First, a quick note about these WitCHes. Any reasonable mathematician looking at such text extracts would immediately see the mathematical flaw(s) and would wonder how such half-baked nonsense could be published. We are aware, however, that for teachers and students, or at least Australian teachers and students, it is not nearly so easy. Since school mathematics is completely immersed in semi-sense, it is difficult to know the rules of the game. It is also perhaps difficult to know how a tentative suggestion might be received on a snarky blog such as this. We’ll just say, though we have little time for don’t-know-as-much-as-they-think textbook writers, we’re very patient with teachers and students who are honestly trying to figure out what’s what.
* Cambridge is a good target, since they are the most respected of standard Australian school texts. We will, however, be whacking other publishers, and we’re always open to suggestion. Just email if you have a good WitCH candidate, or crap of any kind you wish to be attacked.
The above excerpt is indicative of the text’s entire chapter on complex numbers. It is such remarkably poor exposition, the foundations so understated and the direction so aimless, it is almost impossible to find one’s way back to sensible discussion.
Here is a natural framework for a Year 12 topic on complex numbers:
First, one introduces a new number for which .
One then defines complex numbers, and introduces the fundamental operations of addition and multiplication.
One then at least states, and hopefully proves, the familiar algebraic properties for complex numbers, i.e. the field laws, and so forth. All these properties are obvious or straight-forward to prove, except for the existence of multiplicative inverses; one has to prove that given any non-zero complex there is another complex with .
That is the basic complex algebra sorted, and then one can tidy up. This includes the definition of division , noting the essential role played by commutativity of multiplication.
Then, comes the geometry of complex numbers, beginning with the definition and algebraic properties of the conjugate and modulus , the interpretation of these quantities in terms of the complex plane, and polar form.
Finally, the algebra and geometry of complex numbers are related: the parallelogram interpretation of addition, the trigonometric-polar interpretation of multiplication, roots of complex numbers and so forth.
Must complex numbers be taught in this manner and in this order? No and yes. One obvious variation is to include a formal definition of a complex number as an ordered pair ; as Damo remarks below, this is done as an asterisked section in Fitzpatrick and Galbraith. Though unnervingly abstract, the formal definition has the non-trivial advantage of reinforcing, almost demanding, the interpretation of complex numbers as points in the complex plane. More generally, one can emphasise more or less of the theoretical underpinnings and, to an extent, change the ordering.
But, one can only change the ordering and discard the theory so much, and no more. Complex numbers are new algebraic objects, and defining and clarifying the algebra is critical, and this fundamentally precedes the geometry.
What is the Cambridge order? The text starts off well, introducing with ,and then immediately goes off the rails by declaring that . Then, in brief, the text includes:
(b) complex addition stated, presumably defined, with the inverse introduced but not named;
(c) complex subtraction, followed by an almost invisible statement of the relevant field laws, none of which are proved or assigned as exercises;
(d) scalar multiplication;
(d) the complex plane and “the representation of the basic operations on complex numbers”;
(e) complex multiplication defined, with an almost invisible statement of field laws, none of which are proved or assigned as exercises, and with no mention of the question of multiplicative inverses;
(f) the geometry of multiplication by ;
(g) the modulus of a complex number defined, with algebraic properties (including ) stated and assigned as exercises;
(h) the conjugate of a complex number defined, with algebraic properties stated and either proved or assigned as exercises.
(h) Finally, as excerpted above, it’s on to reciprocals of complex numbers, multiplicative inverses in terms of modulus and conjugate, and division.
(i) This is followed by sections on polar form, de Moivre’s theorem and so forth;
(j) CAS garbage is, of course, interspersed throughout. (Which is not all Cambridge’s fault, but the text is no less ugly for that.)
At no stage in the text’s exposition is there any visible concern for emphasising or clarifying foundations, or for following a natural mathematical progression. There is too seldom an indication of what is being defined or assumed or proved.
What is the point? Yes, one can easily be overly theoretical on this topic, but this is Year 12 Specialist Mathematics. It is supposed to be special. The students have already been introduced to complex numbers in Year 11 Specialist. Indeed, much of the complex material in the Year 11 Cambridge text is repeated verbatim in the Year 12 text. Why bother? The students have already been exposed to the nuts and bolts, so why not approach the subject with some mathematical integrity, rather than just cutting and pasting aimless, half-baked nonsense?
