Complex numbers Introduction to complex numbers in Cartesian

Complex numbers. Introduction to complex numbers in Cartesian (rectangular form).

Introduction to Complex Numbers

Fundamental Problem of the Number System No number satisfies:

COMPLEX NUMBERS - ABSTRACT OR REALITY? So far you are able to operate quite confidently in the field of rational numbers. When we solve equations, for example, using a quadratic formula, there are cases where we have no solutions. Do you recall when it happens? Also, when you are asked the following question: solve the equation x 2=1, you would immediately say that this equation has no solutions, since any number squared is always positive. And you will be correct but only if we solve over the field of real numbers. If we solve the same equation x 2=-1 over the field of complex numbers, there is a solution. You will discover what this solution is later in this lesson. Why do we need to have the answer to this bizarre equation? Who discovered, or invented, complex numbers? Are there any practical applications of complex numbers? Or is it just mathematicians’ new ‘game’?

Objectives: • Define a complex number in Cartesian form. • State real and imaginary parts. • Operations on complex numbers. • Complex conjugate. • Argand diagram.

The imaginary number takes mathematics to another dimension. It was discovered in sixteenth century Italy at a time when being a mathematician was akin to being a modern day rock star, when there was a lot of respect to be gained from solving a particularly 'wicked' equation. And the wicked equation of the day went like this: "If the square root of +1 is both +1 and -1, then what is the square root of -1? " Imaginary numbers are real numbers multiplied by i. If, like many, you find yourself saying "but what's the point? " then think on this. Imagine a world without electric circuits. No circuits, so no computers. No computers, so you wouldn't be reading this now. And while engineers need the imaginary number to analyse electrical waves, physicists need it to calculate the fundamental forces that govern our Universe via quantum mechanics. Do "Imaginary Numbers" Really Exist? You may find the answer following the link below. http: //www. math. toronto. edu/mathnet/answers/imaginary. html

DEFINTION OF A COMPLEX NUMBER is called the real part of a complex number z is called the imaginary part of a complex number z Note that i is not included in the imaginary part, just the number. is called the Cartesian (rectangular) form of a complex number z.

Complex Numbers › Combine real numbers with imaginary numbers Real part Imaginary part 9

is called an imaginary unit so that Therefore it is possible to solve an equation z 2 + 1 = 0 over the field of complex numbers as follows:

Example 1. Find the real and imaginary part of the following complex numbers:

Complex numbers There is no number which squares to make -1, so there is no ‘real’ answer! Mathematicians have realised that by defining the imaginary number many previously unsolvable problems could be understood and explored. What is If ? , what is: A number with both a real part and an imaginary one is called a complex number Eg Complex numbers are often referred to as z, whereas real numbers are often referred to as x The imaginary part of z, called Im z is 3 The real part of z, called Re z is 2 A complex number in the form is said to be in Cartesian form ,

EQUALITITY OF COMPLEX NUMBERS Two complex numbers z 1 = a + bi and z 2 = c + di are equal when their real parts are equal and their imaginary parts are equal:

Addition / Subtraction of complex numbers To add two complex numbers, add real parts and add imaginary parts.

Multiplication of complex numbers. To multiply complex numbers, we expand the brackets and simplify:

Powers of i Geometrical representation on Argand diagram?

Operations on Complex Numbers › Complex numbers can be combined with – – addition subtraction multiplication division › Consider 18

Multiplication of Complex Numbers in Cartesian form OR 3

Division of complex numbers. Recall surds and how we rationalise the denominator. Do you remember conjugate surds? It is a similar idea with complex conjugates.

Powers of i

Square roots of negative numbers over C. On CAS: change the Setting to Complex Rectangular