Negative Numbers

Negative numbers probably don't really exist in the real world, but they are a valuable tool in maths, so I'm going to show you how they're used and what they're for. To begin with, you've probably heard negative numbers being used to describe low temperatures when it goes "below zero". There are several different scales used for temperatures, so this can be a confusing subject, but the one I prefer to use is known as Centigrade or Celsius. With this scale, ice melts at 0C (zero degrees C) and water boils at 100C (a hundred degrees C, the C obviously standing for Centigrade or Celsius). Anything colder than zero is given a negative value, so ten degrees below zero is −10C, and 10 degrees above zero is 10C. 10C is 20 degrees warmer than −10C.
indentThe truth about temperature, however, is that real zero is the point where there is no heat energy left at all, and that is found at −273C. This temperature is actually called zero on the Kelvin scale (a scale which is often used by scientists), and on this scale you will see that ice melts at 273K and water boils at 373K (these are again a hundred degrees apart, so you should realise that the size of a degree is the same on the Kelvin scale as it is in Centigrade). There are no negative temperatures on the Kelvin scale because there is no such thing as negative heat: once something has cooled to zero, it is impossible for it to go any lower. The temperature 0 Kelvin (or −273C) is known as absolute zero.

Another place some people often see negative numbers is in their bank account statements: negative values here mean that rather than you having money in the bank, you actually owe the bank money! Again these are not true negatives: the money is positive, but it's money which the bank has given to you and which you will have to pay back to them at some point, which is not a nice situation to be in because they will charge you a fee called "interest" on the loan, and they'll charge you more and more interest the longer you take to pay them back their money. The amount of interest they'll charge will also depend on how much money you've borrowed from them, so it is possible for people to fall into a trap where the interest charges grow until they are nearly as big as the amount of money they are earning: they may spend the rest of their lives giving almost all the money they earn to the bank, and yet they never manage to get the debt back under control. Don't let that happen to you! Sometimes borrowing money from a bank can actually be a good idea if it enables you invest it in something that will help you earn enough money to be able to pay off the loan quickly so that you can then go on earning money to keep for yourself, but you should never borrow money just so that you can have something in a hurry that you would be better off waiting for. The only other sensible kind of debt is a mortgage (when you buy a house - this is because tax savings will more than balance out the interest charges).

Negative numbers also appear on graphs, as you will see in a moment, and again there is no connection between them and any actual negative values in the real world: the negative values only exist in the maths. Let's have a look at an equation for a circle:-

a2 + b2 = 25

This can also be written out as:-

a x a + b x b = 25

As before, we can try out different values for "a" and see what value "b" has to have for each one. So, let's try zero first: if a=0, we replace every "a" in the equation with the number 0, so that's 0 x 0 + b x b = 25, and that obviously means b x b = 25. We now have to work out what x itself = 25? The number 5 works (5 x 5 = 25), so b=5. But hold on a moment: the number b=−5 also works, because −5 x −5 = 25 as well: a negative number times a negative number always gives a positive answer (this is a rule which you need to learn). There are two right answers for b when a=0, so that means we have to plot two points on the graph instead of one: a=0, b=5; and a=0, b=−5. There is actually a special way to write out the two "co-ordinates" used for plotting a point on a graph: they are always put between brackets with a comma in between to keep them apart, so the two points are written as (5,0) and (−5,0), with the b value stated first because the horizontal co-ordinate, which you find on the graph by looking along the numbered horizontal line, is always written before the vertical co-ordinate, which you find on the graph by looking on the numbered vertical line (that's another rule for you to learn). You can see the two points (5,0) and (−5,0) plotted as red dots in this graph:-

