Is it true that an understanding of mathematics is essential for scientific study? Well the answer is yes and no. You wouldn't get very far in science without a basic understanding of maths, but there are few scientists who would claim to be good mathematicians, as Mike Leahy showed when he had to do some sums to calculate Capraia's latitude and longitude.
The Open University course S104 Exploring Science teaches degree-level science, yet it assumes only a basic knowledge of maths.
'Using a Calculator for Arithmetic' (below) is taken directly from the first Block of S103, an earlier entry-level course. This should give you an idea of the level of mathematical understanding that is needed to take the course. All of the S103 course materials do their best to provide you with new mathematical skills as and when you need them.
Using a calculator for arithmetic
To get you used to working with a calculator, we shall provide directions throughout S103 on how to do each type of calculation when it first arises. Here, we shall cover the standard arithmetic operations:
- addition +
- subtraction -
- multiplication x
- division ÷ or /
We'll use some of the data on UK water use from Table 1 as the basis for the calculations.
Adding numbers with a calculator
Suppose we want to calculate the total use of non-mains water per person. This means adding the last three numbers in the right-hand column of Table 1: 7 + 220 + 52.
Let’s take this a stage at a time. First, 7 + 220, which you can work out in your head to be 227, and which we can represent by the equation:
7 + 220 = 227
In words, this equation is ‘seven plus two hundred and twenty equals two hundred and twenty-seven’.
An equation like this tells us that whatever is on the left-hand side of the equals sign (=) is exactly equal to whatever is on the right-hand side. So what we are saying in the above equation is that 7 + 220 is exactly the same as 227, or in other words, 7 + 220 is equal to 227.
Now work out this addition with your calculator. To do this you press the keys in the order in which they appear in the equation:
7 then + then 220 then = and the answer 227 will appear in the display.
Try this procedure for yourself. Of course, the calculator won’t tell you that it’s 227 litres – you have to provide that part of the answer from a knowledge of the meaning of the numbers that you are adding. You added 7 litres and 220 litres, so the answer must be 227 litres.
The complete equation can therefore be written as:
7 litres + 220 litres = 227 litres
Now try using your calculator to add some pairs of small numbers that you can also add in your head, and convince yourself that the calculator produces the correct answer in each case.
To add more than two numbers, as required to calculate the total non-domestic, non-mains water use from Table 1, you again key in the calculation exactly as it is written. So to work out 7 + 220 + 52, you should key in:
7 then + then 220 then + then 52 then = and the answer 279 appears in the display, so the answer is 279 litres.
The equation for this is:
7 litres + 220 litres + 52 litres = 279 litres
This procedure can be extended to add as many numbers as you wish.
Subtracting numbers with a calculator
For subtraction, you just press the - key instead of + . Take the water-use example from Table 1 again: the difference between the non domestic use of water for electricity generation and for industry is found by subtracting the appropriate numbers taken from the table; that is, 220 - 52.
To work this out with your calculator, you should press the keys in the following order:
You can string together a series of subtractions, in the same way as the series of additions above. You can also mix additions and subtractions in a sequence of operations (which would be useful for working out the effect of a series of credits and debits to your bank account, for instance).
What the calculator doesn’t tell you is the unit of the answer.
The calculations that you have just done involved numbers that have a unit of measurement associated with them, namely litres (of water per person per day). However, your calculator only deals with the numbers and doesn’t deal with the unit. So how do you know what the unit of the answer to a given calculation is going to be?
The answer comes from the fact that the unit must be the same on both sides of an equation, just as the numbers must.
For example, if we want to add 3 litres and 5 litres, then we can write:
3 litres + 5 litres =.........
The calculator (or good old-fashioned mental arithmetic) will tell you that 3 + 5 = 8. Then if we want the same unit on both sides of the equation, the answer must be 8 litres; that is, 3 litres + 5 litres = 8 litres.
An important consequence of this requirement for the unit of measurement to be the same on both sides of an equation is that when we are adding or subtracting quantities, then these quantities must have the same unit. You can’t add 2 litres and 5 gallons; the total amount is neither 7 litres nor 7 gallons. To find the total amount, you need to convert 5 gallons into litres (or 2 litres into gallons) so that you are adding amounts measured in the same unit.
