some common mistakes that people make in algebraic manipulation

and working with numbers. I've called it a "baker's dozen"

because there are thirteen separate topics covered in the course of these

four videos.

This first video covers what can loosely be called

"reading the recipe correctly", that is can you

understand the mathematics that's being asked of you when it's presented

in the form of a question and there are three typical things that can

cause trouble under this heading. Making sure that you use powers,

indexes correctly, how you deal with

zero powers and also what happens

if you attempt to divide by zero. Let's look at the first one

concerning powers.

A mistake I

see fairly commonly when people are using algebra is this one.

"Minus a squared" does not actually turn out to be

"-a multiplied by itself". Now to see that,

let's look at a situation that's similar

using some simple numbers. Here we have -3 in brackets

and we are squaring it. Now it's pretty clear that what we're being asked to do

here

is square the item that's right next to the power of two

The brackets are indicating that it's the number -3.

So in that case we take -3

multiply it by itself. Two minuses

multiplied together make a plus so we end up with

positive 9. Now compare that to

the statement you see here "-3 squared".

It is tempting to think that the minus is attached to the 3

and therefore we are still squaring the number -3

but the correct way to read this

is to note that it's only the 3

that's right next to the power of 2 and therefor that's the only part

to get squared. So dealing with that we have a minus sign

out the front of 3 multiplied by 3

which will, of course, produce -9. Now what's important

in correcting any mathematical error

is having some kind of strategy, using your other knowledge of mathematics,

to get around a potential problem

when you run into it. Now, in this case what I think it would help

understand how to apply the power correctly to a negative number

is to think of -3 as not just -3,

a point on the number line, but think of it as a product:

-1 multiplied by +3, because we know

that the second item there (-1 times 3)

is going to produce the number -3. So in this case if you replace the "-3"

with "-1 times 3" then it's not so

easy to make the mistake of squaring the -1

because now it looks like the -1 is separate from the 3,

the 3 squared and that's the right way to interpret it.

So now it's pretty clear that it's only the 3 that's getting squared

and we'll get the correct answer. So what we doing here is using

some other knowledge we have about the way numerical manipulation works

to help us overcome a potential problem we might be having

interpreting some mathematical notation, because of course "-3"

and "-1 multiplied by 3" a mathematically the same thing.

So they should obey the appropriate rules.

Here's a little exercise,

four little questions. What you might like to do at this point is

pause the video, have a bit of a think about which one of those

four statements is correct. It might be all of them, it might be none of them.

Have a bit of a think and then restart the video to

see whether you're right.

Here's a second set

of problems that are similar to the previous set. Note this time that we're

raising all these numbers to the power of 3. So feel free to

pause the video and have a think about those ones as well.

And here are the correct

answers.

Item number two out of our

baker's dozen is what happens when you have a power of 0.

Quite often

I see people interpreting any number, a,

raised to the power 0 as equalling 0.

Now this isn't true but it's a natural

mistake to make. When we first learn about powers we learned about

whole number powers like 2 (squaring) or 3

(raising to the power 3, or cubing). Whole number powers

are pretty straightforward but when we start dealing with

powers of 1, 0, negative powers, fractional powers

we have to trade a little more carefully, because their interpretation is a bit

more sophisticated.

For this one a good strategy

is to think about index laws that are more familiar to us.

You might be familiar with the index law

that says a raised to the power m divided by the same base, a,

raised to another power n, is simply

the base a raised to the power of m - n,

the difference between the numerator and denominator powers.

Even if you aren't particularly familiar with that formula,

or that shortcut if you like, a good idea

is to see what's actually going on

underneath the shortcut. So here we have a simple example:

6 raised to the power 5 divided by 6 raised to the power 2.

Clearly what that means is I've got

five 6's on the top line of the fraction multiplied together and

two on the bottom. Now if we saw it like that we might naturally

realize that there are some cancellations possible, that is

two the 6's on the top line cancel two on the bottom.

What that leaves us with is the remaining three 6's on the top line,

or 6 to the power 3. And that's a good way to

remind yourself that the shortcut to get from the beginning of that

series of calculations to the end is simply

to take the difference in the powers and that formula you see above is a

shortcut

that goes through that process for you. Let's see what happens if we

change the problem to a situation where the powers are both the same.

S here I've got 6 to the power 2 over itself.

Now, it's not all that difficult

to see that this fraction

clearly must be 1, either through cancellation or simply by recognizing

that when you take a number and divide by itself you must get 1.

