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## Homework Statement

Let z and w be two complex numbers such that zw = 0. Show either z = 0 or w = 0.

*Hint: try working in exponential polar form*

## Homework Equations

z = re

^{iθ}

## The Attempt at a Solution

I have absolutely no clue.

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- Thread starter vesu
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Let z and w be two complex numbers such that zw = 0. Show either z = 0 or w = 0.

z = re

I have absolutely no clue.

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- #3

Hurkyl

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Have you tried that yet? What does the question look like in exponential polar form?Hint: try working in exponential polar form

- #4

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so [itex]zw = re^{i\theta} \times qe^{i\phi}[/itex]

and since neither [itex]re^{i\theta}[/itex] or [itex]qe^{i\phi}[/itex] can be negative, either [itex]z = 0[/itex] or [itex]w = 0[/itex]. Is this correct?

- #5

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so [itex]zw = re^{i\theta} \times qe^{i\phi}[/itex]

and since neither [itex]re^{i\theta}[/itex] or [itex]qe^{i\phi}[/itex] can be negative, either [itex]z = 0[/itex] or [itex]w = 0[/itex]. Is this correct?

Not in the slightest. Does negative mean anything when it comes to complex numbers? Also, when imaginary exponents are involved, [itex]e^{a+b\cdot i}[/itex] can be anything. Basic identities about exponents, one of which should be screaming at you, still apply.

- #6

cepheid

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so [itex]zw = re^{i\theta} \times qe^{i\phi}[/itex]

and since neither [itex]re^{i\theta}[/itex] or [itex]qe^{i\phi}[/itex] can be negative, either [itex]z = 0[/itex] or [itex]w = 0[/itex]. Is this correct?

Correct, except what does them being negative have to do with anything?

Edit: what I mean is, correct answer, totally wrong reasoning.

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Right, whoops. So either z = 0 or w = 0 because of the zero-product property? i.e. "if ab = 0, then either a = 0 or b = 0"? Why is working in exponential polar form at all necessary or even helpful then? :SCorrect, except what does them being negative have to do with anything?

Edit: what I mean is, correct answer, totally wrong reasoning.

[itex]rqe^{i(\theta + \phi)}[/itex] ?Not in the slightest. Does negative mean anything when it comes to complex numbers? Also, when imaginary exponents are involved, [itex]e^{a+b\cdot i}[/itex] can be anything. Basic identities about exponents, one of which should be screaming at you, still apply.

and then since [itex]e^{i(\theta + \phi)} > 0[/itex] either [itex]r=0[/itex] or [itex]q=0[/itex], therefore either [itex]z = 0[/itex] or [itex]w = 0[/itex] ?

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It certainly can be smaller than 0, but can it be exactly 0?[itex]e^{i(\theta + \phi)} > 0[/itex]

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Right, whoops. So either z = 0 or w = 0 because of the zero-product property? i.e. "if ab = 0, then either a = 0 or b = 0"? Why is working in exponential polar form at all necessary or even helpful then? :S

[itex]rqe^{i(\theta + \phi)}[/itex] ?

and then since [itex]e^{i(\theta + \phi)} > 0[/itex] either [itex]r=0[/itex] or [itex]q=0[/itex], therefore either [itex]z = 0[/itex] or [itex]w = 0[/itex] ?

Oh yeah, so it should be [itex]e^{i(\theta + \phi)} \not= 0[/itex] then?It certainly can be smaller than 0, but can it be exactly 0?

I think it's all correct other than that, right?

- #10

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Right, whoops. So either z = 0 or w = 0 because of the zero-product property? i.e. "if ab = 0, then either a = 0 or b = 0"? Why is working in exponential polar form at all necessary or even helpful then? :S

The point is that we already know it works for reals. Let's assume it works for reals. Now prove it works for complex numbers. Relying on what you're trying to prove won't get you anywhere. If you really do need a hint,

Ever heard of the absolute value function and how it works on complex numbers?

- #11

Ray Vickson

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## Homework Statement

Let z and w be two complex numbers such that zw = 0. Show either z = 0 or w = 0.

Hint: try working in exponential polar form

## Homework Equations

z = re^{iθ}

## The Attempt at a Solution

I have absolutely no clue.

You can also do it directly: if z = x + i*y and w = u + i*v with x,y,u,v real, then

zw = (xu - vy) + i*(xv + yu), so we need xu = vy and xv = -uy. You can consider several (easy) cases such as u =/= 0 vs, u = 0, and so conclude that in the first case you *must have* x = y = 0, while in the second case you have either v = 0 (so you are done) or else you have x = y = 0.

RGV

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I'm a little confused, could you explain how this is incorrect?The point is that we already know it works for reals. Let's assume it works for reals. Now prove it works for complex numbers. Relying on what you're trying to prove won't get you anywhere. If you really do need a hint,

Ever heard of the absolute value function and how it works on complex numbers?

[itex]zw = 0[/itex]

[itex]zw = re^{i\theta} \times qe^{i\phi}[/itex]

[itex]zw = rqe^{i(\theta + \phi)}[/itex]

[itex]e^{i(\theta + \phi)} \not= 0[/itex]

[itex]\therefore r=0[/itex] or [itex]q=0[/itex]

[itex]\therefore z = 0[/itex] or [itex]w = 0[/itex]

I guess you could also do it this way to eliminate [itex]i[/itex] altogether so that you're only working with reals? This doesn't involve exponential polar form though so I'm not sure if this is how we're expected to solve it.

