Because complex numbers make the fundamental theorem of Algebra nice and simple rather than complicated and ugly. In turn, this makes the spectral theorem of linear algebra nice and simple rather than complicated and ugly. In turn, this makes a bunch of downstream applications nice and simple rather than complicated and ugly.

You will get a feel for this if you work Axler's problems. More importantly, you will gain an intuition for the fact that if you turn up your nose at complex numbers while going into these application spaces, you are likely to painstakingly reinvent them except harder, more ugly, and worse.

Example: in physics, oscillation and waves A. underpin everything and B. involve energy sloshing between two buckets. Kinetic and potential. Electric and magnetic. Pressure and velocity. These become real and imaginary (or imaginary and real, it's arbitrary). This is where complex numbers -- where you have two choices of units -- absolutely shine. Where you would have needed two coupled equations with lots of sin(), cos(), trig identities, and perhaps even bifurcated domains you now have one simple equation with exponentials and lots of mathematical power tools immediately available. Complex numbers are a huge upgrade, and that's why anything to do with waves will have them absolutely everywhere.

Imagine you're a 16th century Italian mathematician who is trying to solve cubic equations. You notice that when you try to solve some equations, you end up with a sqrt(-1) in your work. If you're Cardano, you call those terms "irreducible" and forget about them. If you're Bombelli, you realize that if you continue working at the equation while assuming sqrt(-1) is a distinct mathematical entity, you can find the real roots of cubic equations.

So I would say that it's less that "Complex numbers were invented so that we can take square roots of negative numbers", and more "Assuming that sqrt(-1) is a mathematical entity lets us solve certain cubic equations, and that's useful and interesting". Eventually, people just called sqrt(-1) "i", and then invented/discovered a lot of other math.

Source: http://fermatslasttheorem.blogspot.com/2006/12/bombelli-and-...

I know people have jumped here to try to explain it to you as a comment, but I highly recommend a recent talk by Freya Holmer. It's a fun little exploration of a seemingly simply question about vectors that motivates the answer you asked. It also culminates with an introduction to Clifford Algebras, which you may or may not want to learn about after the talk.

https://www.youtube.com/watch?v=htYh-Tq7ZBI

That is already a first and most important "why". It may be a little bit too terse, but mathematics didn't evolve by knowing the applications before doing something.

Without doing some playing around from sqrt(-1) and discovering how it connects one thing to another, nobody is able to come up with real applications. You need to at least build a placeholder of the concept in your mind before you can examine what's possible.

So a person with a similar mindset as those who were the first people to use complex numbers, would just try to find a way to express a square root of a negative number and see how it goes. It starts with a limitation of an important tool and tries to close a perceived conceptual gap. The mathematicians themselves that write this book probably all think that way. I wouldn't call myself a mathematician but I didn't need anything more than that sentence to believe someone was motivated enough just from that reason alone.

So really there's a whole audience out there - arguably that a professional mathematician most wants to address - that could appreciate this sentence just as-is. So it's not true that it's natural to think the audience requires further explanation. Whether you should care is another matter. But as this is a book about linear algebra not complex numbers, some others would have accused the author of digression if he granted your wishes.

So I don't think what you're demanding is fair. Maybe it's a reasonable request after the fact, but it's a little too harsh to think it's something the author must have addressed in his text to his intended audience. This kind of inquiry is what in-person teaching is useful for.

I prefer a simpler perspective for complex numbers of "defined latently, then discovered, accepted, named and given notation", other than "invented".

Invented implies some degree of arbitrariness or choice, but complex numbers are not an arbitrary construct.

Zero, negative numbers, and imaginary numbers were all latently defined by prior concepts before they were recognized. They were unavoidable, as existing operations inevitably kept producing them. Since they kept coming up, it forced people to eventually recognize that these seemingly nonsensical concepts continued to behave sensibly under the operations that produced them.

Once addition and subtraction were defined on natural numbers, (1, 2, 3, ... etc), the concept of zero was latently defined. The concept of "nothing" was not immediately recognized as a number, but there is only one consistent way of dealing with 2-2, 5-5, 7-7, etc. Eventually that concept was given a name "zero", notation "0", and adopted as a number.

It was discovered, in that it was already determined by addition and subtraction, just not yet recognized.

