Monday, 30 November 2009

The basics of: Algebra

I intended to start this piece with a definition, but than had a look online and found it needlessly verbose for those trying to understand the basics. There is a very simple way of thinking about algebra, and it is to simply do with letters what you would do with numbers.

This sounds patently absurd and I can imagine the cries
"But I know 2 + 2 = 4, what on earth could x + x equal?"
Of course the point is that you don't know what x equals in this case, indeed the use of x is completely arbitrary. The letters in algebra are simply substitutions for unknown quantities,and we do do the same thing with x.
As we don't know what x means, our job is to either simplify a mathematical statement so that should we ever find out what x means it is easier to deal with, or solve an equation that will tell us what x is.

So I've said that we do the same thing with letters as we do with numbers, so how does this work?We can start by looking at the simple term above, just what does x + x equal? Well with numbers if you add a number to itself, you would have 2 of that number. 4 plus 4 does equal 8, but you could also think of it as
2 X 4 or just two 4's.

This is what we do with our x's. We don't know what we would get if we added 2 of them together, but we do at least know that we have 2 of them, so we can simplify.
x + x = 2x
 That's it! we have 2 x's now, and we say so.

This makes adding like unknowns (the letters) really straightforward, if we have 4x's and 2x's we have 6x's and if we have 5x's and take away 4x's we are left with just the one x
4x + 2x = 6x      6x - 5x = x
 It's just like adding apples! (or any other fruit that you might want to count)

So now to multiplication and division. When dealing with like terms (more about this in a moment), we actually do something very similar to what we do when adding, and we still treat them in the same way we treat numbers!

Looking at numbers for a second,  when multiplying a number by itself we see something like the following...
3 X 3 = 9      5 X 5 = 25      1 X 1 = 1
But of course in algebra we are dealing with an unknown, which we will call y this time to make the point that the choice of letter makes no real difference*.
y X y = ?
Well again as we don't know the value of y, we are going to have to simplify. So how else could we write the above?
When we multiply a number by itself, we call that squaring the number which is denoted by a little superscript 2 as we can see below
3 X 3 = 9 = 32      5 X 5 = 25 = 52
So even though we don't know the value of y multiplied by itself, we do know what it simplifies to
y X y = y2
Simple when you know how isn't it? if you are multiplying more than 2 of the same term, the superscript number is equal to the number of times that you multiply that term so
y X y X y X y = y4       y X y X y = y 3
And if you are asked to multiply together, for example y2 and y5 (the latter would be pronounced y 'to the power of 5' for those that care) you do so by simply adding the superscript numbers, or indices, which would give you the answer y7. If it helps you in doing this, it is worth remembering that y = y1.

Similarly, when dividing two like terms (there it is again) together, you subtract indices.
y8 / y3 = y 5
Now to the point about like terms.

When we talk about like terms,  we are referring to the letters, or unknowns, in our equation or statement. So far we have only dealt with one letter at a time, so we have not had any problems. However, if we have more than one unknown we have to be careful to treat them separately. In the following statement the italic terms can be considered like terms
10x + 3x2 + 7x + 5y + 4x
As we can see all the terms that are simply a number followed by x are italicised we can simplify this statement in the same way as those seen earlier, but we we can only do things with those like terms.
21x + 3x2 + 5y
Now it should be reasonably clear why the y is in plain text, but why the x2 ? The easiest way to consider it is if we go back to numbers.

Now when we calculate the value of 3x2 what we are trying to ascertain the the value of '3 times x times x'. If x in the above statement represents 1 then 3x2 must represent 3 and therefore 3x. If x = 2 then 3x2 = 12 or 6x. If x = 3, then 3x2= 27 and therefore 9x!

It doesn't take very long to see that the value of x2 relative to x will change as x changes, that means it cannot be dealt with in the same way.

In a later post, I intend to have a look at the ways of simplifying more complex statements, and how to expand brackets. Before I head off today though, we're going to look at how to solve a simple equation.

Equations are simply pairs of statements that we know are equal to each other, so for example the following is an equation
5 + 7 = 9 + 3
In the above both sides equal 12 and are equal to each other, we could well have written 5 + 7 = 12 and it still would have been a valid equation.

Now of course there was no solving in the above, we were simply offered a statement that we could easily tell was true. In algebra, equations are often used to work out the value of an unknown based on the truth of the equation. Let's have a look at an example...
x + 12 =  5x - 8
When asked to work out the value of an unknown in these cases it is known as solving for '*' where '*' represents our unknown. In this case we are 'solving for x'. When we are solving an equation like this, what we wish to do is isolate any term with our unknown on one side of the equation, and move those terms without our unknown on the other. We do this by manipulating both sides of the equation carefully.

If we go back to our earlier purely numerical statement we can add a number to both sides and the equation will remain true (both sides will still be equivalent)
5 + 7        = 9 + 3
5 + 7 + 3  = 9 + 3 + 3
15             = 15
Now the value of both statements has changed in this case to 15. This is not important, what is important is that the 2 statements are still equal. If we apply the same logic to our equation with x we can see that if we add 8 to both sides of the equation we can isolate our x term on the right hand side.
x + 12       =  5x - 8
x + 12 + 8 =  5x - 8 + 8
x + 20        =  5x
If this doesn't make immediate sense to you, it may help to think of (5x - 8) as (5x + 0 - 8) of course my comments box is always open to anyone with any queries.

Now we are dealing with a number and an x term on one side of the equation, and just an x term on the other side of the equation. To isolate out number on the left, we can subtract x from both sides in the same way we added 8 to both sides earlier.
x + 20      = 5x
x - x + 20 = 5x - x
20            = 4x
Fantastic! Now we are just left with an x term on one side and a numerical term on the other. We have successfully solved for 4x!
But that wasn't what we wanted to find out, we wanted to know the value of just one x, not 4 of them. So what we must do now is divide both sides of the equation by 4 so that we are left with an x on one side and it's value on the other.
20      = 4x
20 / 4 = 4x / 4
5        = x
So we've done it! We now know that x equals 5!

There is of course a lot more to algebra than this, and I hope to go over some furthering topics in the future.

Thanks very much for reading. Leave me a comment if anything is unclear and I will be glad to help.

1 comment:

  1. You have such a good way of simplifying mathematics aside from algebra. Maths always scared me, but if I'd had a maths teacher even half as good at explaining the subject to me as you, it wouldn't have been such a struggle!