02 - Algebra and geometry in the complex plane
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welcome to the second lecture in our
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courts analysis of a complex kind today
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we'll be talking about algebra and
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geometry in the complex plane we'll
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learn what the complex numbers are how
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to add them how to multiply them how to
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subtract and divide them and some of
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their properties complex numbers is
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simply expressions of the form Z equals
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X plus I Y where x and y are simply real
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numbers for example Z equals 3 plus I
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times 5 would be a complex number often
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that is just written as 3 plus 5i so it
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doesn't really matter if the eye comes
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before or after X this first number
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right here is called the real part
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because there's no I attached to it
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where is y the second number with the I
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in front of it is called the imaginary
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part or insured X's re-of see the real
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part of Z and Y is IMF see the imaginary
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part of Z the set of all complex numbers
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is denoted by a C for a complex plane
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this is like a C with a double bar in it
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so regular C with an extra bar in it
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that's the mathematical symbol for the
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complex claim the real numbers are
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considered a subset of the complex
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numbers those for which that second
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parameter that Y is simply equal to zero
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so a number of the form Z equals x plus
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0i it's considered a real number and so
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the real numbers were kind of subsets of
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the complex numbers are Rahila numbers
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three six seven they're all included in
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the complex numbers the complex plane
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can be identified with r2 and here's how
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that goes we often draw an actual plane
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so like a coordinate system and we use
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the x axis for the x coordinate which is
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the real part of z and we use the y axis
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for the imaginary part of Si and with
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that convention we can then draw a
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complex number Z which is of the form X
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plus iy where the coordinates are here X
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the x-coordinate the real part of Z and
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the y coordinate is the imaginary part
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of Si and thereby the real numbers are
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actually sitting on that x axis and
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numbers on the y axis are called purely
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imaginary numbers now how do we add
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complex numbers suppose we have a number
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Z which is of the form X plus iy and we
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have a second number W of the form u
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plus IB how do we add them so here we're
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defining the addition of complex numbers
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we're defining them by simply adding the
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two numbers that don't have any eyes
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attached to them so the accent you add
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those two numbers are those two real
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numbers and then we add the Y in the V
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and put the eye imprint and so our new
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complex number our resulting complex
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number has a real part which is the sum
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of the two real parts and has an
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imaginary part which is the sum of the
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two imaginary parts in other words the
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real part of Z plus W is simply the real
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part of Z plus the view point of W and
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the imaginary part of Z plus W is simply
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the imaginary part of Z plus the
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imaginary part of W let me give you an
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example suppose we're adding the numbers
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3 plus 5i plus let's say minus 1 plus 2i
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the way to do that would be simply to
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add the 3 the negative 1 3 plus negative
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1 equals 2 in that add the 5 in this
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shoe and that's a 7 pi graphically the
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addition of two complex numbers could be
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represented as a vector addition so
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let's go back to our coordinate system
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suppose we have a vector Z right here
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I'll draw it as a vector even though
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it's really a point in the complex plane
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and suppose we have a second vector W
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which would be right here
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so that's u plus IV
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if we wanted to add these two vectors
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all we would have to do is we move the
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beginning of Z to the end of W and sort
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of complete this parallelogram that you
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can see right here and the vector and
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then nibble the diagonal vector that
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corresponds to the vector Z plus W so
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this point right here is Z plus W and
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what we've done there we have indeed
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added the X of Z and the U of W so from
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here to here that distance is exactly
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the same as from here to here which
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corresponds to the length of U and on
