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I want to know what is the best way to define the imaginary unit $i$. In many books we see $$i^2 = -1$$ But sometimes there is a $$i = \sqrt{-1}$$ there. Isn't the first definition a little problematic? Because if $i^2 = -1$ then $|i| = \sqrt{-1}$implying that $i$ could be both $-\sqrt{-1}$or $\sqrt{-1}$ which honestly i don't know if its true) So,

It is better to define $i$ directly as $\sqrt{-1}$?

embedded_dev
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    Observe that $|i|=1$. The general formula for the absolute value for complex numbers is $\sqrt{z\bar z}$ where $\bar z$ is the conjugate of $z$. If $z$ is real the conjugate is itself, hence if $r$ is a real number then $|r|=\sqrt{r^2}$. – Masacroso Jan 28 '17 at 22:21

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why would $i$ be defined seperately from the other complex numbers? I think it makes more sense to define $\mathbb C$ as the set $\mathbb R\times \mathbb R$ with the operation $(a,b)+(c,d)=(a+c,b+d)$ and $(a,b)\cdot(c,d)=(ac-ad,ad+bc)$.

And then identify the pair $(a,b)$ with the expression $a+bi$. so $(0,1)$ is associated with $0+1i$ which we shorten to $i$.

Asinomás
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    You could add for clarity that in this case $i = (0,1)$ – user4894 Jan 28 '17 at 21:56
  • Defining $i$ is defining complex numbers, so this may be the same. A duplicate anyway. – Dietrich Burde Jan 28 '17 at 21:57
  • My point is that we don't really have to define what $i$ "is", we can treat it the same way as any element of a field. We don't need to "define" arbitrary elements in fields, we just say what they do. – Asinomás Jan 28 '17 at 21:58
  • The reason you have $b$ to begin with is because the imaginary unit, $\sqrt{-1}$, is defined- allowing us to rewrite $a+\sqrt{-x}$ (for $a\in \mathbb{R}$, $x\geq 0$) as $a+\sqrt{x}\sqrt{-1}$ (via law of surds). Then we set $b$ equal to $\sqrt{x}$, resulting in $a+b\sqrt{-1}$, or $a+bi$. From there you get the coordinates $(a,b)$. – Radial Arm Saw Jul 16 '20 at 02:21
  • Also, identifying $(a,b)$ with $a+bi$ means you need a definition for $i$. – Radial Arm Saw Jul 16 '20 at 02:27
  • @RadialArmSaw no you don't, you can explicitly define the operation for all ordered pairs in a general fashion. the pair $(0,1)$ requires no special treatment. – Asinomás Jul 17 '20 at 02:18
  • I honestly don't understand what you are talking about, but the entire construction of the complex numbers can be done trivially starting with the real numbers and using ordered pairs. It can be done by explicitly stating the operations. Verifying field axioms is obviously optional but nonetheless trivial parting from these operations. – Asinomás Jul 17 '20 at 02:21