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Archive for June, 2009

I’m on vacation until September, and my work priority this summer is to prepare to team-teach a class on Technology & Privacy for the Winter 2010 quarter.

If you’re interested in this topic and haven’t read Privacy On the Line by Whitfield Diffie and Susan Landau and The Eavesdroppers by Samuel Dash, they are both excellent.

I’ll probably post every few weeks, but, at the moment, my priorities are elsewhere.

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In graphing real valued functions, each x value chosen is a real number, and each corresponding y value is also a real number.

Because both the x and y values are one-dimensional real numbers, the relationship can be graphed on a plane, showing the x and y values together only requires TWO dimensions.

We can use the graph of the function relationship between x and y values to solve equations.

To solve the quadratic equation

0= x²+5x+3

we can graph the function

y=x²+5x+3

and look to see what value(s) of x make the y be zero.

When we graph the function we look for the x values where the graph crosses the x-axis.  This is because the value of y is zero along the x-axis.

graph3

So the values of x that make y be zero in the graph of (y=x²+5x+3) are approximately x≈-0.70 and x≈-4.31

Frequently, we see quadratic (parabolic) graphs that don’t intersect the x-axis at all:

graph1This is because there are no real values for x that make y be zero.  In the case of the graph above (y=x²+5x+9) all the x values we see along the x-axis make y a positive number.

Does this mean that there aren’t any x values that make y be zero?

No.  If we use the quadratic formula, we can find that there are complex-valued roots that are solutions of the equation

0=x²+5x+9

In this case, (x≈-2.5±1.658i) are the complex values of x that make y zero.

If we could see x values on the Complex Plane, then we would see these roots on the graph, but, as I mentioned in a previous post, this creates some difficulties.

The picture from Wikipedia that I posted recently is a graph of a different function from the ones above.

Even though it’s not the same graph, it does show one method to try to get around the difficulties of graphing relationships in the Complex Plane.

The picture

600px-Color_complex_plot

shows the Complex Plane of all x values.  The y values are interpreted by the color and intensity of each x value on the Complex Plane.  The roots (and asymptotes) for the function in this picture are indicated by the points or holes or peaks (however you want to think of them) that we see in the picture.

I haven’t done enough research with this method of graphing to know all the details of how it is colored and how to tell the roots from the asymptotes, but I find it both visually and mathematically beautiful.

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Analysis does not owe its really significant successes of the last century to any mysterious use of √(-1), but to the quite natural circumstance that one has infinitely more freedom of mathematical movement if he lets quantities vary in a plane instead of only on a line.

Leopold Kronecker, (1894) quoted in Remmert’s Theory of Complex Functions

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This from the Wikipedia entry on Complex Analysis and it’s related to what I was writing about in the last post.

I’ll talk about some of the details later, for now it’s just pretty!

600px-Color_complex_plot

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The fact that the complex numbers are two dimensional is part of what makes them useful in electrical engineering, but, at the same time, it makes it impossible to represent mathematical relationships graphically the way we can with Real Numbers.

What?

In a standard x and y graph, numbers for x are fed into an equation (or formula), a calculation is made and an answer for a corresponding y value is computed.  These pairs of (real number) values are then graphed on the xy axes for a two dimensional visual representation of the relationship between x and y.

This works because each number itself is only one dimensional.  The two dimensional graph shows how the value of each y (or vertical) coordinate depends on each x (or horizontal) value.  In this case, each number is one dimensional, so showing them together requires only two dimensions.

In the case of Complex Analysis, each x value is two dimensional and each y value is also two dimensional – this is FOUR dimensions, which is one more that most humans can comprehend.

Although time is often considered to be a fourth dimension, I think that it is more accurate to say that we experience a fourth spatial dimension over time.  In his book Flatland, the author Edwin Abbott describes how a two dimensional creature would experience the third spatial dimension over time.

If you’re not sure how this works, that last link is worth a look, because it can help us to conceptualize how we might experience a fourth spatial dimension over time.

In the next post I’ll look at what this means for solving equations graphically, and analytic geometry in general.

