TI-Nspire - Complex Numbers

Complex Numbers

The graphing calculator can be a very useful tool for checking your work with complex numbers.
Keep in mind, when working with a graphing calculator,
that there may be more than one way to arrive at an answer.

First, we need to be sure that we have set the calculator to work with complex numbers.

Enable the Use of Complex Numbers

, #5 Settings, #2 Document Setting

Under Real or Complex; Choose "Rectangular"

Hit OK.

The "rectangular" mode is this calculator's form of "a+bi" mode.

Changing the "Document Setting" will keep your calculator in "Complex mode" for upcoming work.
The new settings become "default" settings.
(There is no "Make Default" button on the TI-Nspire CX II calculator.)

Since "Complex" mode includes "Real" mode,
keeping your calculator in "Complex" mode will be advantageous for work with Algebra 2.

Entering Complex Numbers
(How to enter the imaginary i )

NOTE: The letter "i" typed off the alphabetic keys is NOT an activated imaginary "i".

Method 1: Pres the key
and this chart will appear. Choose i.

Method 2: use symbols' chart
Press

Note:

is found above

Method 3: if you type "@i" and hit you will activate imaginary i
The @ comes from the symbol chart and the i is from the keypad. When you hit "enter" the @ symbol will disappear, leaving an activated i.

Now, let's look at the arithmetic of complex numbers:

Examples at the right:

Add: (2 + 4i) + (3 - 2i)

Subtract: (6 - 3i) - (4 + 5i)

Multiply: (3 + 2i) • (4 - 2i)

Divide: (2 + 3i) / (4 - 3i)

shows decimal approximation for the division. In this case, the "approximation" is exact.

Things to remember:
• The calculator automatically removes un-needed parentheses (even though you type them in).
• The imaginary i will always appear on the screen in bold print.
• Under division of complex numbers, the calculator will return a + bi form (meaning it has already done the task of applying the conjugate for division.

Using the calculator to investigate powers of i:

We know that the powers of the imaginary i appear in a cyclic pattern of four:
i , -1, -i, 1, i, -1, -i, 1, ...
The calculator validates this pattern.

To see a listing of several powers at one time, raise i to a list {of the powers},
Remember, the parentheses used for mathematical "lists" are called curly brackets or braces { }.

To activate curly brackets, use

.
The curly braces are "above" the right parenthesis.

Special Calculator Functions for Complex Numbers:

There are also special functions on the graphing calculator to deal with complex numbers
(but you probably won't need a calculator for many of these functions):
#2 Number, #9 Complex Number Tools

1. Complex Conjugate:
conj( returns the complex conjugate of a complex number. conj(2+5i) gives 2-5i

2. Real Part:
real( returns the "a" value in an a+bi complex number. real(2+5i) gives 2

3. Imaginary Part:
imag( returns the "b" value in an a+bi complex number. imag(2+5i) gives 5

4. Polar Angle:
angle( returns the angle, or argument, of the complex number - the angle formed by the positive x-axis (the positive real axis) and the segment from the origin to the complex number point on an Argand diagram.
   angle(2+5i) gives 1.190 (with calculator set in radian mode)
   angle(2+5i) gives 68.199º (with calculator set in degree mode)

5. Magnitude
|   | returns the magnitude (or absolute value) of the complex number.
   abs(5+12i) gives 13
   abs (2 + 5i) gives 5.385164807

(Note: The magnitude of a complex number may also be called its absolute value. It you plot a complex number as a single point, the magnitude (absolute value) represents the distance from the origin to that point. If you plot a complex number as a vector, the magnitude (absolute value) represents the length of the vector.)

Items 6 and 7 will be investigated in a future course.

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