# 1.10. Demonstrate the effect of error in calculations

[responsivevoice_button rate=”0.9″ voice=”UK English Female” buttontext=”Listen to Post”]

Calculation errors can have a serious effect in everyday life; for example if the pharmacist gives a patient an incorrect dosage of very strong medicine, it could have fatal consequences, or an engineer calculates incorrectly, a structure could collapse. A cancer patient having radiation therapy must receive highly accurate doses, which target tumours and leave healthy tissue untouched.

Miscalculation also affects us in many ways, for example if we miscalculate how much money we’ve spent, we could find ourselves broke before the end of the month!

Or we could buy too little paint, and when we go back for more, that colour is out of stock!

Every measurement has a degree of uncertainty associated with it. The uncertainty derives from the measuring device and from the skill of the person doing the measuring.

Let’s use **volume** measurement as an example. Say you are in a chemistry lab and need 7 ml of water. You could take an unmarked coffee cup and add water until you think you have about 7 millilitres. In this case, the majority of the measurement error is associated with the skill of the person doing the measuring.

You could use a beaker, marked in 5 ml increments. With the beaker, you could easily obtain a volume between 5 and 10 ml, probably close to 7 ml, give or take 1 ml.

If you used a pipette marked in 0.1 ml increments, you could get a volume between 6.99 and 7.01 ml quite reliably.

It would be untrue to report that you measured 7.000 ml using any of these devices, because you didn’t measure the volume to the nearest microlitre.

You would report your measurement using significant figures. These include all of the digits you know for certain plus the last digit, which contains some uncertainty.

Measured quantities are often used in calculations. The precision of the calculation is limited by the precision of the measurements on which it is based.

The measurement of a physical quantity is always subject to some degree of uncertainty. There are several reasons for this:

- the limitations inherent in the construction of the measuring instrument or device,
- the conditions under which the measurement is made
- the different ways in which the person uses or reads the instrument

In this section, we will be focusing on how to overcome the calculation errors.

**Rational and Irrational Numbers**

Before we talk about rational and irrational numbers, let’s make clear one other definition. An **integer** is in the set:

{…-3, -2, -1, 0, 1, 2, 3, …}

It is often useful to think of the integers as points along a ‘number line’, like this:

Note that zero is neither positive nor negative. It is just a positive or negative whole number. Thus 454564 is an integer, but 1/2 isn’t.

**A rational number** is any number that can be written as a ratio of two integers (hence the name!). In other words, a number is rational if we can write it as a fraction where the numerator and denominator are both integers. Now then, every integer is a rational number, since each integer n can be written in the form n/1.

For example 5 = 5/1 and therefore 5 is a rational number.

However, numbers like 1/2, 45454737/2424242 and -3/7 are also rational since they are fractions where the numerator and denominator are integers.

**An irrational number** is any real number that is not rational. By “real” number we mean, loosely, a number that we can conceive of in this world, one with no square roots of negative numbers.

A real number is a number that is somewhere on your number line. So, any number on the number line that isn’t a rational number is irrational.

For example, the square root of 2 is an irrational number because it can’t be written as a ratio of two integers.

Other irrational numbers include: square root of 3, the square root of 5, pi, etc.