- TI89

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TI83

HP48G series - Note: For those with HP48G calculators, lots a additional material on a variety of computational topics can be found here.

When adding two complex numbers manually (as opposed to graphically
or with an electronic device) ALWAYS put the numbers in **rectangular**
form first.

To add them, add the real parts together to give the real part
of the sum, and add the imaginary parts to give the imaginary
part of the sum.

In general:

IF

z_{1}= a + jb

and

z_{2}= c + jd

then

z_{1}+ z_{2}= (a+c) + j(b+d)

Graphical addition of complex numbers is very simple, although the accuracy usually suffers a little. Note that in this case, the form of the numbers (polar or rectangular) does not matter so long as you can succesfully plot them in the complex plane. Once the complex numbers are plotted as vectors in the complex plane, you merely perform vector addition to get the sum. You may visualize this in any way you wish (assuming it is valid), for example completing a parallelogram of placing the vectors tail to head. Follow the link below to see graphical solutions to the three examples shown.

Examples:

(3 + j5) + (7 + j3) = 10 + j8

(0.4 - j2) + (-1 + j3) = -0.6 + j1

(-18 - j39) + (7 - j3) = -11 - j42

Graphical Solutions

Subtraction of complex numbers is essentially identical to
addition. ALWAYS put the numbers in rectangular form first.

To subtract, separately subtract the real parts and the imaginary
parts.

In general:

IF

z_{1}= a + jb

and

z_{2}= c + jd

then

z_{1}- z_{2}= (a - c) + j(b - d)

Graphical subtraction of complex numbers is essentially identical to addition, except that the one vector (the minuend) is added to the negative of the other (the subtrahend). Finding the negative of a complex number graphically is explained on the page arrived at via the link to graphical solutions below.

Examples:

(3 + j5) - (7 + j3) = - 4 + j2

(0.4 - j2) - (-1 + j3) = 1.4 - j5

(-18 - j39) - (7 - j3) = - 25 - j36

Graphical Solutions

When multiplying two complex numbers manually, it is considerably
easier to put the numbers in **polar** form first.

To multiply two complex numbers, multiply the magnitudes to give
the magnitude of the product, and add the angles to give the angle
of the product.

In general:

IF

z_{1}= A e^{jq}

and

z_{2}= B e^{jf}

then

(z_{1}) (z_{2}) = AB e^{j(q+f)}Examples:

(5 e^{j3}) (0.3 e^{-j2}) = (1.5 e^{j1})

(0.02 e^{-j4}) (0.3 e^{-j0.5}) = (0.006 e^{-j4.5})

(125 e^{j0.1}) (80 e^{-j0.49}) = (10000 e^{-j0.39})Multiplication, division, and inversion (see below for the latter two) cannot really be done graphically. Although the graph may help you visualize the angle of the solution, there is no method to "construct" the resultant vector as there is for addition and subtraction.

When dividing two complex numbers, put both numbers in **polar**
form first.

Division is very similar to multiplication, except you divide
the magnitudes and subtract the angles.

In general:

IF

z_{1}= A e^{jq}

and

z_{2}= B e^{jf}

then

(z_{1}) / (z_{2}) = (A/B) e^{j(q - f)}Examples:

(5 e^{j3}) (0.4 e^{-j2}) = (12.5 e^{j5})

(0.02 e^{-j4}) (0.4 e^{-j0.5}) = (0.05 e^{-j3.5})

(120 e^{j0.1}) (80 e^{-j0.49}) = (1.5 e^{j0.59})

Inversion of a complex number is really just a special case
of division where the numerator equals 1 (purely real). You should
put the number in **polar** form first.

To invert a complex number, invert the magnitude, and __change
the sign__ of the angle.

In general:

IF

z_{1}= A e^{jq}

then

(z_{1})^{-1}= (1/A) e^{-jq}Examples:

(5 e^{j3})^{-1}= 0.2 e^{-j3}

(0.02 e^{-j4})^{-1}= 50 e^{j4})

Considered in **rectangular** form, the complex conjugate
of a complex number has the **same real part**, but the **imaginary
part** has the opposite sign.

Considered in **polar** form, the complex conjugate of a complex
number has the **same magnitude**, but the **angle** has
the opposite sign.

The complex conjugate is often denoted by using an asterix (z^{*})

In general:

IF

z = a + jb

then

z^{*}= a - jbExample:

(4 - j3)^{*}= 4 + j3

IF

z = A e^{jq}

then

z^{*}= A e^{-jq}Example:

(5 e^{-j2})^{*}= 5 e^{j2}

To find the complex conjugate graphically, simply reflect it across the real axis.

Graphical Solutions to above examples.

Note that if a complex number is added to its complex conjugate, the sum is purely real, with a magnitude twice that of the real part of the original number.

Complex Number Index

ENGR 130 Main Page