How Do Complex Number Multiplication Rules Differ from Real Number Algebra?

[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 I am struggling to understand the final answer of z1z2 = (x1x2 - y1y2) + (y1x2 + x1y2)i

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

 

[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

 

[Mentallic]

Hi alpha01! I too have just started learning complex numbers. Extension 2 mathematics in Australia, year 12 high school. 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 $$b=x_1y_2+y_1x_2$$

 

[alpha01]

Hi alpha01! I too have just started learning complex numbers. Extension 2 mathematics in Australia, year 12 high school. 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 $$b=x_1y_2+y_1x_2$$

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 doing undergrad degree in applied finance at macquarie, its for math130 which is roughly equivalent to 3 unit hsc math.

 

[Mentallic]

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 doing undergrad degree in applied finance at macquarie, its for math130 which is roughly equivalent to 3 unit hsc math.

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

 

[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, Schrödinger equation... etc

 

[Mentallic]

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

 

[LogicalTime]

 

[HallsofIvy]

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

People with imagination?

 

[LogicalTime]

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

mmm semantic ambiguity

 

[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.

 

[slider142]

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.

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.