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Precalculus
Course: Precalculus > Unit 3
Lesson 7: Graphically multiplying complex numbersMultiplying complex numbers graphically example: -1-i
CCSS.Math:
We can multiply complex numbers graphically on the complex plane. We rotate an amount equal to the argument and scale by the modulus of the complex number by which we're multiplying. Created by Sal Khan.
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Okay I get that when you multiply by a complex number, you are going to rotate by the argument of that complex number, scale the modulus of Z by the modulus of this complex number. But I don't get why? Sal hasn't proven why this works(8 votes)- Statement to prove:
The modulus and argument of the product of any two complex numbers is the product of the moduli of the two complex numbers and the sum of the arguments of the two complex numbers, respectively.
Givens:
Let z₁ = r₁ ⋅ e^(iθ₁), z₂ = r₂ ⋅ e^(iθ₂), and z₃ = r₃ ⋅ e^(iθ₃)
Let z₃ = z₁ ⋅ z₂
The moduli and arguments of z₁, z₂, and z₃ are r₁ and θ₁, r₂ and θ₂, and r₃ and θ₃, respectively.
z₁ and z₂ can be any two complex numbers.
z₃ is the product of z₁ and z₂.
Proof:
z₃ = z₁ ⋅ z₂ ; given
= [r₁ ⋅ e^(iθ₁)] ⋅ [r₂ ⋅ e^(iθ₂)] ; substitution
= r₁ ⋅ e^(iθ₁) ⋅ r₂ ⋅ e^(iθ₂) ; associative property
= r₁ ⋅ r₂ ⋅ e^(iθ₁) ⋅ e^(iθ₂) ; commutative property
= r₁ ⋅ r₂ ⋅ e^(iθ₁ + iθ₂) ; product rule of exponents
z₃ = r₁ ⋅ r₂ ⋅ e^[i(θ₁ + θ₂)] ; distributive property
z₃ = r₃ ⋅ e^(iθ₃) ; given
r₃ = r₁ ⋅ r₂ and θ₃ = θ₁ + θ₂ ; comparison of coefficients and powers
The modulus and argument of z₃ is the product of the moduli of z₁ and z₂ and the sum of the arguments of z₁ and z₂, respectively.
∵ z₃ is the product of z₁ and z₂, and z₁ and z₂ can be any two complex numbers
∴ The modulus and argument of the product of any two complex numbers is the product of the moduli of the two complex numbers and the sum of the arguments of the two complex numbers, respectively.(14 votes)
Could an alternative solution be just to multiply the 2 complex numbers? (0-3i)(-1-i). Graphically seams to be correct(2 votes)
Yep, if you just wanted to find the product you could literally use FOIL and multiply the complex numbers together. Multiplying them graphically can let you see the process in a different light, though, and the argument and modulus way of thinking might make some problems easier.(3 votes)
- Say z=-3i. Does (-3i)*(-1-i) equal (-1-i)*(-3i)? If so, (-1-i)*(-3i) = 3i+3i^2 = 3i-3, which does not equal 3*sqrt(2)*i-3*sqrt(2) (Sal's answer). The difference is the sqrt(2), the modulus of (-1-i). Where did I get it wrong? Is complex number multiplication not commutative? Thanks.(1 vote)
Complex number multiplication is commutative, and (-3i)(-1-i) does equal (-1-i)(-3i).
Sal's answer was not 3√2i-3√2, that number was never onscreen. Sal pointed out that the modulus of -1-i is √2, and the modulus of -3i is 3, so the modulus of the product must be 3√2.(1 vote)
Video transcript
- [Instructor] We are
told, suppose we multiply a complex number Z by -1-i. So this is Z right over here. Which point represents
the product of Z and -1-I? Pause this video and see
if you can figure that out. All right, now let's work
through this together. So the way I think about this is, when you multiply by a complex number, you are going to rotate by the argument of that complex number. And you're going to scale the modulus of Z by the modulus of this complex number. Now, let me just think
about that a little bit. So I'm gonna draw another
complex plane here. And so this is my real axis, this is my imaginary
axis, right over here. And -1-I, so that's -1 and then minus 1i. So it would go right over there. It would be that right over here. And so let's think about two things. Let's think about what its argument is, and let's think about
what it's modulus is. So its argument is going to
be this angle right over here. And you might already recognize that if this has a length of one, if this has a length of one, or another way of thinking about, this has a length of one,
this is a 45, 45, 90 triangle. So this is 45 degrees but then of course you
have this 180 before that. So that's going to be 180 plus
45, is a 225 degree argument. So the argument here is going
to be equal to 225 degrees. So when you multiply by this, you are going to rotate by 225 degrees. So let's see this is going
to be rotating by 180 degrees and then another 45. So if you just rotate it by that, you would end up right over here. Now we also are going
to scale the modulus. And you can see two choices
that scale that modulus. And so we know it's going
to be choice A or choice B because choices C or D
you'd have to rotate more to get over there. And so to think about that, we have to just think about the modulus of -1-i, this point right over here and then just scale up this
modulus by that same amount. Well, the modulus is just the distance from zero in the complex plane. So it's going to be this
distance right over here. And you could use the Pythagorean theorem to know that this squared,
if you call this C, C squared is equal to one
squared plus one squared or C squared is equal to two or C is equal to the square root of two. So that's the modulus right over here. Modulus is equal to square root of two which is approximately, it's
a little bit more than 1.4. So let's just call it approximately 1.4. So not only going to
rotate by 225 degrees, we're going to scale the modulus, the distance from the origin by 1.4. So it looks like it's three units from the origin right over here. If you multiply that by 1.4, three times 1.4 is about
four, or it is exactly 4.2. So 4.2 of these units is
one, two, three, four, a little bit further,
you get right over here to choice B and we're done.