Confused about the absolute value of a complex number

[torquerotates]

Let z be the complex number: x+iy. Then |z|^2=x^2+y^2 according to my book. But according to the general definition of absolute value, |a|=(a^2)^.5. So letting z=a=x+iy. |z|^2=z^2=x^2+2ixy-y^2

This is not equal to x^2+y^2. I'm confused.

 

[NoMoreExams]

$$|x| = \sqrt{x^2}$$ is only true for $$x \in \mathbb{R}$$. You can think of Complex numbers as a 2-tuple of Reals.

In general

if

$$x = (x_1, x_2, ..., x_n)$$

then

$$|x| = \left(x_1^2 + x_2^2 + ... + x_n^2\right)^{\frac{1}{2}}$$

This might be of interest to you: http://en.wikipedia.org/wiki/Norm_(mathematics)#Euclidean_norm

 

[PowerIso]

In complex analysis |z| is called the the modulus. Since the complex field is not well ordered (meaning we can't say stuff like 3<5) then we cannot generalize the real value absolute value into the complex field. Instead we define |z| = Sqrt[x^2 + y^2]. This is consistent with the real valued absolute value when y = 0.

 

[Hurkyl]

In complex analysis |z| is called the the modulus.

"absolue value" and "norm" are frequently used too.

cannot generalize the real value absolute value into the complex field.

But, but, you just did! See:

Instead we define |z| = Sqrt[x^2 + y^2]. This is consistent with the real valued absolute value when y = 0.

 

[Tac-Tics]

Let z be the complex number: x+iy. Then |z|^2=x^2+y^2 according to my book. But according to the general definition of absolute value, |a|=(a^2)^.5. So letting z=a=x+iy. |z|^2=z^2=x^2+2ixy-y^2

This is not equal to x^2+y^2. I'm confused.

The idea of the absolute value is related to the (Euclidean) distance of the point. That means, if we treat each component as a dimension in boring-old R^n, the absolute value is the distance from the point in question from the origin. This is the definition you should start with.

For reals, we don't have any complex part, so we're really working on R^1. Take any point x. The absolute value of x, |x|, is the distance between x and 0. In other words, sqrt(x^2).

For the complex numbers, we have another component, the imaginary part, as well as the real part. Take any point c. The absolute value of c, |c|, is the distance between c and 0. We're working in R^2 now, so we use the pythagorean theorem or the distance formula or whatever you want to call it. |c| = |a + bi| = sqrt(a^2 + b^2).

Now, there are other things for which this definition continues to work, too. Quaternions are a system of numbers where we have THREE complex values, i, j, and k, which are all distinct and unique. They satisfy a few interesting properties: i*i = j*j = k*k = i*j*k = -1. Like matrices, quaternions don't commutative on multiplication, and for any two quaternions p and q, q*p need not equal p*q. But the full details aren't important.

Take any quaternion q = x + a*i + b*j + c*k. We have two use four components now, as opposed to 2 for the complex numbers and only 1 for the reals. So we're working with distance in R^4. The formula for distance in R^4 follows suit with R^2: it is the root of the sum of the squares. So |q| = |x + a*i + b*j + c*k| = sqrt(x^2 + a^2 + b^2 + c^2).

It's like magic!

We can even use weirder spaces, such as the linear space of polynomials. The polynomials are functions defined like f(x) = a_n x^n + ... + a_1 x_1 + a_0. We can add two polynomials f and g together to create a new polynomial f+g defined by (f+g)(x) = f(x) + g(x). We can also multiply a polynomial f with a scalar a to get a new polynomial a*f defined by (a*f)(x) = a * f(x).

We can define the "absolute value" of a polynomial (we might call it a "norm" instead) in the exact same way! Any polynomial of order n (where the highest exponent of x is x^n) is defined uniquely by n+1 real numbers. So |f| = |x -> a_n x^n + ... + a_1 x + a_0| = sqrt(a_n^2 + ... + a_1^2 + a_0^2).

So when you think of |x| for a real x as sqrt(x^2), remember that this is only because real numbers only have one real component (themselves). For higher-dimensional objects, you'll have the more general form |x| = sqrt(a_0^2 + a_1^2 + ... + a_n^2).