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I just saw this video on KhanAcademy regarding complex numbers, and Sal mentioned that 4 + 2i can't be simplified. I was wondering why this is the case, or is it just a property of complex numbers?

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  • $\begingroup$ Thanks for linking the video. $\endgroup$
    – Toby Mak
    Sep 21, 2019 at 10:14
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    $\begingroup$ $4 + 2i = 2(2 + i)$. That said, why do you think $2(2 + i)$ is simpler than $4 + 2i$? $\endgroup$
    – N. F. Taussig
    Sep 21, 2019 at 10:14
  • $\begingroup$ You have to remember that simplicity isn't a mathematical property, merely a subjective aesthetic choice. If you prefer reading $2(2 + i)$ to $4 + 2i$, then feel free to write it that way. $\endgroup$
    – Theo Bendit
    Sep 21, 2019 at 10:18

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It can be written this way. It depends what your purpose is. Normally you would simplify to a recognised standard form.

Another expression would be $(1+i)(1-i)(2+i)$ or $i(1-i)^2(2+i)$ which are alternative factorisations into primes - the second involves also the unit $i$ to make it obvious that two of the prime factors are (essentially) the same.

Prime factorisation in complex numbers is an important topic, but is not usually addressed when they are first introduced, because other properties of the complex numbers are more immediately applicable.

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What Sal Khan means is that the answer is purely in terms of real and imaginary terms ($a+bi$ where $a,b$ are real). In the context of the video, this means that there are no ungrouped terms, and no unsimplified powers of $i$.

Of course you can write the number as $2(2+i)$. But complex numbers are usually expressed in the form $a+bi$, even when $a,b$ have some common factor, because $a+bi$ represents the number at the point $(a,b)$ in Cartesian space. To me, the analogy to Cartesian space is simpler to understand than having to imagine $2+i$ and scaling it away from the origin by a factor of $2$.

By the way, your 'simplification' reminds me of polar form. This is where a complex point is represented using the distance from the origin $r$, and the anti-clockwise angle from the $x$-axis $\theta$.

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$$4+2i = 2(2+i)$$

However, the reduced form $a + b\cdot i$ want to avoid using factorization, which is probably what Khan Academy meant.

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