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Formula to Convert Complex Numbers from Polar to Rectangular Form in Excel(vice versa)

Estimated reading: 3 minutes

In this article, we will learn to convert complex numbers from rectangular to polar form and vice versa.

To convert the rectangular form to polar form, we need knowledge of the following theory of complex numbers:

complex number to polar form excel

How to Create a complex Number in Excel:

In Excel, a complex number can be written using the COMPLEX function

=COMPLEX(real_num, imag_num)

Where

real_num: The complex number’s real coefficient. This argument is required.

imag_num: The complex number’s imaginary coefficient. This argument is not required. The default value is zero if omitted.

Excel Function to Calculate Absolute or Magnitude of Complex Numbers

Magnitude of complex number can be found using the formula given below

=IMABS(inumber)

Where, 

inumber: The complex number whose absolute value is to be returned

Excel Formula to Calculate Argument or Phase of Complex Numbers

Angle or phase of  complex number can be determined using the following formula

=IMARGUMENT(inumber)

Where, 

inumber: The complex number whose phase is to be returned

Excel Formula to Convert a Complex Number from Rectangular Form to Polar Form

Suppose a complex number is given in cell , Now we will convert it into polar form using the formula below

=IMABS(B3)&"(cos "&IMARGUMENT(B3)&" + isin "&IMARGUMENT(B3)&")"
complex number to polar form excel example
  • IMARGUMENT function returns the phase of the complex number in radians, we will inject the DEGREES function to convert radians to degree
=IMABS(B3)&"(cos "&DEGREES(IMARGUMENT(B3))&" + isin "&DEGREES(IMARGUMENT(B3))&")"
complex number rect to polar form excel
  • To limit the precision, we will apply the ROUND function. and a degree symbol is added for a better look. Ampersand (&) is added to concentrate all text values.
=ROUND(IMABS(B3),2)&"(cos "&ROUND(DEGREES(IMARGUMENT(B3)),2)&"°"&" + isin "&ROUND(DEGREES(IMARGUMENT(B3)),2)&"°)"
complex number rectangular to polar form excel

Excel Formula to Convert a Complex Number From Polar to Rectangular Form

To  convert a complex number from polar to rectangular form we have to determine the following parts of the rectangular form

X = r.cos(θ)

Y = r.sin(θ)

In Excel, we can apply the following formulas to obtain our purposes: 

  • To calculate the value of X
=ROUND(COS(RADIANS(MID(B3,10,4)))*LEFT(B3,4),0)
complex number  polar to rectangular form excel
  • To calculate the value of Y
=ROUND(SIN(RADIANS(MID(B3,24,5)))*LEFT(B3,4),0)
complex number  polar to rectangular form excel example

Explanation(for Y):

  • MID(B3,24,5): retrieves a substring of five characters from cell B3, beginning at character 24.
  • RADIANS(MID(B3,24,5)): Angles are converted from degrees to radians using the RADIANS function.
  • The formula SIN(RADIANS(MID(B3,24,5))) determines the sine of the obtained angle in radians.
  • From cell B3‘s leftmost four characters, LEFT(B3,4) extracts them (which implies the value of magnitude r). Then multiplied with the SIN(RADIANS(MID(B3,24,5)))
  • The result is rounded to the nearest whole number using the formula ROUND(SIN(RADIANS(MID(B3,24,5)))*LEFT(B3,4),0).

A similar, explanation is also applied for the formula to obtain the value of X

Then Just Apply the COMPLEX Function as Below

=COMPLEX(C3,D3,"i")
complex number  polar to rectangular form excel easy
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