Similarly with negative numbers. They were also latently determined by addition and subtraction. At first subtracting a larger number from a smaller number was considered nonsensical. But starting from the simple acceptance that "5-8" can at least be consistently viewed as the number which added to 8 gives 5, and other similar examples, it was discovered that such numbers had only one consistent behavior.

So they were accepted, given a name "negative numbers" and a notation "-x", short hand for "0-x".

And again, once addition, multiplication, (and optionally exponentiation) were defined, the expressions x*x = -1 (or x = sqrt(-1)) were run into, they were initially considered non-sensical.

But starting from acceptance that it at least makes sense to say that "the square of the square root of -1", is "-1", it was discovered that roots of -1 could be worked with consistently using the already accepted operations that produced them.

The numbers that included square roots of -1 were given a name "imaginary numbers", the square root of -1 given notation, "i", and we got complex numbers that had both real and square root of -1 parts.

I think it's the same reason why negative numbers were invented: It lets you do more with algebra than you could before (some of which, like raising something to a power leading to a sine wave is pretty weird, but turns out to be useful in engineering, etc.), and everything else still "just works" the same way as before.

(Admittedly the applications of negative numbers are much more obvious.)

Part of the issue is that there is no simple Why. To invent one is to either petition history or invent some perspective. It's not a bad idea to do these things, didactically, but they're unnecessary for the material. Or, said another way, there's something to be said for discovering your own "why" through familiarity with the many wonderous properties the complex numbers enjoy.

That said, that's a frustrating answer. An excellent book which does just what I said above and tells a lightly fictionalized "just so" story of the "history and development" of mathematics as an excuse to introduce everything in a motivated fashion is MacLane's Mathematics: Form and Function which I just recommend endlessly.

https://www.amazon.com/Mathematics-Form-Function-Saunders-Ma...

> Why? I wish a textbook tell me why, right there

If you're a programmer, consider whether it's easier to reason about a function that always returns a value, or a function that sometimes returns a value and sometimes throws an exception. The latter is a partial function and typically complicates reasoning because of the exceptional cases, the former is a total function and is fairly trivial to reason about (like multiplication vs. division where you have to consider division by zero).

Before complex numbers, the square root function was partial, but adding complex numbers made it total, so it simplified a lot of theory and enabled new types of analysis. Fortuitously, it also turned out to be very useful when applied to the real world.

Complex numbers are algebraically closed, reals are not. This means, if you write a polynomial with complex coefficients, it will have (only) complex roots. Analogous statement for the reals isn't true.

That terse summary is basically misleading, so I can see why it's not helpful. It would be a much better one line summary to say

"Complex numbers were invented so that we could name (and describe) all roots to a quadratic equation"

of course that also requires further explanation, but at least it's not leading the reader down the wrong path.

Very simply speaking, complex numbers are numbers that always have roots. That's one aspect of it.

I wrote a quick piece [0] about it some time back. Hope that adds to your knowledge.

[0]: https://convoluted.bearblog.dev/complex-numbers

At least it gave you one round of a "why" answer!

It might have been a reason, mathematicians wanting everything well defined etc. But here's a better way to think about it: On real number line, addition defines shifts and multiplication defines scaling. If you are in two dimensions, what is the equivalent? We define a 2 dimensional number such that multiplication defines scaling + rotation. The complex in complex number should not be read as complicated but like duplex or two things intertwined together.

The next question is why bother? What's the point? Turns out that important real life signals, like AC voltage and current, are sinusoidal. And real life electrical machines shift the phase of these signals. By using complex numbers to represent these signals, you can continue to use simple maths of DC circuits to analyze AC circuits. So you'd can still use V = IR, but R of a AC machine like motor will be impedance (generally called Z), represented by a complex number.

I found first few pages of MD Alder's complex analysis for Engineers indispensable in demystifying this complex stuff. Here's a quote from first paragraph "If Complex Numbers had been invented thirty years ago instead of over three hundred, they wouldn't have been called `Complex Numbers' at all. They'd have been called `Planar Numbers', or `Two-dimensional Numbers' or something similar, and there would have been none of this nonsense about `imaginary' numbers"

Because mathematicians like to make up things and theories to feel important, since they are impractical people who don't do anything important in the real world.

Half-joke apart (and I studied math in college, BTW, as my major, with Sanskrit as a minor), complex numbers have many uses in the real world, in engineering and other areas.

See the Applications section of https://en.m.wikipedia.org/wiki/Complex_number