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.










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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.










    share|cite|improve this question
























      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.










      share|cite|improve this question













      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






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      asked Nov 16 at 20:40









      Dovla

      799




      799






















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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?






          share|cite|improve this answer





















          • 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











          Your Answer





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          1 Answer
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          1 Answer
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          active

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          active

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          active

          oldest

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          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?






          share|cite|improve this answer





















          • 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















          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?






          share|cite|improve this answer





















          • 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













          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?






          share|cite|improve this answer












          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?







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 16 at 20:45









          gimusi

          88k74393




          88k74393












          • 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










          • @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


















           

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