# Confused about the absolute value of a complex number

• torquerotates
In summary, the absolute value of a complex number is the distance from the point in question from the origin. This is the definition you should start with. However, for higher-dimensional objects, you'll have the more general form |x| = sqrt(a_0^2 + a_1^2 + ... + a_n^2).
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.

$$|x| = \sqrt{x^2}$$ 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

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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.

PowerIso said:
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.

torquerotates said:
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).

## 1. What is the absolute value of a complex number?

The absolute value of a complex number is a measure of its distance from the origin on the complex plane. It is also known as the modulus of the complex number and is always a positive real number.

## 2. How is the absolute value of a complex number calculated?

To calculate the absolute value of a complex number, you need to find the distance between the complex number and the origin on the complex plane. This can be done using the Pythagorean theorem, where the real part of the complex number represents the length of one side of the triangle and the imaginary part represents the length of the other side.

## 3. Can the absolute value of a complex number be negative?

No, the absolute value of a complex number is always a positive real number. This is because it represents the distance from the origin, which is always a positive value.

## 4. What is the significance of the absolute value of a complex number in mathematics?

The absolute value of a complex number has many important applications in mathematics, such as in solving equations, graphing functions, and representing vector quantities. It also helps in understanding the behavior and properties of complex numbers.

## 5. How does the absolute value of a complex number relate to its conjugate?

The absolute value of a complex number is equal to the absolute value of its conjugate. This means that the distance between a complex number and its conjugate on the complex plane is the same. This property is known as the conjugate symmetry property of complex numbers.

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