Let's work out a few more points. If a=3, then 3 x 3 + b x b = 25, and that means 9 + b x b = 25. We can move the nine to the other side of the equals to get b x b = 25 − 9, and then we can take the 9 away from the 25 to get b x b = 16. This time we have to work out what x itself = 16? The answer is 4, so b=4. This gives us the point (4,3). Remember that the value for "b" comes before the value for "a" because the the horizontal co-ordinate is written before the vertical one. There is another possible answer again, and that's b=−4 (because −4 x −4 = 16), so we also have the point (−4,3). The two points (4,3) and (−4,3) can be seen above plotted on the graph as light blue dots.
indentIf you try a=−3 instead of a=3, you will find that −3 x −3 + b x b = 25 turns into 9 + b x b = 25 and on into b x b = 25 − 9 and again into b x b = 16, so you will come up with b=4 and b=−4 for the other co-ordinate just as you did when "a" was positive 3, giving the points (4,−3) and (−4,−3), and these are shown on the graph as black dots.
indentIf you try a=4, then you get 4 x 4 + b x b = 25, so that translates to 16 + b x b = 25, which then turns into b x b = 25 − 16, and that in turn becomes b x b = 9, so b=3, or b=−3. This gives us the points (3,4) and (−3,4), as shown on the graph in yellow dots. Do the same thing starting with a=−4 and you'll end up with the dark blue dots.
indentYou can see that all these dots are falling on a circle, and if you try a=5 and a=−5 you'll get the points shown in green as well. The numbers for any of the other points on the circle are harder to work out because they are no longer going to be whole numbers, so it's easiest to use a calculator. Click this button to make one appear under the graph . You can now go back to the graph and type a number into the first box, click the first button to multiply it by itself (this is known as "squaring" a number), click the second button to take it away from 25, and then click the third button to find out what times itself = that number (this is known as finding the "square root" of a number). Then look at the graph to see if that number (in box 4) and the one you started with (in box 1) give you a point on the graph that lands on the gray circle (note that the symbol means that a number can be either positive or negative, but be aware that most calculators don't show you this, so you normally have to remember that there are two possible answers). See what happens if you start with a number bigger than 5 (you should know that "NaN" stands for "Not a Number"). You can see now that the equation produces all possible points on the circle, and no points anywhere else. Make sure you try the calculator with values like 1.5 as well as whole numbers, and also try negative numbers.

You now have a better idea about how negative numbers are used. The important rules to remember when multiplying are these:-

  • positive x positive = positive.
  • negative x negative = positive.
  • negative x positive = negative.
  • positive x negative = negative.

The same rules also work for dividing, but not for adding or subtracting (which we'll get to in a moment).

When you use the calculator above with a number bigger than 5, you'll notice that it results in a negative number after you click the "25 −" button, and there is never a real answer when you then click the "Root" button. This is because there is no number which can be multiplied by itself to make a negative number: a negative times a negative will always lead to a positive. If you do maths at university level you will meet "imaginary numbers" which are an attempt to represent square roots of negative numbers, and they do actually have real uses in maths (and physics), but you needn't worry about them just yet.

Adding with negative numbers

When you add a negative to a positive number (e.g. 5 + −7, which means much the same as 5 − 7) what you should do is take the smaller number away from the bigger one (7 − 5 = 2) and then give the answer the same sign as the bigger number (so in this case it's the negative sign from the −7, making the answer −2). If the bigger number is positive (as with 7 + −5), then the bigger number has a hidden positive sign (the sum is really +7 + −5), so the answer will have a positive sign (+2) which can also be left out (so it's simply 2).

Taking away with negative numbers

When taking away, there are other things to think about, because taking away a negative number is the same as adding that number without its negative sign (e.g. 5 − −7 is the same as 5 + 7). You just have to stop and think carefully about each sum to make sure you're doing the right thing with it. If you meet a sum like −5 + 7, try to rearrange it to get rid of the negative number: in this case the order can be changed to 7 + −5, and then you can see that it's the same as 7 − 5, so the answer's obviously 2. Here's another sum: 5 − 7. You can see that the answer will be negative, so it's just a matter of taking the smaller number (5) away from the bigger one (7) and making the answer negative (−2). If this sounds like the same rule as for adding negative numbers, it's because it is the same rule: 5 − 7 is the same as 5 + −7.

If both numbers are negative, then you need to keep your wits about you even more, treating each sum as a puzzle. −5 + −7 is really the same sum as −5 − 7, so the answer's −12. Let's try another one: −5 − −7 means the same as −5 + 7, and that can be turned round to make 7 − 5, so the answer's 2. You'll get lots of practise with this, so you'll get the hang of it soon enough.

What's next?

Well, we're going to play with scientific calculators (don't worry if you don't own one because your computer can do the job instead, though of course it's always nice to own the dedicated gadget as well). You've finished with primary maths now: it's time to move up!