z = -7+13i M = abs(z) %magnitude Ph = angle(z) %phase angle Ph2 = atan2(imag(z),real(z)) %phase angle

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) 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) 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

See this newsgroup thread where Bruno Luong gives advice on how to do this calculation without using the symbolic toolbox: