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**R->P and P->R for the 35s, Reloaded**

*Posted by Frido Bohn on 26 Jan 2010, 4:06 a.m.*

It is true that the HP35s does not offer a direct access to polar-rectangular conversion. However, you may do this with the capabilities of the calculator by taking advantage of the display of complex numbers as real numbers with an imaginary part (x, *i*y) or in polar notation (r, "theta" degrees).
Be aware of the angular representation: degrees, radians etc accessible via "MODE". Theta is accessible as a blue arrowed key in the second row, 4th column with the appearance of a strikethrough "O".

To convert rectangular to polar coordinates (assuming RPN mode, and angular degrees):

1. Choose "DISPLAY"; item 10 ".0" for the display in polar notation.

2. Enter the rectangular coordinates as x*i*y; press "ENTER"

For example type in "8*i*6", "ENTER" and you will get 10"theta"36.87 instantaneously which represent the polar notation with an r of 10 and an angle of 36.87°.

For the polar to rectangular conversion just do the opposite:

1. Choose "DISPLAY"; item 9 "9" for the display of complex numbers "x*i*y"

2. Enter the polar coodinates as r"theta"°; press Enter

For example "7.07"theta"45" gives "5*i*5" (5 on x-axis and 5 on y-axis).

Another way to yield polar coordinates from rectangular notation is to type in "x*i*y" and calculate "ABS" to obtain r, and "ARG" to obtain the angle.

Note: "x*i*y" has to be duplicated in the stack as the complex number is destroyed by the calculation of the absolute or argument. After typing "ABS", type "x<>y" then type "ARG". Of course, I should disclose a more elegant approach using the "LAST X"-command.

Addendum:

The conversion of rectangular to polar notation earned only a weak hint in the HP35s User's Guide. On page 9-6 you will find that results of complex numbers are dependent from the option set in "DISPLAY".

If you refer to the training modules for the HP35s (HP35s Training Modules, as per 02-Feb-2010), you will find in the chapter about complex numbers (Part 1) on page 2 a note how to use complex numbers for the conversion, although no example is given.

Interestingly, the apparent lack of a comprehensible rectangular to polar conversion is a main complaint issued by users in the comments on the official site of the HP35s.

Although it has nothing to do with the rectangular to polar conversion, the handling of negative roots by this calculator is whimsical. It might have been quoted elsewhere, but I will repeat this because it fits so well into the context of imaginary numbers.

Ever tried to get a result of the square root of -2 by typing "2" "+/-" "SQRT X"? The answer you get is "SQRT(NEG)", which of course is not false, but is not just what you expected.

Also the use of "2" "+/-" "ENTER" "2" "ROOT X,Y" (accessible via yellow arrow above "SQRT X") is vain. All you get is "INVALID y^x". The 3rd option would be to raise the power of 0.5 by typing "2" "+/-" "ENTER" "0.5" "Y^X". Again the result is " INVALID y^x ". Now try the complex-number notation "2" "+/-" "i" "0" "ENTER" "0.5" "y^x", and miraculously you will get the correct answer "0*i*1.414". Try the same use of complex notation of a real negative number with "SQRT" or "ROOT X,Y" and you will get "INVALID DATA".

The conclusion is that the HP35s is able to handle the root of negative numbers, but you have to use complex notation and the "y^x"-command. The other commands related to the raise of powers or handling of roots are not useful in this context. For me it appears that the internal circuitry for "SQRT" and "ROOT X,Y" came from the simplest calculator design, whereas the "y^x"-command has a more noble origin. Try "1" "e^x" "PI" "I" "*" "y^x" and you get somewhat like "-1.0*i*-1.267E-12" which is of course a fair approach of such a calculator to Euler's identity.

Kind regards

Frido

*Edited: 2 Feb 2010, 5:47 a.m. *