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When I say a argand plane , in my textbook it says the plane having a complex number assigned to each of its point. I interpreted a meaning for this statement down below , let me know if it’s correct

Second is does it mean that when I talk about a plane. It is like x and z axis is one plane , x and y axis , y and z axis . So , it has to be 2D. x,y and z axis all together can’t be an argand plane. Is this correct ?

Confusion in ordered pairs :

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So , there is no squaring of numbers done here like R ^2 as written . That is my confusion.

Srijan
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2 Answers2

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The expression “Argand plane” simply means the set $\Bbb C$ of complex numbers, when we see it as a plane, that is, when we identify each complex number $a+bi$ ($a,b\in\Bbb R$) with the point $(a,b)\in\Bbb R^2$. So, yes, it is a $2$-dimensional thing.

José Carlos Santos
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  • Why does a,b belongs to R ^2 sir – Srijan Jan 24 '21 at 08:16
  • Set “ C” symbol that you wrote , is it some special symbol for complex numbers? – Srijan Jan 24 '21 at 08:17
  • Does that R^2 means distance ? Why not write square root symbol over it – Srijan Jan 24 '21 at 08:18
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    @user102532 $\mathbb R$ means the set of real numbers and $\mathbb{R}^2$ means the set of ordered pairs of real numbers (like $(x, y)$ coordinates). $\mathbb C$ is the set of complex numbers. This is all standard notation. – Deepak Jan 24 '21 at 08:22
  • @Deepak pls check the edit – Srijan Jan 24 '21 at 08:26
  • Yes, $\Bbb C$ is the set of all complex numbers. And $\Bbb R^2$ is the set of pairs $(a,b)$ of real numbers. I made no reference whatever to distance. – José Carlos Santos Jan 24 '21 at 08:36
  • @JoséCarlosSantos my only question why R^2. And not R. Even if it is convention , then why R for a+ib but R^2 for a,b – Srijan Jan 24 '21 at 08:46
  • For any set $X$, we denote the set of all pairs of elements of $X$ by $X^2$. Saying that both elements $a,b\in\Bbb R$ is equivalent to saying that the single pair $(a,b)\in\Bbb R^2$. – Berci Jan 24 '21 at 08:58
  • @Berci ok. So it does not mean square of real numbers – Srijan Jan 24 '21 at 09:00
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    @user102532 No. There are no squares whatsoever in my answer. As I told you in the comments, $\Bbb R^2$ is the set of pairs of real numbers. – José Carlos Santos Jan 24 '21 at 09:03
  • @JoséCarlosSantos I am getting what you mean to say that R^2 does not mean square of that R value. Right ? – Srijan Jan 24 '21 at 09:17
  • why do we write ^2 this symbol over real numbers to represent a,b .? Since it is also used as 2^2 = 4. – Srijan Jan 24 '21 at 09:18
  • It's because of cardinalities. If a set $X$ has $n$ elements then the set of its ordered pairs $X^2$ has $n^2$ elements. – Berci Jan 24 '21 at 09:30
  • @user102532 It's because, for any set $S$, the expression $S^2$ means the set of pairs $(a,b)$ of elements of $S$. It has nothing to do with computing squares of numbers. – José Carlos Santos Jan 24 '21 at 09:50
  • @JoséCarlosSantos thank you very much. I got it – Srijan Jan 24 '21 at 10:34
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    @user102532 It is not strictly relevant to the question, but as to why the "exponent" of $2$ is used in different ways - it's just that mathematical notation is sometimes context dependent. In this case, it's not even ambiguous because $\mathbb {R}^2$ pretty much always has the meaning of the set of ordered pairs of real numbers and never has the meaning of the square of numbers. Compare this to other ambiguous -bordering on abuse of- notation like $\sin^2 x$ (which means square of the sine) vs $\sin^{-1} x$ (which generally means the inverse function of sine, not the reciprocal of sine). – Deepak Jan 24 '21 at 13:35
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It says Argand plane, not any volume, is n't it?

The Argand diagram contains points $z= x+iy$ and lines in the the complex plane.

Narasimham
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