3AComplexNums


 * Complex Numbers **

Complex numbers were first developed between 1545 and 1572 by Italian mathematicians (Rafael Bombelli and Girolamo Cardano) as a purely theoretical idea. These ideas were adopted and have been developed since then into a large area of study within mathematics.

Complex numbers are now used extensively in physics and engineering in areas such as electric circuits and the study of electromagnetic waves. They are based around an “impossible” concept – the square root of negative numbers.

Since the square root of a negative number can’t exist in reality, we have to “imagine” it. For this reason, they are often called __**imaginary numbers**__.

**i**

By defintion math . \qquad i=\sqrt{-1} math

We don’t need to define any other values because all other square roots of negative numbers can be written in terms of **i**.

math \\ . \qquad \text{Eg: } \sqrt{-16}=\sqrt{16}\times\sqrt{-1}=4i \\. \\ . \qquad \text{Eg: } \sqrt{-10}=\sqrt{10}\times\sqrt{-1}=\sqrt{10}\,i math

Complex Numbers

We use C (or Z) to represent the set of Complex Numbers.,

A complex number is given by: //** z = x + iy **//
 * we often use **z** to represent a complex number, **z Î C**
 * **x** and **y** are any real numbers, **x, y Î R**
 * **x** is the __**Real**__ part of z:
 * **x = Re(z)**
 * **y** is the **__Imaginary__** part of z:
 * **y = Im(z)**


 * Example **

For the complex number: z = 2 + 3i State the real part: Re(z) and the imaginary part: Im(z)


 * Solution:**
 * Re(z) = 2
 * Im(z) = 3

NOTE : The imaginary part of z is 3 and NOT 3i.

Write as a Complex Number

From above we saw that: math \\ . \qquad \sqrt{-16}=\sqrt{16}\times\sqrt{-1}=4i \\. \\ . \qquad \sqrt{-10}=\sqrt{10}\times\sqrt{-1}=\sqrt{10}\,i math

So if we can write the following in proper complex number notation

math \\ . \qquad \quad 7 - \sqrt{-9} \\. \\ . \qquad = 7 - 3i \\. \\ \text{and} \\. \\ . \qquad \quad \sqrt{12} + \sqrt{-12} \\. \\ . \qquad = 2\sqrt{3} + 2\sqrt{3} \, i math


 * Note that surds should always be simplified and left in exact (surd) form.

Powers of i

math . \qquad i^2 = -1 math Therefore math . \qquad i^3=i^2 \times i=-i math and math . \qquad i^4=i^2 \times i^2 = -1 \times -1 = 1 math etc

For another site that explains i, go here: MathIsFun

Equal Values

Two complex numbers are equal if their real parts are equal AND their imaginary parts are equal. math . \qquad z_1 = z_2 \quad \textbf{ if}\\ . \qquad \qquad Re(z_1)=Re(z_2) \\. \\ . \qquad \qquad \quad \textbf { AND } \\. \\ . \qquad \qquad Im(z_1)=Im(z_2) math


 * Example **

Find a, b when .. 2a + 3bi = 4 + 15i


 * Solution:**

... ... ........ 2a = 4 ... {Real parts are equal} ... ... so ... . a = 2

... ... ...... 3b = 15 ... {Imaginary parts are equal} ... ... so .... b = 5

All numbers are complex numbers

A complex number is given by: z = x + iy, .. x,y Î R

If y = 0, then z is a real number

eg. The real number 4, is equal to the complex number 4 + 0i ... ... 4 = 4 + 0i

So ANY number can be written as a complex number.

The Complex Plane (Or Argand Diagrams)

A Complex Number can be represented as a point on the Complex Plane.


 * The horizontal axis shows x = Re(z)
 * The vertical axis shows y = Im(z)

The Argand Diagram works much like the normal Cartesian plane except that
 * a point on an Argand Diagram represents a single complex number
 * a point on the Cartesian Plane represents a pair of numbers

Complex Numbers on the Calculator

Your calculator can work with complex numbers.
 * Select Main from the menu.
 * On the bottom row of the screen, the third entry across should be REAL. Click on it to change it to CPLX.
 * Use **i** from the row of variables on the math keyboard. (not from the abc keyboard)
 * The ACTION menu, COMPLEX sub menu has **Re** and **Im** as options.

(Don’t forget to change your calculator back to REAL mode when you want to work in the Real number system).


 * Example **

1) .. Use the calculator to find: .. i 4 – 2i 2 + 1

... ... {answer should be 4}

2) .. Use the calculator to find: .. Re(3 + 4i)

... ... {answer should be 3}

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