest is fundamental to how we save our money and pay back mortgages and other loans. Indices can help us to quickly work out compound interest calculations. For example, if you invest £500 in a bank account which is paying 5% compound interest per year, then after 3 years you will have $£550 * 1.03^{3} = £578.81$ (Assuming you haven’t made any withdrawals from your account.)

Earthquakes and the Richter Scale

You might have heard of the Richter Scale – this used to be the main way to measure earthquakes, whereas more up to date measurement scales are now used instead.

All of these Earthquake magnitude scales are based on logarithms. Logarithms are a special way of displaying indices, and what this means is that if you have an earthquake which is magnitude 5 then it is 10 times more powerful than an earthquake with magnitude 4. If you compare an earthquake with magnitude 6 to an earthquake with magnitude 9 then there is a $10^{3}$ difference in strength.

Standard Form – Big Numbers and Small Numbers

If you want to talk about very big numbers or very small numbers then you will need Standard Form. Very large numbers often appear in science, particularly when it comes to space.

The mass of the sun is $1.988 × 10^{30}$ kg

Very small numbers are found all over the natural world and can be used to measure things like viruses and bacteria.

The mass of a single dust particle is $7.53 * 10^{-10}$ kg

Growth and Decay of Populations

Imagine you have a type of bacteria which divides once per hour. If you start off with a single bacterium then after 24 hours there will be 24 divisions. This will result in $2^{24} = 16,777,216$ bacteria! Indices are very important in biology.

Indices are also important in calculating exponential decay, for example radioactive decay.

The binary number 1010 is equal to the decimal number 10.

Surds

Paper Sizes

If you take an A4 piece of paper then the ratio of the long edge to the short edge is $\sqrt{2}$ . This same ratio holds if you take other A sized pieces of paper such as A0,A1,A2,A3, etc. There is a good reason that $\sqrt{2}$ was chosen as the ratio between the length and width. It means that when you cut an A4 piece of paper in half, you will get exactly two pieces of A5 paper, and when you cut an A5 piece of paper in half you will get two pieces of A6 paper and so on. Convenient both for manufacture of the paper and also for binding books and creating different arts and crafts.

Golden Ratio

Some people believe that using the Golden Ratio produces the most beautiful art and architecture, although this is very much a personal preference. The Golden Ratio also occurs naturally in nature such as in the spirals of sunflowers and in pinecones. At the heart of the Golden Ratio is a Surd:

The value of the Golden Ratio is: $\frac{1+\sqrt{5}}{2}$

Find out more about the Golden Ratio.

Did you know? There is also a lesser known Silver Ratio, and even a Bronze Ratio!

Projectile Motion

If you throw a ball, or launch any kind of projectile, then the position of the object can be modelled using a quadratic equation. Often when you solve a quadratic equation you will be left with an answer which includes a surd.

Engineering

Being able to work with surds is really important in the study of engineering. For example, being able to find the length of a catenary chain bridge support involves doing advanced calculations with a square root.

Trigonometric Ratios

Many exact Trig Ratios can be expressed in terms of surds. Trigonometry is useful in architecture, engineering and the sciences, the applications are endless!