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the other hand the same is true on the y
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axis so addition of complex numbers
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corresponds to a factor addition the
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modulus of a complex number is the
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distance of that number from the origin
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or the length of the vector from the
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origin to the point Z so again if I draw
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my complex plane and suppose I have a
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point Z equals X plus iy then this is
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the x-coordinate and here's the
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y-coordinate and suppose I'm asking how
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far is the from the origin so I'm asking
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for this length right there how long is
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that and I'm going to call that the
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modulus of
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at length well I can see there's a right
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triangle right here and in order to find
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the length of the hypotenuse in that
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right triangle all I need to do is I
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square the lengths of the two legs but
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how long are those legs this like right
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here has length x squared well that's
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the square of this leg this leg has
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length Y so Y squared and together that
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gives me the length of the hypotenuse
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squared so Z squared is x squared plus y
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squared or in other words Z itself is
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the square root of x squared plus y
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squared now how do we multiply complex
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numbers here's the motivation let's
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pretend for a second that they behave
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just like normal numbers that we know
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from elementary school so suppose I have
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a number X plus iy and I wanted to
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multiply that with u plus IV and suppose
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all the regular rules help so I can just
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distribute this out the way I'm used to
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then I would get x times u I'm
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multiplying this number with this number
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plus I XV from here and then I get I why
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u in an I squared Y D so we'll get
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exactly these four terms now if I indeed
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has the property that I squared is
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negative one then this should really
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just be a minus one here now this is a
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motivation so this is the thought that
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goes into this following definition if
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this is supposed to be true then the
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part of this number that is real will
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consist of this X u because there's no I
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in there and this number right here
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because I squared is minus 1 so minus YV
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is also a real number so the real part
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of the product of these two complex
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numbers should then meet Xu minus YV and
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the imaginary part would be this
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together with this so XV + y u and
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indeed that's how we define our
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multiplication
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so let's look at an example suppose we
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wanted to multiply 3 plus 4i by negative
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1 plus 7i so what I have to do according
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to this formula I have to take the X and
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multiplied with a hue that gives me a
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minus 3 then I have to take the Y just
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the 4 multiply it by V and put a
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negative sign in front of the product so
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4 times 7 is 28 - 28 then I'm going to
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get the complex part by taking X and
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multiplying it with V so 3 times 7
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that's this 21 and finally I take Y
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times u so this is y and U is here so 4
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times negative 1 is negative 4 then I
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simplify -3 minus 28 is minus 31 and 21
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minus 4 is 17 so the result of this
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product is again a complex number and at
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this negative 31 plus 17 I now one can
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check that with this definition the
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usual properties of multiplication still
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hold even though we're now multiplying
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complex numbers and not just real
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numbers and those are for example at Z 1
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times Z 2 times Z 3 if you want to
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multiply three numbers it doesn't matter
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if you multiply the first in the second
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first and then take the result and
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multiply it by the third or you multiply
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the second and the third and then the
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first one times that be solved this is
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called associativity you can move these
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parentheses around and decide which
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multiplication gets executed first
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multiplication is also commutative which
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means it doesn't matter if I write this
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factor first and then this factor or if
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I write this the other way around
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so in fact this would be the same thing
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as negative 1 plus 7i times 3 plus 4 you
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should check this out
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get out a piece of paper and apply this
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formula to this product and verify that
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indeed you do get the same result
