Check if these statements are true for all complex nubmers
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I have these two statements and I should check whether they are true in $mathbb{C}$:
$frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert}iff(exists k in mathbb{R}^+) vec{0z_1}=kvec{0z_2} $- $arg(z_1)=arg(z_2)ifffrac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert}$
Well, I know these statements are "generally true", but since we are looking at whole complex set, meaning it includes $0$, and argument is not defined for $0$, I am not sure if $iff$ still holds.
complex-numbers
asked Nov 16 at 20:40
Dovla
799
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up vote
1
down vote
favorite
I have these two statements and I should check whether they are true in $mathbb{C}$:
$frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert}iff(exists k in mathbb{R}^+) vec{0z_1}=kvec{0z_2} $- $arg(z_1)=arg(z_2)ifffrac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert}$
Well, I know these statements are "generally true", but since we are looking at whole complex set, meaning it includes $0$, and argument is not defined for $0$, I am not sure if $iff$ still holds.
complex-numbers
asked Nov 16 at 20:40
Dovla
799
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have these two statements and I should check whether they are true in $mathbb{C}$:
$frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert}iff(exists k in mathbb{R}^+) vec{0z_1}=kvec{0z_2} $- $arg(z_1)=arg(z_2)ifffrac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert}$
Well, I know these statements are "generally true", but since we are looking at whole complex set, meaning it includes $0$, and argument is not defined for $0$, I am not sure if $iff$ still holds.
complex-numbers
asked Nov 16 at 20:40
Dovla
799
I have these two statements and I should check whether they are true in $mathbb{C}$:
$frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert}iff(exists k in mathbb{R}^+) vec{0z_1}=kvec{0z_2} $- $arg(z_1)=arg(z_2)ifffrac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert}$
Well, I know these statements are "generally true", but since we are looking at whole complex set, meaning it includes $0$, and argument is not defined for $0$, I am not sure if $iff$ still holds.
complex-numbers
complex-numbers
asked Nov 16 at 20:40
Dovla
799
asked Nov 16 at 20:40
Dovla
799
asked Nov 16 at 20:40
Dovla
799
asked Nov 16 at 20:40
Dovla
799
asked Nov 16 at 20:40
Dovla
799
799
add a comment |
add a comment |
1 Answer
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We need to assume $z_1,z_2neq 0$, otherwise the expression $frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert}$ is not defined.
Then for the first one we need to prove that for $z_1,z_2neq 0$
$$frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert} implies z_1=frac{|z_1|}{lvert z_2rvert}z_2=kz_2 $$
and
$$z_1=kz_2 implies frac{z_1}{lvert z_1rvert}=frac{kz_2}{lvert kz_2rvert} =frac{z_2}{lvert z_2rvert} $$
Can you proceed with the second one?
answered Nov 16 at 20:45
gimusi
89.1k74495
I understand, I was just not sure generally speaking if these statements, as they are(so without assuming $z_1,z_2neq 0$), are true or false
– Dovla
Nov 16 at 21:00
@Dovla They are true for $z_1,z_2neq 0$ otherwise they are false.
– gimusi
Nov 16 at 21:13
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
We need to assume $z_1,z_2neq 0$, otherwise the expression $frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert}$ is not defined.
Then for the first one we need to prove that for $z_1,z_2neq 0$
$$frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert} implies z_1=frac{|z_1|}{lvert z_2rvert}z_2=kz_2 $$
and
$$z_1=kz_2 implies frac{z_1}{lvert z_1rvert}=frac{kz_2}{lvert kz_2rvert} =frac{z_2}{lvert z_2rvert} $$
Can you proceed with the second one?
answered Nov 16 at 20:45
gimusi
89.1k74495
I understand, I was just not sure generally speaking if these statements, as they are(so without assuming $z_1,z_2neq 0$), are true or false
– Dovla
Nov 16 at 21:00
@Dovla They are true for $z_1,z_2neq 0$ otherwise they are false.
– gimusi
Nov 16 at 21:13
add a comment |
up vote
2
down vote
accepted
We need to assume $z_1,z_2neq 0$, otherwise the expression $frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert}$ is not defined.
Then for the first one we need to prove that for $z_1,z_2neq 0$
$$frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert} implies z_1=frac{|z_1|}{lvert z_2rvert}z_2=kz_2 $$
and
$$z_1=kz_2 implies frac{z_1}{lvert z_1rvert}=frac{kz_2}{lvert kz_2rvert} =frac{z_2}{lvert z_2rvert} $$
Can you proceed with the second one?
answered Nov 16 at 20:45
gimusi
89.1k74495
I understand, I was just not sure generally speaking if these statements, as they are(so without assuming $z_1,z_2neq 0$), are true or false
– Dovla
Nov 16 at 21:00
@Dovla They are true for $z_1,z_2neq 0$ otherwise they are false.
– gimusi
Nov 16 at 21:13
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
We need to assume $z_1,z_2neq 0$, otherwise the expression $frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert}$ is not defined.
Then for the first one we need to prove that for $z_1,z_2neq 0$
$$frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert} implies z_1=frac{|z_1|}{lvert z_2rvert}z_2=kz_2 $$
and
$$z_1=kz_2 implies frac{z_1}{lvert z_1rvert}=frac{kz_2}{lvert kz_2rvert} =frac{z_2}{lvert z_2rvert} $$
Can you proceed with the second one?
answered Nov 16 at 20:45
gimusi
89.1k74495
We need to assume $z_1,z_2neq 0$, otherwise the expression $frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert}$ is not defined.
Then for the first one we need to prove that for $z_1,z_2neq 0$
$$frac{z_1}{lvert z_1rvert}=frac{z_2}{lvert z_2rvert} implies z_1=frac{|z_1|}{lvert z_2rvert}z_2=kz_2 $$
and
$$z_1=kz_2 implies frac{z_1}{lvert z_1rvert}=frac{kz_2}{lvert kz_2rvert} =frac{z_2}{lvert z_2rvert} $$
Can you proceed with the second one?
answered Nov 16 at 20:45
gimusi
89.1k74495
answered Nov 16 at 20:45
gimusi
89.1k74495
answered Nov 16 at 20:45
gimusi
89.1k74495
answered Nov 16 at 20:45
gimusi
89.1k74495
89.1k74495
I understand, I was just not sure generally speaking if these statements, as they are(so without assuming $z_1,z_2neq 0$), are true or false
– Dovla
Nov 16 at 21:00
@Dovla They are true for $z_1,z_2neq 0$ otherwise they are false.
– gimusi
Nov 16 at 21:13
add a comment |
I understand, I was just not sure generally speaking if these statements, as they are(so without assuming $z_1,z_2neq 0$), are true or false
– Dovla
Nov 16 at 21:00
@Dovla They are true for $z_1,z_2neq 0$ otherwise they are false.
– gimusi
Nov 16 at 21:13
I understand, I was just not sure generally speaking if these statements, as they are(so without assuming $z_1,z_2neq 0$), are true or false
– Dovla
Nov 16 at 21:00
I understand, I was just not sure generally speaking if these statements, as they are(so without assuming $z_1,z_2neq 0$), are true or false
– Dovla
Nov 16 at 21:00
@Dovla They are true for $z_1,z_2neq 0$ otherwise they are false.
– gimusi
Nov 16 at 21:13
@Dovla They are true for $z_1,z_2neq 0$ otherwise they are false.
– gimusi
Nov 16 at 21:13
add a comment |
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