To practise using your calculator for addition and subtraction, try the following calculations:
(a) 46 + 78; (b) 83 + 29; (c) 94 litres + 136 litres;
(d) 283 + 729; (e) 56 - 35; (f) 463 metres - 89 metres;
(g) 274 grams - 168 grams; (h) 38 + 92 - 61.
Multiplying numbers with a calculator
For multiplication of two numbers, or for multiplying a whole series of numbers, you need to press the calculator keys in the order written for the calculation, just as with addition and subtraction.
So if the daily use of water in washing machines is 16 litres, then you find the weekly use by multiplying by 7 (the number of days in a week); that is, 16 x 7.
So you key in:
If we had wanted to know the amount used in washing machines by a person during their 75-year lifetime, then the appropriate sum would be 16 litres per day x 365 days per year x 75 years. Keying this as:
Note that generally in science, numbers are printed with a space between thousands and hundreds, not a comma.
Dividing numbers with a calculator
As with the other three arithmetical operations discussed above, if you can write down a division sum as a series of numbers and symbols, then you can simply key them into the calculator in the order in which they are written.
For example, if your annual use of water was 54 750 litres, then you can calculate your daily use by dividing this by 365, so the calculation is 54 750 ÷ 365, which is often written as 54 750/365).
To do this on the calculator, you need to press the keys as follows:
54750 ÷ 365 =
and the answer is 150. So your daily use of water would be 150 litres.
To practise using your calculator for multiplication and division, try the following calculations:
(a) 48 x 21; (b) 95 x 24; (c) 761 x 13; (d) 293 litres x 212;
(e) 94 ÷ 47; (f) 392 ÷ 49; (g) 378 metres ÷ 54;
(h) 24 x 32 ÷ 8; (i) 245 x 76 ÷ 20.
The answers to both sets of questions can be at the bottom of the page.
Working out longitude
Let’s try the calculation that the castaways used to determine the longitude of their island location in the first series of Rough Science. By watching the pendulum swinging, Mike Leahy worked out that Greenwich Mean Time noon occurred 2,496 seconds past noon on the island. What we have to determine is how many degrees of longitude this time interval represents.
The Earth revolves once on its axis every 24 hours, and during this time it goes through 360 degrees of longitude, as shown in the article ‘How to measure latitude and longitude’.
If we were to divide the world into 24 one-hour time zones, how many degrees of longitude would there be to each time zone?
If 360 degrees is divided into 24 segments then each segment would correspond to ‘360 divided by 24’ degrees; in other words, 15 degrees.
What this means is that the world rotates by 15 degrees every 60 minutes.
How long does it take the world to rotate by one degree then?
If the world takes 60 minutes to rotate 15 degrees then to rotate by one degree it will take one-fifteenth of this time; in other words, ‘60 divided by 15’ minutes, or four minutes.
So one degree of longitude corresponds to four minutes, or 240 seconds.
Our castaways reckoned that noon on the island occurred 2,496 seconds before noon by Greenwich Mean Time, which corresponds to (2,496 seconds divided by 240 seconds) degrees...
in other words, 10.4 degrees of longitude.
Since noon on the island occurred before noon by Greenwich Mean Time, the location must be east of Greenwich. The castaways’ calculation of Capraia’s longitude was within 0.9 degrees of their true location, which is pretty impressive!
Modules 1 and 4 of K507, Breakthrough to Mathematics, Science and Technology, The Open University, 1998
Graham A., Teach Yourself Basic Maths, Hodder & Stoughton
Graham L. and Sargent D., Countdown to Mathematics (volume 1), Addison Wesley Publishers Ltd
Northedge A. et al., The Sciences Good Study Guide, The Open University
Answers to maths questions
(a) 124; (b) 112; (c) 230 litres [Here, the two quantities that you added have the unit litres, so you need to include the word litres in the answer.]; (d) 1 012; (e) 21; (f) 374 metres; (g) 106 grams; (h) 69 [You do not have to press = after the first part (38 + 92), though if you do, the final answer is the same; check this for yourself.]
(a) 1 008; (b) 2 280; (c) 9 893; (d) 62 116 litres [Remember to include the unit in the answer to balance the unit in the question.]; (e) 2; (f) 8; (g) 7 metres; (h) 96; (i) 931.