[itex]z = a + ib[/itex]

[itex]w = c + id[/itex]

[itex]|zw| = |z||w| = 0[/itex]

[itex]\sqrt{a^2 + b^2} \times \sqrt{c^2 + d^2} = 0[/itex]

[itex]\therefore a[/itex] and [itex]b = 0[/itex]

or [itex]c[/itex] and [itex]d = 0[/itex]

[itex]\therefore z = 0[/itex] or [itex]w = 0[/itex]

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[tex]

w = \frac{1}{z} \cdot 0 = ?

[/tex]

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This has confused me even more. :Szexists. But, then:

[tex]

w = \frac{1}{z} \cdot 0 = ?

[/tex]

also I cringe every time I re-read this:

I was very very tired.so [itex]zw = re^{i\theta} \times qe^{i\phi}[/itex]

and since neither [itex]re^{i\theta}[/itex] or [itex]qe^{i\phi}[/itex] can be negative, either [itex]z = 0[/itex] or [itex]w = 0[/itex]

- #15

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[itex]zw = rqe^{i(\theta + \phi)}[/itex]

[itex]e^{i(\theta + \phi)} \not= 0[/itex]

[itex]\therefore r=0[/itex] or [itex]q=0[/itex]

It's this step. To justify it, we need to rely on the very identity we are trying to prove, resulting in circular reasoning. Hint: find some way to introduce an absolute value into this equation.

- #16

HallsofIvy

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Because this is what you are being asked to prove! That is, you are asked to prove that the "zero product property" holds for complex numbers as well as real numbers.Right, whoops. So either z = 0 or w = 0 because of the zero-product property? i.e. "if ab = 0, then either a = 0 or b = 0"? Why is working in exponential polar form at all necessary or even helpful then? :S

NO! This not true. For example, [itex]e^{i\pi}= -1[/itex]. But, as you were told before, there is no "less than" or "larger than" relation on the complex numbers.[itex]rqe^{i(\theta + \phi)}[/itex] ?

and then since [itex]e^{i(\theta + \phi)} > 0[/itex]

It is, however, true that [itex]e^{ix}[/itex], for x real, is never 0.either [itex]r=0[/itex] or [itex]q=0[/itex], therefore either [itex]z = 0[/itex] or [itex]w = 0[/itex] ?

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Uh... I understand what you're saying about what's wrong with my proof but I really don't know what to do even with the hint.It's this step. To justify it, we need to rely on the very identity we are trying to prove, resulting in circular reasoning. Hint: find some way to introduce an absolute value into this equation.

I mean I know [itex]r = |z|[/itex] and [itex]q = |w|[/itex] so [itex]rq=|zw|[/itex] but I don't know where that gets me.

- #18

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Try taking the absolute value of both sides.

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Ray Vickson

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Uh... I understand what you're saying about what's wrong with my proof but I really don't know what to do even with the hint.

I mean I know [itex]r = |z|[/itex] and [itex]q = |w|[/itex] so [itex]rq=|zw|[/itex] but I don't know where that gets me.

I gave you an alternative suggestion in my previous posting. Have you looked at it? Have you tried to complete the argument I gave?

RGV

- #20

vela

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It's not circular reasoning. It's safe to assume here that for real numbersIt's this step. To justify it, we need to rely on the very identity we are trying to prove, resulting in circular reasoning. Hint: find some way to introduce an absolute value into this equation.

- #21

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It's not circular reasoning. It's safe to assume here that for real numbersxandy, that xy=0 implies that x=0 or y=0, and this is what the proof is relying on asrandqare both real numbers. The OP is being asked to show the same property holds for complex numbers.

Well, the step I was quoting assumes that, for some real numbers r, q, and ##\psi##,

$$r\cdot q\cdot e^{\psi\cdot i}=0$$

and says that

$$r = 0\lor q = 0\lor e^{\psi\cdot i}=0$$

As ##e^{\psi\cdot i}## isn't necessarily real, this step's invalid. There are a few ways, however, to proceed from here. My favourite is taking the absolute value of both sides, but one could also write ##e^{\psi\cdot i}## as ##a+b\cdot i## for real numbers a and b and work from there.

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vela

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- #23

cepheid

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|exp[i(theta + phi)]|

= |1*exp[i(theta + phi)]|

= 1

And therefore the complex exponential part can never be 0, because it must lie on the unit circle in the complex plane?

- #24

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Would it be valid then to reason that

zw = 0

case 1: z≠0

divide by z

∴w = 0

case 2: w≠0

divide by w

∴z = 0.

case 3: w=z=0

0*0 = 0.

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Sorry, I'm not really sure what to do with your suggestion and I'm still trying to figure out if either of the proofs I have is correct!I gave you an alternative suggestion in my previous posting. Have you looked at it? Have you tried to complete the argument I gave?

RGV

Is this proof okay if the other one isn't? (even though this doesn't involve exponential polar form, that was only a hint, not a requirement)

let [itex]z = a + ib[/itex] and [itex]w = c + id[/itex]

[itex]zw = 0[/itex]

[itex]|zw| = 0[/itex]

[itex]|z||w| = 0[/itex]

[itex]\sqrt{a^2 + b^2} \times \sqrt{c^2 + d^2} = 0[/itex]

[itex]\therefore a[/itex] and [itex]b = 0[/itex]

or [itex]c[/itex] and [itex]d = 0[/itex]

[itex]\therefore z = 0[/itex] or [itex]w = 0[/itex]

Thoroughly confused at this point.

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