**Introduction**

Complex numbers have been a recurring feature on scientific calculators, and for most calculators, the calculations we can readily do on them have been limited - particularly on non-graphing scientific calculators.

If your non graphing calculator has complex number calculations, chances are this is the limit of what the calculator can do:

* Arithmetic (+, -, x, ÷)

* Square, but not square root

* Cube, but not cube root

* Rectangular/Polar conversions

* A separate function for magnitude, angle, real, and imaginary part extraction

The point of this blog is to give you some tricks to extend the calculation of complex numbers:

* Powers and principal root

* Exponential and Natural Logarithm (next week)

* Sine and Cosine (next week)

For any calculators using reverse polish notation (RPN), such the HP 35S, HP 50g, HP 15c, or even apps such as the GO-25 SCI RPN; I will present a "generalized" routine that you can use to perform the calculations. Please adjust as necessary what how your calculator works best.

Sources:

HP 41C Math Pac, Hewlett Packard, 1979

** Square of a Complex Number **

By simple algebra, we can calculate the square of a complex number as:

(a + bi)^2 = (a^2 - b^2) + 2abi

Remember that i = √-1, so i^2 = -1.

Thankfully, scientific calculators allow this calculation simply by typing in (a + bi)^2.

For calculators using RPN, try this routine: starting with b stored in Register 2 (or letter if you have a HP 35S, 48, 50g...family) and a stored in Register 1

RCL 2

RCL 1

X

2

X (imaginary part)

RCL 1

2

y^x

RCL 2

2

y^x

- (real part)

** Power (and Principal Root) of a Complex Number**

By De Moive's Theorem, we can calculate the power of any complex number as:

(a + bi)^n = r^n (cos (nθ) + i sin (nθ))

Where: r = a^2 + b^2, θ = arctan (b/a), and n is a real number

Note abs(a + bi) = r and arg(a + bi) = θ (arg is named angle in the TI calculators)

On calculators such as the Casio fx-115ES, TI-36X Pro, and the Sharp EL-W516; unless n = 2 or n = 3, straight input will cause an error message. Use the above formula as the alternate. The nice thing of both the fx-115 and the TI-36X Pro is that we can store complex numbers to variables.

So the key strokes for the fx-115ES would look somewhat like this:

MODE 2

a + bi SHIFT STO A

Then input:

Abs(A)^(n) × (cos(n arg(A))+ i sin(n arg(A)))

For the TI-36X Pro:

a + bi sto→x [x y t z a b c d key]

Then input:

Abs(A)^(n) × (cos(n angle(A))+ i sin(n angle(A)))

For the EL-W516, complex numbers cannot be stored to variables. In addition, there is angle/argument function.

MODE 3

abs(a + bi)^n × (cos(n tan^-1(b ÷ a)) + i sin(n tan^-1(b ÷ a)))

A Sample RPN Routine for (a + bi)^n

Let a be stored in Register 1

Let b be stored in Register 2

Let n be stored in Register 3

RCL 2

RCL 1

→P

RCL 3

y^x

STO 0. (temporary register)

x<>y

RCL 3

×

Enter

SIN

RCL 0

×

x<>y

COS

RCL 0

×