3.11

# Estimation, rounding and standard form

A brief look through any newspaper headlines and you will soon encounter something such as this:

“Cash-strapped entrepreneur, 33, launched a greeting cards company with just £30 in the bank - and now turns over £1.5 MILLION a year!”

When we examine this headline we soon realise that the numbers are there just to give a sense of scale. No one believes that the “cash-strapped entrepreneur” had exactly £30 in the bank and that they now have a turnover of exactly £1.5 million. The £30 is just telling us that they did not have very much money and the £1.5 million is telling us that they are now very successful and have a lot of money. The numbers are just there to give us a sense of scale.

For the next part of the course, we are going to consider the accuracy to which we need to give numerical answers. When is the exact answer required and when will an estimate be sufficient?

We’re going to look at a few questions. Before you go any further, answer the following questions and note down your answers. You may know the answers, you may be able to guess the answer or you may have to look up the answer. We’ll then look through possible ways of responding.

### Questions

1. How many teeth does an adult human usually have?
2. What is the usual number of bones in an adult human body?
3. What was Usain Bolt’s average speed in mph when he broke the 100 metre world record?
4. What is the value of $\pi$ (pi)?
5. What is the distance, in km, from the earth to the moon?
6. What is the number of atoms in the human body?

We’re now going to look at the answers you noted down and whether it is necessary to get the answer exactly correct. Let us consider not just the answers, but to what degree of accuracy you gave your answers.

### 1. How many teeth does an adult human usually have?

Usually 32 including wisdom teeth.

### 2. What is the usual number of bones in an adult human body?

Usually 206.

You may have known, or found, the exact answers to these two questions. If you guessed, a good guesses would be a whole number somewhere around 30 for the first answer and a whole number about 200 for the second. Did we need the exact answer? In both cases there is some variation from person to person so perhaps a rounded answer is probably good enough to give us an idea.

### 3. What was Usain Bolt’s average speed in miles per hour when he broke the 100 metre world record?

I calculated this to be about 23.4 mph (to 1 d.p.).

### 4. What is the value of $\pi$ (pi)?

For Usain Bolt’s sprint, the answer my calculator gave was 23.350065670146. The exact answer for pi goes on for ever and ever. In both cases a rounded answer to one or two decimal places is accurate enough. This is shown with 1 d.p. or 2 d.p. at the end of the value.

### 5. What is the distance, in km, from the earth to the moon?

The answer I got was 384,400km. I do not think that this is the exact distance from the earth to the moon. I expect this answer to have been rounded to four significant figures.

Significant figures help to represent the size of a number, without presenting the detail of more accurate, lower order place values. The number of significant figures is shown with the suffix s.f..

In the case of the distance from the Earth to the Moon, the exact distance varies and just over 384,000 km (3 s.f.) is a good enough approximation for most purposes. If you are planning to land on the moon I expect you would want to be a little more accurate.

### 6. What is the number of atoms in the human body?

The answer I found is seven billion, billion, billion. In other words a very big number. This number is seven followed by twenty seven zeros: 7,000,000,000,000,000,000,000,000,000

This number is not an exact answer but a figure given to one significant figure. We often write very larger numbers like this, and very small numbers, in standard form rather than writing so many zeros. Standard form is a number greater than zero and less than ten multiplied by a power of 10.

In standard form this number is $7 \times 10^{27}$.

## Teaching resources

To understand more about why we round to decimal places and significant figures we recommend you watch the short video entitled Rounding: Snails vs Rockets on the STEM Centre website. There are also further resources available:

## Discussion

What might influence our decision to use approximations and rounded answers?

Share your thoughts below. You might want to consider one or two of the examples above and where a higher or lower degree of accuracy would be more appropriate.