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In this video I'm going to try[br]to motivate the study of complex

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numbers by explaining how we can[br]find the square root of a

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negative number. Before we do[br]that, let's record some facts

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about real numbers.

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On the diagram here, I've drawn[br]what we call a real number line.

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And every real number has its[br]place on this line. Now I've

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marked the whole real numbers[br]from minus nine up to plus nine,

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so all the positive numbers to[br]the right hand side. The

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negative numbers are to the left[br]hand side. Every real number has

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its place on this line, so the[br]integers, positive integers,

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negative inches, integers are[br]here. We could also put the

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fractions on as well. So for[br]example the real number minus

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1/2 would lie somewhere in here.

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Decimal numbers, like 3.5 would[br]be somewhere in there.

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And even numbers like pie with[br]some, some place someone here as

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well. So as pie is going to be[br]in there somewhere. So the point

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is that all real numbers have[br]their place on this real number

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line. Let's look at what happens[br]when we square any real number.

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Suppose we take the number 3 and[br]we square it.

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When we square 3, remember we're[br]multiplying it by itself, so 3 *

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3 the answer is 9.

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What about if we take the number[br]minus three and square that?

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Again, when we square it, we[br]multiplying the number by

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itself, so it's minus 3[br]multiplied by minus three.

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And here, if you recall that[br]multiplying a negative number by

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a negative number yields a[br]positive result, the answer

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minus three times minus three is[br]plus 9 positive 9.

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Now the point I'm trying to make[br]is, whenever you Square a

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number, be it a positive number[br]or a negative number. The answer

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is never negative.

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In fact, unless the answer,[br]unless the number we started

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with zero, the answer is always[br]going to be positive. You can't

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get a negative answer by[br]squaring a real number.

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Now, over the years,[br]mathematicians found this

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shortcoming a problem and they[br]decided that will try and work

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around that by introducing a new[br]number, and we're going to give

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this new number at the symbol I,[br]and I'm going to be a special

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number that has this property[br]that when you square it, the

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answer is minus one.

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So I is a special number such[br]that the square of I I squared

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is minus one. Now that clearly[br]is a very special number because

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I've just explained that when[br]you square any positive or

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negative real number, the answer[br]can never be negative. So

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clearly this number I can't be a

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real number. What it is, it's an[br]imaginary number. We say I is an

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imaginary number. Now that might[br]seem rather strange. When you

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first meet, it's starting to[br]deal with imaginary numbers, but

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it turns out that when we[br]progress a little further and we

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do some calculations with this[br]imaginary number, I lots of

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problems in engineering and[br]physics and applied mathematics

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can be solved using this[br]imaginary number I.

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Now using I, we can formally[br]write down the square root of

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any negative number at all. So[br]supposing we want to write down

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an expression for the square[br]root of minus nine square root

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of a negative number.

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What we do is we write the minus[br]nine in the following way. We

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write it as plus 9 multiplied by

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minus one. And then we split[br]this product as follows. We

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split it as the square root of

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9. Multiplied by the square root[br]of minus one.

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So the square root of 9. We do[br]know the square root of 9 is 3

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and the square root of minus one[br]is going to be I because I

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squared is minus one. So I is[br]the square root of minus one.

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The words now we can formally[br]write down the square root of

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minus nine is 3 times I.

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Let's give you another example.[br]Suppose we wanted the square

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root. Of minus Seven, we do it[br]in exactly the same way we split

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minus 7 into 7 times minus one,[br]and we write it as the square

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root of 7 multiplied by the[br]square root of minus one.

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Now the square root of 7. We[br]can't simplify, will just leave

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that in this so called surd form[br]square root of 7 and the square

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root of minus one.

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We know is I.

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So the square root of minus[br]Seven. We can write as the

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square root of plus Seven times.[br]I, so the introduction of this

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imaginary number I allows us to[br]formally write down an

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expression for the square root[br]of any negative number.

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Now using this imaginary number[br]I, we can do various algebraic

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calculations, just as we would[br]with normal algebra. Let me show

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you a couple of examples.[br]Supposing were asked to

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calculate I cubed.

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Or I cubed we can write as I[br]squared multiplied by I.

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We already know that I squared[br]is minus one, so I squared

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becomes minus 1 multiplied by I.

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Which is just minus one times I[br]or minus I so we can simplify

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the expression I cubed in this[br]way just to get the answer.

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Minus I let's look at another[br]one, supposing we have an

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expression I to the four, we[br]could write that as I squared

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multiplied by I squared.

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And in each case I squared is[br]minus one. So here we have minus

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1 multiplied by minus one and[br]minus one times minus one is

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plus one. So I to the four is[br]plus one and in the same way we

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can start to simplify any[br]expression that involves I and

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powers of I. You'll see[br]this imaginary number I

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in lots more calculations[br]in the following videos.