Now, finally and briefly, some specific comments on the specific nonsense excerpted above.
division of complex numbers has already appeared in the text, in the list of (unproved) properties of the modulus.
the algebraic manipulation of is unfamiliar and unmotivated and, as is admitted way too late, is undefined. There is a place for such “let’s see” calculations – what mathematicians refer to as formal calculations –but they have to be framed and be motivated much more carefully.
There is no need here for a “let’s see” calculation. The critical and simple observation is that is real. It is then a short step to realise and to prove that acts as, and thus is, the multiplicative inverse of .
Having finally admitted that has not been defined, the text goes on to not define it again. The text states the multiplicative inverse of , but it is not clear whether this statement amounts to a definition or a conclusion.
Division of complex numbers is then defined with needless subscripts and, more importantly, with no mention of the fundamental role of commutativity of multiplication.
Throughout, the use of conjugate and modulus is muddying rather than clarifying.
At no stage is it made clear why makes sense in contrast to, for example, the non-sense of for a matrix.
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.
Tweel is one of the all-time great science fiction characters, the hero of Stanley G. Weinbaum’s wonderful 1934 story, A Martian Odyssey. The story is set on Mars in the 21st century and begins with astronaut Dick Jarvis crashing his mini-rocket. Jarvis then happens upon the ostrich-like Tweel being attacked by a tentacled monster. Jarvis saves Tweel, they become friends and Tweel accompanies Jarvis on his long journey back to camp and safety, the two meeting all manner of exotic Martians along the way.
A Martian Odyssey is great fun, fantastically inventive pulp science fiction, but the weird, endearing and strangely intelligent Tweel raises the story to another level. Tweel and Jarvis attempt to communicate, and Tweel learns a few English words while Jarvis can make no sense of Tweel’s sounds, is simply unable to figure out how Tweel thinks. However, Jarvis gets an idea:
“After a while I gave up the language business, and tried mathematics. I scratched two plus two equals four on the ground, and demonstrated it with pebbles. Again Tweel caught the idea, and informed me that three plus three equals six.”
That gave them a minimal form of communication and Tweel turns out to be very resourceful with the little mathematics they share. Coming across a weird rock creature, Tweel describes the creature as
“No one-one-two. No two-two-four”.
Later Tweel describes some crazy barrel creatures:
“One-one-two yes! Two-two-four no!”
A Martian Odyssey works so well because Weinbaum simply describes the craziness that Jarvis encounters, with no attempt to explain it. Tweel is just sufficiently familar – a few words, a little arithmetic and a sense of loyalty – to make the craziness seem meaningful if still not comprehensible.
But now, here’s the puzzle. The communication between Jarvis and Tweel depends upon the universality of mathematics, that all intelligent creatures will understand and agree that 1 + 1 = 2 and 2 + 2 = 4, and so forth.
But why? Why is 1 + 1 = 2? Why is 2 + 2 = 4?
The answers are perhaps not so obvious. First, however, you should go read Weinbaum’s awesome story (and the sequel). Then ponder the puzzle.
Thanks to those who have posted so far. Everyone is circling with the right ideas, but perhaps people are searching for something deeper than intended. Anyway, for this first update (to which people are free to object in the comments), here is our suggested, simplest answer to why 1 + 1 = 2:
“1 + 1 = 2” is true by definition.
To take a step back, what does 2 mean? It depends slightly on how you think of the natural numbers being given, but there are really only (ahem) two, similar choices. If you accept that addition is around then 2/two is simply a new symbol/name that stands for 1 + 1.
Or, more fundamentally, we can follow Number 8 and go Peano-ish, in which case 2 is defined as S(1), as the “successor” of 1. But then we have to define addition, and the first(ish) step for that is to define n + 1 = S(n); that is, 1 + 1 is defined to be S(1), which we have decided to call 2. There’s a good discussion of it all here.
With 1 + 1 = 2 done (modulo objections), why now is 2 + 2 = 4?
It’s probably close enough to round this one off. To clearly state why 2 + 2 = 4, we first have to clearly state what 2 and 4 and + are. So, as discussed above, 1 + 1 = 2 by definition (more or less). And, similarly, we define 3 = 2 + 1 and 4 = 3 + 1. So, the question of why 2 + 2 = 4 comes down to understanding why
2 + (1 + 1) = (2 + 1) + 1
So, our question amounts to a simple instance of the associative law of addition. And, how do we know the associative law is true? Naively, we can accept that’s the way numbers work. Or, we can go Peano-ish again, and the above example of associativity becomes part of the definition of addition.
In summary, to know that 1 + 1 = 2 all we need is the notion of natural numbers, of counting. To know that 2 + 2 = 4, however, requires the notion of addition.