# Multiplying complex numbers

1. Nov 22, 2008

### alpha01

i have just started on complex numbers today and have read that the "algebraic rules for complex are the same ordinary rules for real numbers"..

when multiplying 2 complex numbers (z1 and z2) i can see easily that:

(x1+y1i)(x2+y2i) = x1x2 + y1x2i + x1y2i +y1iy2i

however im struggling to understand the final answer of z1z2 = (x1x2 - y1y2) + (y1x2 + x1y2)i

This does not appear to be consistent with "normal" algebraic rules.

2. Nov 22, 2008

### rock.freak667

$$(x_1+iy_1)(x_2+iy_2)=x_1y_1+iy_1x_2+ix_1y_2+i^2x_2y_2$$

recall that i2=-1

$$=(x_1y_1-x_2y_2)+i(x_2y_1+x_1y_2)$$

EDIT: I expanded incorrectly, but that wasn't the point....LaTex that looks like a picture confuses me apparently :S

Last edited: Nov 23, 2008
3. Nov 22, 2008

### Mentallic

Hi alpha01! I too have just started learning complex numbers. Extension 2 mathematics in Australia, year 12 highschool. How about you?

Anyway to the point:
so we have $$z_1z_2=(x_1+iy_1)(x_2+iy_2)$$
Expanding we get: $$x_1x_2+ix_1y_2+iy_1x_2+i^2y_1y_2$$
But remember that the definition of i is that $$i=\sqrt{-1}$$
so this means $$i^2=-1$$
now the expanded form can be simplified:
$$x_1x_2+ix_1y_2+iy_1x_2-y_1y_2$$
and we simply collect all real and unreal terms. Factorise i out of the unreal terms so that we can express it in the form a+ib
i.e. $$(x_1x_2-y_1y_2)+i(x_1y_2+y_1x_2)$$

therefore $$a=x_1x_2-y_1y_2$$ and $$b=x_1y_2+y_1x_2$$

4. Nov 22, 2008

### alpha01

i was aware of the definition i=sqrt(-1) but didnt notice it in there. thanks for pointing that out.

i am also in aus, i am doin undergrad degree in applied finance at macquarie, its for math130 which is roughly equivalent to 3 unit hsc math.

5. Nov 23, 2008

### Mentallic

Applied finance and you are dealing with complex numbers? I never thought there were applications for this topic.

rock.freak667 you have made an error in your expanding. But nonetheless I think the OP has got the point

6. Nov 23, 2008

### LogicalTime

there are lots of applications, one of the biggest is the solution to the harmonic oscillator differential equation

complex numbers really useful in differential equations which are used to describe everything- newtons laws, Maxwell equations, Schrodinger equation... etc

7. Nov 23, 2008

### Mentallic

Who would've ever known that the imagination can have physical applications.

8. Nov 23, 2008

### LogicalTime

9. Nov 24, 2008

### HallsofIvy

Staff Emeritus
People with imagination? :tongue2:

10. Nov 24, 2008

### LogicalTime

is imagination imaginary? if so, does that mean it doesn't exist?

mmm semantic ambiguity

11. Nov 24, 2008

### Mentallic

No no no. While I said that, I didn't mean imagination in general. Of course, imagination is the backbone of invention.

I have only been exposed to complex numbers through quadratics. When looking at a quadratic that does not touch the x-axis whatsoever, but has imaginary roots. Well this just seems silly to me and can never have a real world use
But that frizzy-haired man would probably tell me otherwise.

12. Nov 24, 2008

### slider142

Interestingly enough, quadratics were not the reason for the acceptance of "imaginary numbers" as viable mathematical objects of study. It was only due to their appearance and utility in solving higher degree polynomials where purely real methods were rather contrived that their study expanded, and their analysis gave us incomparable tools for modern physics, electrical engineering, and solutions of differential equations, which led to the study of topology and differential geometry.