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.
Update
Thanks to Damo for their hard work below.
The main problem with the above excerpt is that it should not exist. It is pointless to introduce complex numbers with more than a sentence on complex roots, and it is almost impossible to do so in a sensible manner. The nonsense of the text’s approach is encapsulated by the equation
This equation is best thought as false and, in the context of the excerpt above, must be thought of as meaningless. As is, thus, the discussion leading up to this equation.
How did they get there? To begin, i is introduced as a number for which i2 = -1, which is fine and good at the school level. Then, they note that the equation x2 = -1 has the two solutions i and -i, which is significantly less fine; since general complex numbers, and -i in particular, have not yet been defined, the notation -i is thus far meaningless, as is the notion of squaring this number. Still, if the sentence were more carefully worded, it would be reasonable in an introductory paragraph. The cavalier attitude to definition and meaning, however, is the sign of much worse to come.
The text continues by “declaring” that √(-1) = i, and then heads on its merry calculating way. But the calculation is complete fantasy. The declaration amounts to a (bad) definition of a specific root which cannot, in and of itself, tell us what any other root means or how it might be manipulated. So, √(-4) is as yet undefined, and the manipulation of this quantity is unjustified, as yet unjustifiable, and is best thought of as wrong.
In the real context we use √x to distinguish the positive root but it is fundamental that complex roots are multiple-valued. And, for the polynomial focus of VCE mathematics, multiple values are perfectly fine and perfectly natural. The quadratic formula remains true without change and the purportedly troublesome identity
is always true (modulo the understanding that if x is a positive real then√x is now ambiguous). Moreover, with this natural interpretation, the text’s declaration that √(-1) = i is false, as is the equation √(-4) = 2i.
Admittedly, at some point it is valuable, and essential, to introduce principal values of roots, by which the text’s equation can be interpreted to be true. But principle roots are intrinsically awkward, must be introduced with great care and should only be introduced when there is a purpose. Which is not on page 1 of a school text, and arguably not ever in a school text.
Apart form the utter pointlessness and utter meaningless of the excerpt, we note:
- The text conflates the introduction of imaginary numbers in the 16th century with the introduction of the symbol i in the 18th century.
- The text implies 0 is an imaginary number, which is ok though a little peculiar.
- The real numbers and imaginary numbers are not subsets of
. - The characterisations
and 
are grandiose and pointless.
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 number we now denote by i with the introduction of notation for that number, and specifically the introduction of the symbol i. It is true, however, that “mathematicians” first used i in 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.)
Is it wrong to “define” Re(z)=a and Im(z)=b? Is this not a bit backwards? Aren’t these labels, rather than definitions?
I wouldn’t say “wrong”, but yes, it’s clumsy. Probably better to say a is defined to be the real part of z, and is denoted by Re(z).
I noticed a small error which helped reveal a much larger mistake.
In the “Note” which discusses sqrt(a x b), I think they have neglected some values of a and b. It should hold for the real numbers a and b, where a>=0 and b>=0.
This led me to a larger error.
The author’s explain here that this is the only time that we can say sqrt(ab)=sqrt(a)*sqrt(b). However, they have disregarded this when they a moment earlier state that sqrt(4*-1)=sqrt(4)*sqrt(-1).
By defining i as i^2=-1, then we are instead able to say that sqrt(4*i^2)=sqrt(4)*sqrt(i^2) without having to contradict ourselves in the next sentence.
Am I on the right track for this Witch?
Definitely the right track. I’ll reply in some detail later.
Luke, you have nailed one of the major errors in the excerpt. There is no indication in the text, for any z and w, why the equation
√(zw) = √z x √w
might or might not be true.
To be fair, the text never states that this equation only holds for z and w positive [by which they mean nonnegative] real numbers. The text only states that the equation fails to hold for z and w both negative.
But, the text gives no indication why the equation should hold for z negative and w positive. Underlying this is the fundamental question: what does √z mean?
What does √z mean? A square root of z is a number w such that w^2=z. z will always have two square roots w and -w. For a positive real number z we are (somewhat) familiar with referring to the square roots of z as being √z and -√z, where √z is a positive real number and is called the principal square root, while -√z is a negative real number. With this understanding, it makes sense that √(4*9)=√4*√9=6. However, to make sense of the square roots of a negative real number z, we need to think of z as being a complex number. They appear to be assuming that √z refers to a principal square root in the same way that it does with real numbers, but they haven’t considered what the principal square root of a complex number is, which makes the whole thing kind of meaningless. One way of thinking about √(zw)=√z x √w : if a is a square root of z and b is a square root of w, then ab is a square root of zw. So, for √(-4 x -1) = √(-4) x √(-1) : if you multiply a square root of -4 by a square root of -1 you get a square root of 4. Which is true.
