Welcome to NRICH.

 
Graphing Complex Solutions
By Alexandra on November 2, 1998:

The first place I came across complex numbers was when I came across graphs such as y = x2 + 1 which have no real solutions. I was told then that I'd find out about the complex solutions when we did complex numbers. The mental picture I had for 'no real solutions' was of a graph which did not intercept the x-axis.

When we started complex numbers and did the Argand plane, I found that if I made the x-axis into an Argand plane by adding an imaginary axis, then plotted y = x2 + 1 for all real and imaginary values (not complex ones) I got two curves. One was the conventional one in the xy plane (min (1,0) parabola) and the other was a parabola max (1,0) intercepts on the imaginary x-axis of 1 and -1. (These are the two complex solutions we get algebraically). The new bit of curve was a reflection in x = 1 and rotation of 90 degrees round the y-axis of the original part.

OK - but I can't do that to all curves because they give complex y values as well as x and I only have 3 dimensions. Nor can I do complex starting values in y = x2 +1. I did try making the y-axis a plane too so the two axes `shared' an imaginary axis (but +ve i for x was -ve i for y so the 90 degrees both went anticlockwise)but it made a mess. It was too hard to draw the surface in 3D.

Can you give me any idea what my surface would look like? 3 or 4D or anything; that is, if the surface exists at all.

By Simon Munday (sjm78) on November 5, 1998:

Hi Alexandra,

I'm afraid I've got some bad news. You can't imagine the surface you desribe. Basically what you are attempting to do is imagine four lines that are all perpendicular to each other. This is impossible in the three dimensions we inhabit, so it's pretty unlikely that you're imaginative enough to see it!

However, your surface does exist, at least in an abstract sense. Technically speaking, it lives in a two dimensional complex inner product space, and is itself the union of perpendicular one dimensional subspaces. I'll try to explain what some of that garbage means. The space can be thought of the set of ordered pairs of complex numbers (just like coordinates), with two operations, addition and multiplication by scalars (eg real or complex numbers) which always give another member of the space (we say the space is closed under these operations). There is an inner product defined on the set, which is just like the scalar product of two vectors. This allows you to say what you mean by "perpendicular". A subspace is a subset of the space which is a space in its own right under the same operations, and two spaces are orthogonal if the inner (scalar) product of any member of one of the spaces with any member of the other is zero. Don't worry if you didn't follow all that - it's pretty technical stuff. What you need to remember is that most of the ideas are just abstract generalisations of things you already know about (the x-y plane, co-ordinates, scalar products, etc.).

Your function is described by the set of ordered pairs (z, f(z)), exactly as it is when you plot a graph of real numbers in the x-y plane. In fact there are a lot of similarities between the set of ordered pairs of real numbers, and that of complex numbers. In fact, there's not much point just trying to deal with two dimensional spaces. Much of the same theory applies when the dimension is any finite number n. It sounds weird, I know, but when you just think of the space as the set of ordered n-tuples (n numbers arranged on a row or column, like a vector) of real or complex numbers, it seems a bit more tame. You can actually get similar objects that are infinite-dimensional, which is very bizarre indeed.

There is a very rich theory of these and related objects, often without the inner product (In this case the space is called a vector space). It has strong links with things like simultaneous equations and matrices, and is very beautiful. Mathematicians study in particular the functions between vector spaces that preserve their structure, ie f(x+y)=f(x)+f(y) where x and y are members of the space, and f(ax)=af(x) where a is a scalar and x is a member of the space. These are called linear maps, and are extremely important (and nice to deal with).

If you do a degree in maths (which I would recommend, if only because you wouldn't have to write essays), you'll learn an awful lot more about these things. If you want to find out more in the interim about higher dimensions and some of the things I've been desribing, look at any book with a title like "Linear Algebra" or "Vector Spaces and Geometry" or something. You just might find it interesting.

As a little footnote, I do know a way of drawing a four-dimensional cube. Simply draw two cubes from the same perspective and so that they overlap, and join the corresponding corners. It's just like constructing a cube by drawing two overlapping squares. Have fun...But don't expect it to look like anything you've ever seen!

Simon

PS Having read through what I've just written, I think it's probably unlikely that you've got much idea what I'm talking about. However, I hope I've given you a flavour of the type of thing mathematicians do, which is generally very unlike what you do at A-level, and much more interesting.

By Alexandra on :

Hello Simon!

I'm afraid I'm sufficiently curious about this to ask more questions. The second paragraph leaves me completely lost! Is there any way you can explain in simpler terms? - for instance we only did scalar products last week and so I'm still a bit fuzzy in my comprehension of them.

Terms I don't understand that seem critical are
`inner product', - and `defined on the set'
saying what I mean by perpendicular - as in why is it a problem and why do I need to?!?!
`orthogonal' - from how you define it this seems like two planes being perpendicular, but curvy planes which confuses!
Next paragraph. You say there are a lot of similarities between the set of ordered pairs of real numbers and that of complex ones. What are these?

I should probably tell you that we haven't done set theory yet, are covering intersections between 2 planes in vectors, and have `finished' complex numbers for Further Maths AL.

Thank you!

