What does it mean to take the absolute of an complex number? And why is it always positive?

Hi,

I have a number which is part imaginary or complex.

-0.993061325012476 + 1.03511742045580e-16i

When I use the function abs(x) on this number like so:

abs(-0.993061325012476 + 1.03511742045580e-16i)
ans =
      0.99306
      

Then I receive a positive number.

Can someone kindly explain to me how this works and if the abs(x) function makes it positive or if the abs(x) function simply never returns negative numbers. If so why?

Or is it maybe the case that I need to be better versed in complex numbers in general?

Thanks!

Answer 1

[Steven Lord]

I think one potential source of confusion is that the abs of this particular complex number looks an awful lot like its real part.

n = -0.993061325012476 + 1.03511742045580e-16i;
R = abs(n)

R = 0.9931

That's not due to abs simply taking the real part of the complex number, it's due to the fact that the imaginary part of this number is extremely small. If you were to plot this number you'd see that it's very close to being on the real line.

setupFigure();
% This plots using the real part of n as the x coordinate
% and the imaginary part of n as the y coordinate. I made the marker
% larger so it stands out
plot(n, 'bo', 'MarkerSize', 10)

If we plotted another point, one whose real and imaginary parts were both not small, you'd see the abs of that number was the distance between it and the origin. For the point n2 the red dotted line is the line whose length is measured by taking abs(n2).

setupFigure();
n2 = 1+1i;
plot(n, 'bo', 'MarkerSize', 10)
plot(n2, 'ks', 'MarkerSize', 10)
plot([0 n2], 'r:', 'LineWidth', 2)

fprintf("The red dotted line is %g units long.\n", abs(n2)) % sqrt(2)

The red dotted line is 1.41421 units long.

function setupFigure()
% Set up the plot to look nice
figure
axes('XAxisLocation', 'origin', 'YAxisLocation', 'origin')
axis([-1, 1, -1, 1])
xlabel('real(n)')
ylabel('imag(n)')
xticks(-1:0.25:1)
yticks(-1:0.25:1)
grid on
hold on
end

Answer 2

[Torsten]

abs(z) where z=x+i*y is the distance of the point (x/y) from the point (0/0), thus always >=0 and =0 only if z = 0.

[lil brain]

Thanks Torsten but does that mean that the distance actually does not change when using abs? Other then from negative to positive? What does the abs do in this case? Thanks!

[Torsten]

Why should anything change ?

z=x+i*y is a static point in the x-y plane. If you interprete C as R^2, then this point z is P=(x/y). And abs(z) returns the distance of P from (0/0), thus sqrt(x^2+y^2).

Answer 3

[Walter Roberson]

syms a b real
c = a + b*1i

c = 

d = abs(c)

d = 

With the real and imaginary components each being real, their squares are going to be non-negative numbers, and the sum of non-negative numbers is non-negative, and the square root of a non-negative number is non-negative

[lil brain]

Does that mean that the value/distance actually does not change when using abs? Other then from negative to positive?

[lil brain]

What does the abs function do in this case? Thanks!

[Walter Roberson]

The abs() function does the calculation shown for complex values, except possibly it might internally use hypot() for better accuracy with small values.

[Walter Roberson]

absolute value is the Euclidean magnitude function. If you represent the points as ordered tuples of components, < x1, x2, x3,... xn > for a vector in n dimensions, then absolute value is sqrt(sum of (x(K)^2), K=1 to n). In the case of complex numbers which you could represent as the tuple < xreal, ximaginary > then sqrt(xreal^2 + ximaginary^2)

This is a formal Distance, and formal Distance are never negative.

Perhaps you might be wondering about the actual implementation. The actual implementation for scalar real IEEE floating point values is delegated to hardware. The algorithm is

1. check if the value represents NaN. If it does then the result is the same as the input 2. Otherwise, zero the most significant bit of the IEEE representation of the number. This works even for infinity and negative zero.

IEEE floating point representation uses "separate sign" which is 0 for non-negative values and 1 for negative values.

At the moment I do not know what the formal algorithm is for IEEE scalar complex values; the model is sqrt of the sum of squares of the components

[Walter Roberson]

For complex numbers, it appears that hypot() is used internally. hypot() is a modified sum of squares algorithm that detects when the squaring would underflow or overflow and uses alternate calculations in those cases.