# Books on Elementary Complex Numbers

This result came up in my diff eq class the other day:

If i = x^2 then x = [(sqrt(2)/2) + (sqrt(2)/2)i]^2

While there aren't a lot of use for complex numbers in this class, I still feel stupid for not knowing it. Another trick that I'd like to learn about is the "complexifying the integral" trick that was mentioned in this video:

AFAICT, some of this stuff people learned in high school.. WTF.. I don't remember anything like that in high school. What books did they use for this? I don't have enough time to read an advanced undergrad level complex analysis book. I'd just like enough knowedge of complex numbers to know what's going on.

High school doesn't focus too much on complex math. It's not useful for most people.

50% of all complex mathematics is abuse of Euler's formula.

Another 10% is remembering that logarithms are not full inverses of exponentials (they are only partial inverses). This means if e^x = e^y, it's not necessarily true that x = y.

Oh, and you have to be able to convert between linear and polar forms of complex numbers. The Linear form makes addition, subtraction, integration, and derivation easy. The polar form makes multiplication and division easy.

And then there's little bits of calculus information that's useful. Integrals are done over a path (since the notion of an "interval" does not exist in complex numbers). Differentiability as a condition is much, much stronger. Stuff like that.

Just punch through the difficulties. No one learns much about complex numbers in high school.

High school doesn't focus too much on complex math. It's not useful for most people.

50% of all complex mathematics is abuse of Euler's formula.

Another 10% is remembering that logarithms are not full inverses of exponentials (they are only partial inverses). This means if e^x = e^y, it's not necessarily true that x = y.

Oh, and you have to be able to convert between linear and polar forms of complex numbers. The Linear form makes addition, subtraction, integration, and derivation easy. The polar form makes multiplication and division easy.

And then there's little bits of calculus information that's useful. Integrals are done over a path (since the notion of an "interval" does not exist in complex numbers). Differentiability as a condition is much, much stronger. Stuff like that.

Just punch through the difficulties. No one learns much about complex numbers in high school.

I don't mind "punching though" it but it seems like an area that I would be caught off guard if a curve ball were thrown. I bet theres more tricks out there than just the square root of i and Euler's formula.. I'd just like to know where they're at :)

I would have thought that your requirements would be met by the chapter devoted to complex numbers in many intermediate maths, science and engineering books.

The Chemistry Maths Book, by Erich Steiner

A course in Pure Mathematics by Maggie Gow

Electrical Technology by Hughes

All have good clear chapters.

The 'Demystified' series have many good volumes, but sadly not as yet one on complex numbers. However
Trigonometry Demystified has a good entry chapter.

I would not recommend looking beyond this to whole books on complex until your general background has caught up.

Then you can look for books on 'complex analysis', as mathematics using complex numbers is called.

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I think there is...

My list is obviously out of date. Thanks for the info.