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negative 31 plus 79 finally our
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multiplication is distributive so if I
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have three complex numbers and I have to
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multiply z1 by the sum of t2 and t3 I
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could also multiply this out and
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multiply z1 and z2 and multiplies you
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want to see 3 and add them up that's
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called the distributive law so all these
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laws still hope now what exactly is ah
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that number five so far it's kind of
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just a symbol that's been sitting there
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while I is certainly the same as zero
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plus one times I and so I can now
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calculate I squared because I is 0 plus
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1 times I and the other is also 0 plus 1
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times I and I can use the rule of
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multiplication that I just defined to
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find out what this is so according to my
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rule of multiplication I have to take
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this 0 and that 0 and multiply them and
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that gives me this number right here
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then I have to take this one in that one
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multiply them and put a negative sign in
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front of it that's what I get right here
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then I have to take this 0 and that one
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and multiply them and finally I have to
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take this one and that 0 then multiply
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them in these two form the imaginary
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part together so 0 minus 1 is negative 1
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and all the rest is 0 so I squared is
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indeed negative 1 so the multiplication
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we define has the property that we
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wanted it to have I squared equals
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negative 1 so what is I cubed
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well I'm cubed is I squared times I by
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the associativity we just realized it
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doesn't matter which one I multiply
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first I squared is negative 1 so I cubed
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is negative 1 times I so it's negative I
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I to the fourth well I could write that
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as I squared times I squared each of
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these I Squared's as we just showed is
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negative 1 so that's negative 1 times
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negative 1 and they get a 1 what side of
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the fifth well that's I to the fourth
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times I I to the fourth we just showed
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us 1 so it's 1 times
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which is to say either the six well that
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is I to the fifth times I I to the fifth
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of us I I squared is negative one and it
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keeps going so what if we noticed I is I
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I squared is negative one I keep this -
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I at the fourth is one and then these
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four numbers keep repeating so I I
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squared is negative one I cubed is
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negative I I to the fourth is 1 I to the
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fifth this high I to the six is negative
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one I to the seventh is negative I I to
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the 8th this one and it keeps repeating
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like that so there's really just four
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different states that a power of I can
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have so how do you divide complex
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numbers suppose we are given a Z X plus
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iy and then W u plus IV so do complex
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numbers how do I divide Z by W and of
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course I'm going to need W to be nonzero
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I can't divide by 0 so here it's a great
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trick and I'll show that to you with
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actual number so Z over W would be X
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plus iy over u plus IV let's do this
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whole calculation with actual numbers so
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suppose Z is 3 plus 4i
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and suppose W is negative 1 plus 7i and
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I wanted to calculate this quotient so I
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wanted to calculate 3 plus 4i and divide
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that by negative 1 plus 7i how do I do
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that what the idea is to multiply top
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and bottom by this number u minus IB so
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almost the same as the bottom but I'm
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multiplying it by the bottom number with
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a plasterer placed by a minus so in our
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example here I would have still my top 3
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less for AI divided by negative 1 plus
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7i but now I would multiply top and
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bottom by negative 1 minus 7i so it's
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almost the same at the bottom but I'm
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replacing this plus here with a negative
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sign now when I multiply this out on top
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I have a regular multiplication of two
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complex numbers so I'm going to get here
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3 times negative 1 is negative 3 then 4
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times negative 7 is negative 28 but I'm
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going to have to make it a plus because
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that's how the multiplication goes and
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then plus I times and I need to multiply
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the 3 by the negative 7 so that's
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negative 21 and the 4 by the negative 1
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so minus 4 and what happens in the
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denominator I have au squared plus V
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squared so I'm just going to multiply