This post from last week touched on these ideas.

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Gentlemen, this is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means.  But we have proved it, and therefore we know it must be the truth. -Benjamin Pierce, American mathematician

As we saw in the last post, using the Real Number system can lead to equation whose solution is not a real number.

For example:

x²=-1

We know that if we square a positive number, we get a positive number.

And, if we square a negative number, we also get a positive number.

So, what kind of number will give us a negative answer when we square it?

A COMPLEX NUMBER!

The book An Imaginary Tale, by Paul Nahin is a great source of information about complex numbers and their applications, although it can be somewhat technical for non-mathematicians.

One of the first mathematicians noted to have worked successfully with complex numbers was Girolamo Cardano, a 16th century Italian mathematician.  Cardano was one of several Italian mathematicians involved in the efforts to find formulas that would solve cubic and quartic equations in the same way that the quadratic formula is used to solve quadratic equations.

In the process of using the formula to solve cubic equations, Cardano found a step in the solution process in which negative numbers appeared under square root symbols.  Unsure as to how to proceed, Cardano figured that these numbers (if they were to interact with the Real Number system) must work in a similar way.

In An Imaginary Tale, Paul Nahin says

The fact that Cardan[o] had no fear of imaginaries themselves is quite clear from the famous problem he gives in Ars Magna, that of dividing ten into two parts whose product is forty.  He calls this problem “manifestly impossible” because it leads immediately to the quadratic equation x²-10x+40=0, where x and 10-x are the two parts, an equation with the complex roots…of 5+√(-15) and 5-√(-15).  Their sum is obviously 10, because the [square roots] cancel, but what of their product?  Cadan[o] boldly wrote “nevertheless we will operate” and formally calculated

(5+√(-15))(5-√(-15)) = (5)(5)-(5)√(-15)+(5)√(-15) -√(-15)√(-15)=25-(-15)=40

The geometric interpretation of complex numbers was developed in the centuries following Cardano’s work and led to the conception of two number lines crossed with each other to create the complex number plane.  This is one of the defining characteristics of a complex number –

it is two dimensional.

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The most basic form of mathematics is counting.

Most cultures count (although some don’t).  Thus, the simplest, most basic set of numbers is the counting numbers (1, 2, 3, ….) also known as the Natural Numbers, symbolized by a capital N.

An issue arises in algebra if we create an equation using only natural numbers whose solution is not a natural number –

x+5=2

This leads to the creation of the Integers, symbolized by a capital Z (for the German word for number – zahlen) which includes all the positive and negative whole numbers along with zero – (…, -3, -2, -1, 0, 1, 2, 3, …).

Again, a problem arises if we use integers to write an equation whose solution is not an integer.

2x=-1

This leads to the creation of the Rational Numbers (symbolized by a capital Q for quotient), which is the set of all numbers that can be represented as the ratio of two integers.  We can see a connection between the operations of multiplication and division and the set of Rational Numbers Q .

Again, we can create an equation using rational numbers whose solution is not a rational number.

x²=2

This was actually a famous problem for the Pythagoreans, who believed that all numbers were rational and could be represented as the ratio of two whole numbers.  However, the diagonal of a square whose sides are length 1 will have a diagonal whose length is the square root of 2.  When the Pythagoreans realized that this number was not rational, it caused them great concern.

The existence of irrational numbers requires the existence of a set of numbers that includes both rational and irrational numbers – the Real Numbers.  This allows us to include such irrational numbers as square and cube roots as well old favorites like pi and relative newcomer e.

I think that the best way to think of the Real Numbers is on the number line.  Every Real Number corresponds to a point on the number line and every point on the number line corresponds to a real number.  In fact, even the technical mathematical definition of Real Numbers can be visualized on the number line.

Also, we can see a connection between the processes of finding roots and exponentiation to the need to broaden the conception of numbers to the Real Numbers.

The process of finding roots leads not only to the realm of the irrational numbers, but also to that of the Complex Numbers – symbolized with a capital C.

We’ll look at the complex numbers in the next post…

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