Well, WitCH 1 is still not satisfactorily resolved, and Tweel’s puzzle is also still out there. But, we may as well get another ball rolling.

The second in our *What is this Crap Here* series comes from Cambridge’s textbook Specialist Mathematics VCE Units 3 & 4 (2018). Enjoy, and please get to pondering, and posting.

I will admit to using this book but never having seen this glaring issue!

Defining i=sqrt(-1) is quite different to defining i such that i^2=-1.

The +,- just adds to the sense that the author is confused. I’m hesitant to blame the author completely though as this may have been something insisted upon by the publisher or possibly even a reviewer… (or, even possibly taken from a VCAA curriculum document directly)

Thanks, Number 8. Yes, falsely declaring i = √(-1) is where the rot begins, though not where it ends.

Like sqrt(2) as a decimal in WitCH 1, it never really ends…

Don’t you refer to a and b as the real part and imaginary parts of a complex number? Not real and imaginary numbers. So bi would still be a complex number but the real part being 0 and not written.

On a side note, could you say it as no real part if a=0? I guess on the complex plane it 0 has a position and maybe you cannot say it has no real part. However, I do remember using that phrase at uni. So I’m not sure.

Hi Potii. Both are acceptable. So, for the complex number 3 + 4i, 3 is the real part and 4 is the imaginary part, but it’s still ok to say 4i is an imaginary number. In fact that expression jars with me; i’m used to “imaginary” as being a synonym for complex, and I would refer to 4i as purely imaginary in that context. But the textbook’s usage appears to be common. I don’t quite understand your second point, though note that the number 0 is both real and imaginary.

You answered my second part with 4i being said to be purely imaginary. That’s just what I was wondering, that you can say there is no real part if only there is only 4i or -13i.

The mapping of C to R is really getting to me on this page. Like – who cares?!? Plus, since Re and Im are not functions, it smells of lacking understanding…

Thanks, Number 8. Well, Re and Im *can* be thought of as functions as the text has written. (As can modulus and conjugate and argument and pretty much everything that can be interpreted in an input-output manner.) But your underlying point is absolutely correct, that it is utterly pointless and distracting to think of Re and Im in this manner in this context. It is just being pompously formal and gratuitously obfuscating, in a manner that Cambridge has turned into an art form.

Weren’t imaginary numbers introduced before the 18th century? I remember reading about roots of negative numbers coming up when solving cubics. A quick internet search says that Heron of Alexandria knew about them and Rafael Bombelli created rules for multiplication with complex numbers. Both of whom lived before the 18th century.

Yes, you’re correct. The application of imaginary/complex numbers began in the 16th century, with Cardano and Bombelli. In an earlier passage, the Cambridge text briefly and badly refers to Cardano’s work.

The Cambridge passage excerpted above is not exactly wrong in the manner you suggest but it is close; it is vague, muddled and just short of meaningless. The wording confuses the introduction of the

numberwe now denote byiwith the introduction ofnotationfor that number, and specifically the introduction of thesymbol i. It is true, however, that “mathematicians” first usediin this way in the 18th century, where mathematicians = Euler and 18th century = 1777. (Other symbols were used beginning around 1760, but not to my knowledge much earlier than that.)