HP 35s Manual

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HP 35s Working with complex numbers – Part 2
Complex numbers
Practice working problems involving complex numbers

hp calculators
HP 35s Working with complex numbers – part 2
hp calculators - 2 - HP 35s Working with complex numbers – part 2 - Version 1.0
Complex numbers
Complex numbers occur in problems facing several disciplines, from quantum mechanics to working with magnetic
fields. They are also useful in modeling the flow of a fluid around a pipe. They even show up in the solution of a
differential equation that models the up and down movement of a car’s shock absorber. They are also used to describe
the inductance and capacitance of electrical circuits, for example, using the formula E = I x Z, where E is voltage, I is
current, and Z is impedance. In many electricity and electronics areas, the “i” of an imaginary number is usually
represented as “j” to avoid any confusion with the variable “I” which represents current in electronics formulas.
To distinguish complex numbers from real numbers, the HP 35s has a dedicated 6key, which is pressed between the
real and imaginary part of a complex number. Because the HP 35s holds an entire complex number in one stack
register, the entire 4-level stack can hold 4 complex numbers at once.
In RPN mode, the HP 35s has two “complex number” modes available. The first is the standard xiy mode, where the real
portion is input, the key pressed, and then the complex number portion is input. The second is by entering the complex
number in “polar” format or a magnitude r, then the theta symbol, followed by an angle, or simply rΘa. These are
selected using the z8menu choices 9 and 10 as shown in figure 1. To choose option 9 once z8has
been pressed, press 9. To choose option 10, press the decimal point followed by a zero.
Figure 1
In algebraic mode, the HP 35s has three “complex number” modes available. The first two modes are the same as for
RPN and are described in the preceding paragraph. The third mode which is only available in algebraic is the x+yi mode.
It is selected using the z8menu choice 11, as shown in figure 2. To choose option 11 if you have already
pressed z8, press the decimal point followed by a one.
Figure 2
Note that changing the display mode changes any previously entered complex numbers to the new format. This means
that to convert from polar to rectangular coordinates, for example, all that is needed is to change how a polar form
complex number is displayed.
The HP 35s provides a new level of ease of use when dealing with complex numbers.
Practice working problems involving complex numbers
Example 1: Compute (2+3i) * [(7-6i) + (4+5i)]. Use the x+yi display mode in algebraic mode
Solution: Put the HP 35s into algebraic by pressing 94. Then press z8Ë
46Õy44_6Õ46

hp calculators
HP 35s Working with complex numbers – part 2
hp calculators - 3 - HP 35s Working with complex numbers – part 2 - Version 1.0
Figure 3
Answer: 25 + 31i. Figure 3 shows the display in algebraic mode.
Example 2: Extract the X and Y coordinates of the complex number 5Θ30. Use degrees mode.
Solution: Changing the display mode to z89will convert the complex number to an X and Y form.
However, to extract the X and Y values, it is necessary to leave the display in polar form using the
º^ function to extract the magnitude and the ¹=function to extract the angle. Once
extracted, the X and Y coordinates can be computed for further use as follows:
X = r COS Θ
Y = r SIN Θ
In either RPN or algebraic mode:
9(Sets degrees mode)
z8Ë0
5{?0
In algebraic mode:
º^ºÍÕyn¹=ºÍ
Figure 4
5{?0º^ºÍÕyk¹=ºÍ
Figure 5
In RPN mode:
º^ºÍ¹=ny
Figure 6

hp calculators
HP 35s Working with complex numbers – part 2
hp calculators - 4 - HP 35s Working with complex numbers – part 2 - Version 1.0
wº^ºÍ¹=ky
Figure 7
Answer: The X coordinate is approximately 4.33 and the Y coordinate is 2.5. Figures 4 and 5 show the display in
algebraic mode. Figures 6 and 7 show the display in RPN mode.
Example 3: Use the HP 35s to verify the triangle inequality for two complex numbers, which states that if z and w are
any two complex numbers, then |z + w| ≤|z| + |w|.
Use the complex numbers z = 3 + 1i and w = -1+2i
Solution: 9(Sets degrees mode)
z8Ë0
In RPN mode, compute |z + w| first.
611_6
Figure 8
The magnitude of the resulting complex number will be approximately 3.6. It is computed by pressing:
º^
Figure 9
In RPN mode, now compute |z| + |w|.
61º^
1_6º^

Figure 10
Answer: The results verify the triangle inequality. The individual magnitudes of the two complex numbers added
together is larger than the magnitude of the result from adding the complex numbers together.