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  • Welcome to the first of four videos which will go through

  • 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

  • satisfies the equation. So we'd better do that as well.

  • So we go back to the original equation you see at the top of the screen

  • and we substitute x equals 0 into it.

  • Since it's a single value we don't need to do anything particularly

  • sophisticated

  • other than work out what the left and right hand side are.

  • So quite clearly if you put x equals 0 into the left hand side you get 0.

  • If you do the same thing to the right hand side of that equation, that is

  • take 0 squared, clearly you also get 0.

  • What this means is that x equals 0 is also

  • a solution to this problem, one that was invisible to us

  • until we checked the value of 0

  • separately. The reason we had to check 0 separately

  • was because the algebra we performed earlier

  • was only valid if x did not did not equal 0.

  • So the correct answer to this problem is that

  • x equals 1 or 0 are solutions

  • to this problem.

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Welcome to the first of four videos which will go through

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