Damo, I couldn’t have said it better myself (but I’ll try).
Indeed, and the simplest way of showing why you can’t have both
in that rule (when using the principal square root – but of course the principal square root cannot possibly be defined at this early stage) is to show some simple paradoxes:
But it surprise me that they would place restrictions on using the rule
but say nothing about the rule 
(which does require the stricter restriction 
):
They obviously want students to use the former rule to simplify things like
, and I’d have thought that this would be ‘reinforced’ by showing that simplifying something like 
is problematic unless it’s treated as 
.
JF, one can’t do anything until one defines the operations of multiplication and taking roots. Whatever one may want of roots, the above discussion in the text is, at that stage of the text, meaningless.
Indeed, and yet in every current VCE textbook I’ve seen, the very first questions in the very first exercise ask to ‘simplify’ things like
etc. It’s like they just copy each other.
I suppose you *could* get into principal roots which goes a fair bit beyond square roots, but surely there is an easier way to just avoid all this mess by using a more considered original definition for the number i?
(I’m not even pretending to be an expert, so this is a genuine question raised as an act of genuine curiosity, not making a suggestion as to how it could be done, because I’m sure my attempt would also be problematic)
Thanks, RF. Hilariously, the text *did* start with a considered definition: it began with i2 = -1, which is exactly where one should begin. There was then simply no need to mention square roots at all. Not on Page 1.
As to how to define roots if one must/desires, your non-expert pondering makes a lot more sense than the text. Yes principal roots is one natural approach. The other natural approach, which Damo mentions above, is to consider √z to be double-valued. For solving polynomial equations and the like, the latter approach suffices and is a lot simpler.
Fair point. I just noticed the other day that with cube roots of negatives, Wolfram chooses the complex number with the real imaginary part as the default answer (unless the full solution set is asked for) in a number of cases – although considering some high schools are using this program, I did wonder how many teachers would be able to trouble-shoot such a situation…
…then I opened another bottle of wine and forgot about most other things for a while.
So how come purely real and purely imaginary numbers (i.e. the numbers on the horizontal and vertical axes) are not a subset of complex numbers? Just asking this cos after reading this post it reminded me of the back-to-back error on the past two Specialist exams, where, if I’m reading it correctly, the problem was VCAA didn’t realise that real numbers are complex.
Hi, Craig. Your premise is wrong – the axes are subsets – but it’s a good question.
There are really two issues:
(i) a subtle issue discussed a bit in the comments on the blog;
(ii) A not-subtle issue, which is the basis of the 2021 and 2022 exam screw-ups.
First, (ii). If you talk about
being “complex”, or if you write 
, that does not mean that 
must have a non-zero imaginary part. So, for example 3 + 0i is a perfectly legitimate complex number. The world of complex numbers most definitely includes the guys on the axes.
That was the screw-up with the 2021 and 2022 exam questions. In 2021 they assumed writing
meant that 
and 
couldn’t be like 3 + 0i$, couldn’t be on the real axis. The 2022 error was basically the same, and arguably worse, but it’s distracting and off the point to untangle further here.
Now to (i).
The question is whether the real number 3 is the exact same thing as the complex number 3 + 0i?
It is best to think of the answer as “yes”, especially if you’re a not-super VCE student, or explaining things to a not-super VCE student. However, if you really do the defining of complex numbers properly, then the answer is “no”: technically, the world of real numbers and the world of complex numbers are their own separate worlds.
That may seem weird, and it is worth it for a Specialist teacher to unweird it for themselves (so they don’t make similar stuff-ups to VCAA). But it is subtle. And, it has no bearing on point (ii): within the complex world, the guys on the axes are definitely complex numbers.
Hi Marty,
Would you mind explaining why 3 ≠ 3+ 0i?
Hi Anonymous. I’ll explain below why 3 ≠ 3 + 0i, but please keep in mind my prior comment: it is best in the VCE context to regard 3 and 3 + 0i as being the same. Now, here’s the explanation, of why, in pedantic truth, they’re not the same …
First, a very brief history. Complex numbers were stumbled upon in the 16th century, when mathematicians were trying to solve cubic equations. Weirdly, it turns out you sometimes need square roots of negatives to solve cubics, even if all the solutions are real. The mathematicians had no idea what the hell was going on, but they knew it worked. That tradition continued for over two hundred years, with very very powerful mathematics being produced, but still with no clear understanding of what complex numbers really were.
