# 8.6: Complex Variables

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The number $$\sqrt{-1}$$ is predefined in MATLAB and stored in the two variable locations denoted byiandj. This double definition comes from the preference of mathematicians for using $$i$$ and the preference of engineers for using $$j$$ (with $$i$$ denoting electrical current).iandjare variables, and their contents may be changed. If you typej = 5, then this is the value forjandjno longer contains $$\sqrt{-1}$$. Type inj = sqrt(-1)to restore the original value. Note the way a complex variable is displayed. If you typei, you should get the answer

i =
0+1.0000i.

The same value will be displayed forj. Try it. Usingj, you can now enter complex variables. For example, enterz1 = 1+2*jandz2 = 2+1.5*j. Asj is a variable, you have to use the multiplication sign*. Otherwise, you will get an error message. MATLAB does not differentiate (except in storage) between a real and a complex variable. Therefore variables may be added, subtracted, multiplied, or even divided. For example, type inx = 2, z = 4.5*j, and z/x. The real and imaginary parts of z are both divided byx. MATLAB just treats the real variablex as a complex variable with a zero imaginary part. A complex variable that happens to have a zero imaginary part is treated like a real variable. Subtract2*jfromz1 and display the result.

MATLAB contains several built-in functions to manipulate complex numbers. For example, real (z) extracts the real part of the complex number z. Type

≫ z = 2+1.5*j, real(z)

to get the result

z =
2.000+1.500i

ans =
2

Similarly, imag(z) extracts the imaginary part of the complex number z. The functions abs(z) and angle(z) compute the absolute value (magnitude) of the complex number z and its angle (in radians). For example, type

≫ z = 2+2*j;
≫ r = abs(z)
≫ theta = angle(z)
≫ z = r*exp(j*theta)

The last command shows how to get back the original complex number from its magnitude and angle. This is clarified in Chapter 1: Complex Numbers.

Another useful function, conj (z), returns the complex conjugate of the complex number z. If z = x+j*y where x and y are real, then conj (z) is equal to x-j*y. Verify this for several complex numbers by using the function conj (z).

This page titled 8.6: Complex Variables is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Louis Scharf (OpenStax CNX) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.