Simple Stuff About Complex Numbers

 

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If you are new to electrochemical impedance, or at least new to doing math with impedance values, here are a couple of simple facts about the arithmetic of complex numbers. More hints can be found at "Computing Impedance Values". Bard & Faulkner has a useful math appendix also.

Before you rush off to write a computer program to do arithmetic with complex numbers, consult Numerical Recipes There are tips and techniques to make things more "computable," i.e.,  how to compute values without crashing your program! See Sec 5.4.

The symbol i or j is used to represent the square root of -1. American electrochemists prefer j since it is not confused with current ( i ).

j=sqrt(-1); j^2=-1

The magnitude of a complex number is generally thought of as a positive (real) number and is the square root of the sum of the squares of the real and imaginary parts. It is sometimes called the modulus and often shown as | (a+jb) |.

The complex conjugate of a complex number simply negates the imaginary part.

To add or subtract two complex numbers, add or subtract the real and imaginary parts individually.

(a+jb)+-(c+jd)=(a+-c)+j(b+-d)

To multiply two complex numbers, multiply all cross terms and then collect real and imaginary components. Remember that j2=-1.

(a+jb)(c+jd)=(ac-bd)+j(ad+bc)

To invert a complex number, multiply top (numerator) and bottom (denominator) by the complex conjugate of the denominator. The is the equivalent of multiplying by one (1), so the value is unchanged.

1/(a+jb)

To divide one complex number by another, remember that it is the same as multiplying by the inverse of the denominator, and to combine the two hints, above.

(c+jd)/(a+jb)

 

 

 


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