Then, in the early 19th century, a Norwegian mathematician, Caspar Wessell, basically figured out what complex numbers were. It was Wessell who came up with the idea of picturing complex numbers as points on the “complex” plane, with a Real direction and an Imaginary direction. Thought of in this way, the Real axis consists of the real numbers, and 3 really is the exact same thing as 3 + 0i.
BUT, modern mathematicians don’t think of complex numbers this way.
In brief, modern mathematicians never define things by pictures: we always want things to be defined in terms of sets and numbers and the like. It’s perfectly reasonable for you to ask why we bother, and I’m happy to take supplementary questions. But for now accept it: for modern mathematicians, Wessel’s intuitive picture of the complex plane was not good enough.
So what do we modern guys do? If you think of a complex number like 3 + 4i in the plane, then that just amounts to a pair of coordinates, (3,4). For mathematicians, that is precisely what complex numbers are. A typical complex number is an ordered pair (a,b) of real numbers. Of course we think of it as a + bi, and we picture it, in our minds or on paper, in the standard way. But what the complex number is, what it is defined to be, is (a,b).
So, now compare the real number 3 to the complex number 3 + 0i. The second is defined to be the pair of numbers (3,0). And the pair (3,0) is not the same thing as the single number 3. Two numbers in a list is not the same as one number, even if there’s a 0 involved. QED.
Now there are lots of very natural questions you might now ask. Such as: Who gives a stuff? Why does it matter? Are you guys all nuts?
I’m happy to answer all further questions. But that, as briefly as I can explain it, is why 3 ≠ 3 + 0i.
Thanks for the reply, Marty.
I don’t quite understand why there’s this distinction between a point on the horizontal axis on the complex plane and a real number (have I really been a “not-super (ex) VCE student” this whole time??), but I believe you that there’s high-tech maths where it matters, as you said to Anonymous.
I’ve always thought of the complex numbers as being these numbers that were out there this whole time, but we’ve only just known about that vertical strip called the (real) number line, and the complex plane is just the number line except 2D, or something like that.
Maybe I had this ‘real numbers are complex’ stuff reinforced in my head because of this book (Complex Variables and Applications by Ruel V. Churchill), I’ve attached the relevent page. [Edit: the image came out a little small, here’s an imgur link https://i.imgur.com/LISQ94g.jpg%5D
I looked in Spivak’s Calculus and he says (paraphrasing) that real numbers can’t be complex because of the ordered-pair definition, but the Churchill book uses it too so… different strokes I guess.
Thanks, Craig, and everyone. The questions are good. The Cambridge text is so bad, these questions cannot even be properly seen.
In brief, Spivak is right and Churchill is wrong. But to step through it slowly:
*) Your approach/view that the complex numbers were always “out there” is (for us guys) very natural and very intuitive, and what most students/teachers will do, and how mathematicians traditionally thought about it. (Less clearly before Wessel and more clearly after.)
*) However, modern mathematics has taken a very different approach, to complex numbers and pretty much to all mathematics creatures. Mathematicians no longer hunt for mathematics like zoologists hunting for and classifying new species in a jungle. Rather, mathematicians think of everything having to be created, to be carefully *defined*, step by step. (They may still do this with the sense of discovering the ideas that were always out there. But, nonetheless, everything must be carefully defined, not simply accepted as existing.)
*) Churchill does exactly this, by defining a complex number to be a pair (a,b).
*) BUT, Churchill also wants his readers to think of the real number a as being the same thing as the complex number (a,0). They are not the same thing. Obviously. Don’t overthink it: a pair of numbers cannot be the exact same thing as a single number.
*) There is good reason for Churchill to do this. He is writing a text on complex numbers, not on the foundations of mathematics, and for the majority of the relevant students it will be pointless distracting and confusing to think of 3 and 3 + 0i as different.
*) Nonetheless, Churchill is being slippery with his “is to be identified with” language, and his equality sign in (2) is false.
*) There are proper mathematical ways to take care of this, to make mathematically solid the idea of “to be identified with”. Spivak discusses this briefly, and more directly and honestly than Churchill. (The key notion is of an “isomorphism”: a mathematically precise way of deciding when two worlds can be “identified”.)