How to convert complex numbers to polar form?
Formula in cell D4:
Complex numbers are usually presented in this form
z = x + yi
or
z = x + yj
However, complex numbers can also be represented in polar form:
z = r*(cos θ + isin θ)
In other words, theta θ in the polar form is calculated using the IMARGUMENT based on complex numbers.
Pythagorean Theorem
r2Â = x2Â + y2
To calculate the absolute value we can use this formula:
r = √(x2+y2)
Excel has a function that does this for you, the IMABS function calculates the absolute value based on complex numbers.
Explaining formula
Step 1 - Calculate theta θ
The IMARGUMENT function calculates theta θ which is an angle displayed in radians based on complex numbers in rectangular form.
Function syntax: IMARGUMENT(inumber)
IMARGUMENT(C4)
becomes
IMARGUMENT("12+9i")
and returns
0.643501108793284
Step 2 - Calculate the absolute value
The IMABS function calculates the absolute value (modulus) of a complex number in x + yi or x + yj text format.
Function syntax: IMABS(inumber)
IMABS(C4)
becomes
IMABS("12+9i")
and returns
15
Step 3 - Join calculations with text
The ampersand character lets you concatenate values in an Excel Formula.
IMABS(C4)&"(cos "&IMARGUMENT(C4)&" + isin "&IMARGUMENT(C4)&")"
Step 4 - Evaluate the formula
IMABS(C4)&"(cos "&IMARGUMENT(C4)&" + isin "&IMARGUMENT(C4)&")"
becomes
15&"(cos "&0.643501108793284&" + isin "&0.643501108793284&")"
and returns
15(cos 0.643501108793284 + isin 0.643501108793284)
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