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Is the complex conjugate of a number (or a real multiple of it) the only complex number which, when multiplied with the original number, gives a real number?

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I don't understand the question. I mean you can always multiply by $0$, and get a real number! –  voldemort Aug 29 at 16:55
    
@Anthony. If zero is excluded, then the answer is yes –  imranfat Aug 29 at 16:58
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"or multiples of it"... $0$ is a multiple of $\bar z$! –  Najib Idrissi Aug 29 at 16:59
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Presumably you mean real multiples of it? –  Mathmo123 Aug 29 at 16:59
    
Multiply by a non zero real multiple of the complex numbers inverse? Or do you want to get every real number... –  snulty Aug 29 at 18:55

6 Answers 6

up vote 8 down vote accepted

Let $z,w \in \mathbb C$ and suppose $zw = k \in \mathbb R\setminus \{0\}$. Without loss of generality, $|w|=1$ so that $\frac1w = \overline w$. Then $$z = \frac kw = k\overline w$$So $w = \dfrac 1k \overline z$

If $zw = 0$ and $z\ne 0$, then $w = 0 = 0\cdot\overline z$.

However, if $z=0$, then $\overline z = 0$. But $zw = 0 \in \mathbb R$ for all $w \in \mathbb C$, and if $w \ne 0$, then $w$ is not a multiple of $\overline z = 0$.

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4  
Better yet, if $z \in \mathbb{C}$ is non-zero, and $w \in \mathbb{C}$ is any complex number such that $zw = k \in \mathbb{R}$, then $$ w = \frac{k}{z} = \frac{k}{|z|} \overline{z}, $$ so that $w$ is necessarily a real multiple of $\overline{z}$. –  Branimir Ćaćić Aug 29 at 17:03

Since Complex multiplication of $z$ by $w:=c+id$ involves rotation by $arg(w)$ , w must have $$argw +argz= k\pi ; k \in \mathbb Z$$

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Suppose we are given a complex number $a+bi$ and we wish to find another complex number $c+di$ such that $(a+bi)(c+di)$ is real. Since $(a+bi)(c+di) = (ac-bd)+(ad+bc)i$, we need $ad+bc = 0$. If we know that $b \neq 0$ (i.e. the given number isn't real), then we have $c = -\dfrac{a}{b}d$. Thus, $c+di = -\dfrac{a}{b}d+di = -\dfrac{d}{b}(a-bi)$, which is a real multiple of the conjugate of $a+bi$.

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Various answers have already provided symbolic proofs of the theorem you needed, but if you want a visually intuitive reason, consider:

When you multiply two complex numbers you multiply their distances from the origin and add their angles relative to the real axis. So how can you end up on the real axis? One way is to multiply by the complex conjugate so your angles sum to zero. Every other way is equivalent to multiplying by the complex conjugate first (to get to the real axis), then multiplying by some real number to move back and forth along the real axis.

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If $z=re^{i\phi}$ and $w=se^{i\psi}$ where $r,s\in [0,\infty)$ and $\phi,\psi\in[0,2\pi)$ then: $$zw=rse^{i(\phi+\psi)}\in \mathbb R\iff rs=0\vee \phi+\psi\in \{0,2\pi\}\iff rs=0\vee w\text{ is multiple of }\overline{z}$$

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Let $z$ be you complex number; you want to find all $z'\in\Bbb C$ such that $zz'\in\Bbb R$. There are two relevant cases:

  1. $z=0$, for which every $z'\in\Bbb C$ will do.
  2. $z\neq 0$ in which case $zz'=r\in\Bbb R$ is equivalent to $z'=\frac rz$ (with still $\in\Bbb R$). Then your solution set is formed by all real multiples of $\frac1z$. Since $\overline z=\frac{|z|^2}z$ is a nonzero real multiple of $\frac1z$, this is the same as the set of all real multiples of $\overline z$.
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