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this through here Megan if one times
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negative 1 is 1 and I have a 7 times a
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negative 7 that's negative 49 but I'm
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going to make that a plus a plus 49 and
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then I catch plus I times negative 1
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times negative 7 that is 7 and 7 times
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negative 1 is negative 7 and now see
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what happened so on top I have in this
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example here 28 minus 3 that's 25 plus I
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times and here I have minus 21 minus 4
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is minus 25 in my denominator I have a
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50 and the imaginary part went away and
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so in other words this simplifies now to
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a complex number I have no more eyes in
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the denominator and so I can simplify
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this 25/50 simply 1/2 plus
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i times and here I have a minus 1/2 it
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could have pulled the minus sign out and
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put it right here but this looks exactly
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like our complex numbers slope in other
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words when I divide complex numbers the
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result is again a complex number in the
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trick we used was to make the
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denominator reveal and how do we do that
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we make the denominator real by
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multiplying the denominator u plus IV by
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u minus IB we also multiply the
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numerator by that so that we would have
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equality so we just multiply top and
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bottom by u minus IB but the treat was
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that that made the imaginary part cancel
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out and so the denominator became purely
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real so that we could then separate real
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and imaginary part of the result so here
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you step result is a formula for the
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quotient of two complex numbers in
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particular if we just wanted to
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calculate 1 over Z so not Z over W is
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just 1 over a complex number it's 1 over
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X plus iy and again we would do the same
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thing we would multiply top and bottom
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by X minus iy and if we do that in the
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bottom that imaginary parts would cancel
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out and we would just end up with x
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squared plus y squared as long as of
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course Z is not 0 so 0 plus 0 I would be
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back couldn't be divided by that but
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this is 1 over C so this quantity X
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minus iy where we just replace the plus
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with a minus that was really important
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in the previous calculation and it's so
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important that we give it a special name
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so if Z equals X plus iy is a complex
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number then we call that complex number
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will be replaced the plus with a minus
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the complex conjugate of Z and it's
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denoted with a Z with a bar on top of it
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you see that here since u bar is the
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complex conjugate of C so let's look at
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some of its properties
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if you conjugate twice you end up with
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the original number so if you take the
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conjugate here which is X minus I Y and
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you take another conjugate of that all
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you do is you flip the sign in front of
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the amount
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Neri part you're going to end up with X
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minus minus iy in minus and minus plus
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and it's X plus iy and we're back to the
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origin where we started with so taking
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the conjugate twice gets you back to you
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started what does taking the conjugate
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mean graphically let's go back to our
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coordinate system and then suppose I
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have a complex number Z right here which
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is X plus iy then where is the complex
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number X minus iy well here's X and
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there's y well now I'm going to retain
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the X for the complex conjugate but I'm
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making the Y negative I'm taking minus y
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instead of Y and so in other words I'm
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going to get this number right there for
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the complex conjugate X minus iy so the
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complex conjugate is the number Z but
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it's reflected with respect to the x
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axis now here's a property of complex
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numbers if I take the complex conjugate
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of a sum of two complex numbers I might
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just as well take the conjugate of Z
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first and then the conjugate of W and
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then add those two numbers up it's a
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property one easily verifies the
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absolute value of the conjugate is the
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same as the absolute value of Z well you
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can kind of see that in my picture right
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here but this length right there is the
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length of C and down here is the length
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of the conjugate of C and because this
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is just a reflection of that
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these of course have the same length law
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to see in my picture but because it's a