The beauty the index law above

is that it also applies in this situation. So here I'm taking 6 to the power of 2 - 2,

which is, of course,

6 to the power 0 and here we see a practical example

of what a number raised to the power 0 will produce.

Because of the consistency of all the mathematical rules we're using,

6 to the 0 has to equal 1.

So the correct interpretation

of a 0 power is to replace it

with 1. If you have trouble remembering that

you can always go back to simpler rules of mathematics

and derive the logically consistent result.

The third item in our first part of the baker's dozen

is what happens when you divide by 0. Most people have an idea

that something interesting happens when you attempt to divide by 0

but we're not terribly clear about what exactly the implications are.

Now, you've probably seen stated

the division by zero is a process that is

"not defined", which is a tricky concept in some ways.

Most mathematical operations produce something, some kind of number

or formula or expression. Not to be confused, of course,

with taking 0 and dividing it by other numbers.

So "0 divided by 3", which is equivalent to the fraction

"0 over 3" is a perfectly respectable number.

0 is a number, it sits on the number line. It's a defined quantity.

The trouble starts in this situation:

when you attempt to divide any other number by 0.

And the most you can say

about those two statements is a word

"undefined". It's a rare occasion where a mathematical operation

doesn't produce a mathematical answer. Now most of us are aware of

those

restrictions. We're not always aware of

why it's necessary to declare that division by zero

isn't defined, simply not allowed, and

a very good example that convinced me was an algebraic proof.

So here we have a simple

statement, "variable a equals

variable b" and I'm now going to manipulate that

equation using some operations. So firstly I'm

going to multiply both sides by a. So now I have that "a squared equals a times b".

I have also

subtracted b squared from both sides

which produces the equation you see here.

Now both sides of this equation can

be factorised. On the right hand side I have a common factor of b

and on the left hand side I've got the difference two squares.

So factorising both of those produces

these expressions here. I've taken the common factor of b out of the right hand side

and also used the standard method for factorising a difference of two squares.

Now you'll notice that a - b is a common factor

of the left and the right hand side, so I'll divide

both sides by a - b to make the expression

on both sides simpler. Now at the top we stated that a and b

were the same, so I'm its going to replace a with b

and I'll simplify a little bit so that 2b = b,

then I'll divide by b and

I've just shown that 2 is equal to 1. That's bad news.

Mathematically, one of the bedrock principles of all

mathematics is that different numbers

are different. 2 does not equal 1.

And we've performed a series of algebraic steps

that appear to have proved just that.

So the fabric of mathematics is in jeopardy into

we work out what's gone wrong. In fact one of the steps I took in that

proof

was illegal, incorrect. Have a bit of a think for a moment

about which step may be the culprit.

Well, the offending step in this process is that one,

the division by a - b. You've probably done

division by expressions like that several times already in your mathematical

career.

Why is this one wrong? Well the answer is because

we've deliberately stated that a and b are the same.

Therefore, dividing by a - b is inevitably

dividing by 0. Since that step

is not defined then none of the following steps

are allowed. The process is incorrect from that point onwards.

So you get the idea now

that division by 0 has to be not only

declared to be illegal

it must never happen at all and here's a visual

representation what that means

in case you having trouble with the algebra. You allow division by zero to

exist,

you let that slip past you G then a

Kraken will rise from the ocean and drag your ship of mathematics to the bottom

of the ocean.

Now that was a fairly contrived example we just saw.

Here's a more practical problem where division by 0 may turn out to give us

a hard time.

We're asked to solve this equation for x:

"x times (2 - x) equals x squared".

What is the solution or solutions (if any) to that problem?

Well a good way to tackle a problem like this

is to recognize that there's a common factor of x in the left and the right hand side.

So we may as well divide both sides by x to make our job easier.

Now we have that "2 - x equals x".

Rearranging that, putting the x's together,

eventually produces the answer

"x equals 1". So

we might be inclined to claim that the solution to this problem

is that x equals 1 and that certainly is a perfectly satisfactory solution to the

problem.

Substituting x = 1 into that equation at the top

will give you a left and right hand side that are both the same.

But there is a problem with this process.

We divided both sides at the equation

by a variable, x. Remember

a variable x can take any value at all along a number line

which includes 0. So what you should get into the habit of doing when

you're working on a problem like this and you do divide by

a variable or an expression involving variables,

is make sure that you specify

that that expression or variable can't equal 0.

So, in this case, that division of both sides by x is only valid

if we restrict x to values other than 0.

What that means is that so far in our solution to the problem

we have considered possible values of x

from minus infinity to infinity but excluding 0,

which means we haven't tested whether 0 itself