Alexandra

By Simon Munday (sjm78) on November 26, 1998:

Hello again,

I guess I didn't explain things very well, but I'm glad I got you interested anyway! The basic idea behind all this stuff is that of generalization. You've probably been told that a "vector" is something like ``a directed line segment'', which lives in 2 or 3-d space. Now that is all very well and nice, but mathematicians are less interested in these things specifically than in their algebraic properties (physicists and engineers are the ones who are really interested in that sort of vector, because they are very useful in describing things like velocity, angular momentum, etc.). Now mathematicians are not content to simply study the algebraic properties of vectors themselves. What they really want to know is, ``What is the most simple set of rules we can think of so that anything obeying these rules has the algebraic properties that we expect these vectors to have?'' and also ``What other objects have the same algebraic properties as vectors''. These are really the same question.

What I mean by studying the algebraic properties is that we have a set (which is exactly what you think it is, ie a collection of objects) and some operations that tell us how we can combine the elements of our set. For example, the set of real numbers and the operations addition and multiplication, have the algebraic properties of something called a "field", and the integers with the operations of addition and multiplication, have the properties of something called an "integral domain". Can you see any algebraic differences between the real numbers with addition and multiplication and the integers with addition and multiplication?

We call the set of rules satisfied by the set and the operations the axioms of the system we are considering. For example some of the axioms for a field F with operations + (addition) and * (multiplication) are:

if x, y are in F then x+y is in F,
there is an element i, called the additive identity, such that x + i = i + x = x for any x in F, (i is normally written 0),
if x is in F, then there is an element y in f such that x + y = i (y is normally written -x)
addition is commutative and associative.

There are similar rules for multiplication - what are they, and what are the differences (if you take the field F to be the familiar real numbers with addition and multiplication) ? Finally and rather crucially, the two operations in a field are linked by the distributive law:

if x,y and z are in F then x*(y + z) = x*y + x*z.

So you have an example of a set of axioms for a certain algebraic system (or structure). The real numbers with + and * form a field (in fact this field has some extra properties not required for something to be a field, but that doesn't matter), as do the rational numbers with the same operations, and the complex numbers with the analagous operations.

Now mathematicians in the 19th Century worked out the axioms that any algebraic system with the same algebraic properties as ordinary vectors should have, and called any system that satisfies these rules a "vector space". So obviously the set of ordinary vectors in three dimensions is a vector space. What other things are vector spaces? Well, think about what algebraic properties the set of ordinary vectors in three dimensions has. You can add two vectors to get another vector, you can multiply a vector by a real number to get another vector. There is an additive identity vector (0), each vector v has an additive inverse - v, etc. Notice that the vectors are very closely associated with the field of real numbers. A "vector space over a field F" is formally a set of things that satisfy are certain pretty long set of rules including the ones that I've just mentioned, where instead of real numbers, you take numbers from your field F (the elements of a field are usually called "numbers") to multiply your vectors by. The set of ordered triples (x, y, z) of real numbers is a vector space over the real numbers, as is the set of ordered n-tuples (x1, x2, ..., xn) of real numbers for any natural number n. (Check to see that these have the properties they ought to.) You can consider the set of continous real-valued functions defined on the interval [a,b] a vector space over the real numbers (don't worry if you don't understand all the terms here); the set of ordered n-tuples of complex numbers a vector space over the complex numbers; the set of real/complex sequences (x1, x2, x3,...) satisfying a recurrence relation like xk + 3xk+1 = 4xk+2 a vector space over the real/complex numbers; and many, many more.

I hope that helps you to see in what way the two sets I mentioned (the set of ordered pairs of real numbers and the set of ordered pairs of complex numbers) are similar - they are both vector spaces (albeit over different fields). You can use vector spaces to define the concept of dimension abstractedly, so that for example you are not restricted to less than 4 dimensions when doing geometry, but I won't go on about that now.

You were confused about what I meant by "inner products", so I will attempt to explain them too. Remember that I didn't give any kind of multiplication operation for two vectors, like you get for two numbers in a field. A vector space doesn't automatically have such a thing, but a more sophisticated structure called an "inner product space" does. An inner product on a vector space V over the real/complex numbers is a function that maps two vectors onto a real/complex number, where VxV is the set of ordered pairs of elements of V, which has certain properties that are not very important at the moment. This makes V an inner product space (note that it is still a vector space, because it still satisfies all the axioms of a vector space) . The scalar product that you have just learnt is the example of an inner product that "inspires" the general definition, and it turns the vector space of ordinary vectors in three dimensions over the real numbers into an inner product space. Now you will soon learn that two vectors are orthogonal (perpendicular) if their scalar product is zero. This is no accident - the scalar product has been defined to make that the case. But often, we don't intuitively know what it means for vectors in a certain vector space to be orthogonal (eg vector spaces of functions - what does it mean for two functions to be orthogonal??), so if we have an inner product defined on the vector space, (so that it is an inner product space), we define two vectors to be orthogonal if their inner product is zero, ie we decide to take the sentence "x and y are orthogonal" to mean "the inner product of x and y is zero." That is why you have to "say what you mean by orthogonal".

There is an awful lot more I could tell you about vector spaces and their properties, and particularly about the special maps between them, the linear maps, (see previous reply), but I realise that I've said rather a lot already. I hope this explanation makes more sense than the last one did, although I've necessarily missed large chunks out. Find a book on linear algebra if you want to know the exact list of axioms for a vector space, and what things like dimension actually are. Or you could get back to me, if you want. The one thing you should understand above all the details is that mathematics is a lot about generalizing familiar, intuitive ideas, like sets, numbers, vectors, functions, shapes, etc. by looking at the logical consequences of the simplest rules that define these objects, without worrying about the physical or intuitive