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reflection they have the same length now
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here's a really neat property Z times
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its conjugate that's what we were making
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used off in our calculation of the
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quotient this X plus iy times X minus iy
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when I multiply 3 I get x times X which
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is x squared I get Y times minus 1 which
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is minus y squared what
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together with a negative sign that I get
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from the I squared so plus y squared and
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I get x times minus I Y the next x plus
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iy that cancels out so I get x squared
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plus y squared which is their length of
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Z squared so Z times its conjugate is
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actually the length of Z squared and
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that means we can rewrite the formula
22:35
for 1 over Z 1 over Z now becomes a
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multiply top and bottom by Z conjugates
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so it's G times Z conjugate and in the
22:43
numerator epsy conjugate and the
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denominator simplifies to length of Z
22:48
squared so 1 over Z is the conjugate of
22:51
Z divided by the length of these squared
22:53
now what is it the case that the
22:56
conjugate of a number is equal to the
22:58
number itself we learned that the
23:00
conjugate is simply a reflection with
23:03
respect to the x axis so when is the
23:06
reflection with respect to the x axis
23:08
the same as the number itself so when is
23:11
this number equal to its own reflection
23:15
only if the number was on the real axis
23:17
to begin with so this is the case if Z
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is a real number
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now here's another interesting thing to
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note if you add a number in it's complex
23:34
conjugate what you end up doing this X
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plus iy plus X minus iy so you add the X
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and the X you get 2x and then you add
23:42
the complex parts the iy and the minus
23:45
iy they cancel each other out so you get
23:47
something real in other words if I
23:50
divide this equation by 2 I get the X
23:52
which is the real part of Z is equal to
23:55
Z plus it's conjugate over 2 so we find
23:58
a nice formula for finding the real part
24:01
of a complex number we add the number to
24:05
its conjugate and similarly if you take
24:07
a complex number and subtract the
24:09
conjugate then you get 2 I times the
24:12
imaginary part so if you want the
24:14
imaginary part by itself without the I
24:16
you're going to have to divide by 2 I so
24:20
this is also a nice formula
24:22
now you can verify that the length of Z
24:26
times W is the length of C times length
24:28
of W so length and multiplication behave
24:31
well with respect to each other
24:34
you can also verify that the conjugate
24:38
of Z over W is the conjugate of Z over
24:41
the conjugate of W and finally that the
24:45
length of Z is 0 if and only if Z was
24:49
here and now this is pretty easily
24:50
verified because the length of Z
24:53
remember let's look at the square of the
24:56
length that's just x squared plus y
24:59
squared and x squared plus y squared
25:03
when is that equal to 0 these are two
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real numbers when cannot be 0 they both
25:07
have to be 0 so this is 0 if and only if
25:12
X is equal to 0 and Y is equal to 0 and
25:19
that means if and only if the number Z
25:23
itself
25:24
what's the number 0 now we have some
25:27
inequalities that are really helpful one
25:30
of them is pretty obvious by looking at
25:32
a picture let me draw the complex plane
25:35
right here and there's a complex number
25:41
Z right there then this is the real part
25:48
of Z this length right there it's the
25:51
real part of Z whereas this length right
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here is the norm or the length of Z and
25:58
this length right there
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is the imaginary part of Z so this is
26:03
the real part of Z and that's the
26:04
imaginary part of Z and since they form
26:06
a triangle I see that this length here
26:10
is certainly greater than or equal to
26:14
either one of those two legs and that's
26:17
what expressed here the real part of Z
26:19
is less than or equal to the absolute
26:21
value of Z and the same is true for the
26:24
imaginary part and because that's true
26:27
in absolute value we get the other
26:29
inequality easily as well in inequality
26:32
that's also very nice as the triangle in
26:35
the quad
26:35
Adie the length of ZD plus W is bounded
26:38
above by the length of C plus the length
26:41
of W let me show you in a picture why
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that is the case so let's add two
26:48
complex numbers so you can W so remember
26:51
how we do that we do a vector addition
26:54
so here's Z and suppose we wanted to add
26:58
to that W then what we have to do we
27:02
have to complete this parallelogram and
27:06
this is our point Z plus W and it can be
27:12
represented by this diagonal this is the
27:16
length of Z plus W and if I look over
27:21
here
27:22
this length here that was the length of
27:24
Z and here I see the length of W because
27:28
it's the same vector as this one down
27:30
there and again these form a triangle my
27:33
my triangle didn't really meet very
27:34
nicely over here these three lengths
27:37
form a triangle and the shortest path
27:41
from the origin to is V plus W is
27:43
certainly this diagonal and a slightly
27:46
longer path is to go along Z first and
27:48
then a long w so Z plus and W is greater
27:52
than or equal to the direct path Z plus
27:55
W that's what the triangle inequality
27:57
says there's a reverse triangle
28:00
inequality that follows from the
28:01
triangle inequality
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it's the minus W is greater than or
28:05
equal to Z minus W we'll go into much
28:09
detail there finally a little outlook
28:12
into the future this is the fundamental
28:15
theorem of algebra you may have heard of
28:17
it suppose we have complex numbers a 0 a
28:20
1 through a N and this a and this last
28:23
one is a nonzero number and you look at
28:25
this polynomial so for example again if
28:29
you'd like to have an example in mind
28:30
you could look up the polynomial P of Z
28:33
equals 5 Z cubed plus 4z squared minus
28:41
2z plus 14 so in this case you know 3
28:47
will be the largest end
28:48
have here so a3 would be 5 a2 is 4 a 1
28:53
is negative 2 and a 0 is 14 so this is
28:56
called the degree 3 polynomial and this
28:59
theorem says that such a polynomial has
29:02
n roots in this case has 3 roots in C so
29:05
3 these for which P of C can be equal to
29:08
0 and it can be factored as a n times u
29:12
months the first rudiment is the second
29:14
fruit and so forth we'll be able to
29:16
prove this theorem later on in this
29:18
course now in our this was totally not
29:22
true this is a special fact about
29:23
complex numbers for example look at the
29:26
polynomial P of x equals x squared plus
29:28
1 so we have a degree 2 polynomial but
29:32
we're only considering it for real
29:33
numbers and not for complex numbers this
29:36
never ever crosses the x axis so if i
29:39
graph this polynomial x squared plus 1
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is always above the x axis an ever
29:44
crosses the x axis so it's never you go
29:47
to 0 and so we can't factor it either it
29:49
has no real roots but if you looked at
29:52
that as a polynomial in C so Z squared
29:56
plus 1 instead of x squared plus 1 and
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we're allowed to not only plug in real
30:00
numbers for the X but also complex
30:02
numbers then I can find two complex
30:04
numbers namely I and negative I for
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which this term right here is actually
30:09
equal to 0 I squared plus 1 is 0 and
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minus I squared plus 1 is also 0 and so
30:16
it actually factors so this is a theorem
30:18
about complex numbers and we will be
30:21
able to prove this later on using
30:23
complex analysis in this course