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Complex numbers.

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?.

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Complex numbers.

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

  2. Introduction to Complex Numbers

  3. Fundamental Problem of the Number System No number satisfies:

  4. 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 x2=-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 x2=-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’?

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

  6. 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

  7. 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 iis not included in the imaginary part, just the number. is called the Cartesian (rectangular) form of a complex number z.

  8. Imaginary part Real part Complex Numbers • Combine real numbers with imaginary numbers

  9. is called an imaginary unit so that Therefore it is possible to solve an equation z2 + 1 = 0 over the field of complex numbers as follows:

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

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

  12. EQUALITITY OF COMPLEX NUMBERS Two complex numbers z1 = a + bi and z2 = c + di are equal when their real parts are equal and their imaginary parts are equal:

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

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

  15. Powers of i Geometrical representation on Argand diagram?

  16. Operations on Complex Numbers • Complex numbers can be combined with • addition • subtraction • multiplication • division • Consider

  17. How would we go about adding two Complex Numbers for example:

  18. Now let’s see what happens when we subtract Complex Numbers.

  19. Multiplication of Complex Numbers in Cartesian form OR 3 OR

  20. 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.

  21. Powers of i

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

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