Title: Complex Numbers Quadratic Equations in the Complex Number System
1 Section 1.3
Complex Numbers Quadratic Equations in the Complex Number System
2 IMAGINARY NUMBERS Definition The number i called the imaginary unit is the number such that i2 -1. 3 COMPLEX NUMBERS If a and b are real numbers and i is the imaginary unit then a bi is called a complex number. The real number a is called the real part and the real number b is called the imaginary part. 4 COMPLEX NUMBER ARITHMETIC If a bi and c di are complex numbers then 5 COMPLEX CONJUGATES 6 PRODUCT OF CONJUGATES 7 DIVIDING COMPLEX NUMBERS To perform the division we multiply the numerator and denominator by the conjugate of the denominator. Then simplify the complex number into standard form. 8 PROPERTIES OF CONJUGATES
The conjugate of the conjugate is the number itself.
The conjugate of a sum is the sum of the conjugates.
The conjugate of a product is the product of the conjugates.
9 POWERS OF i i 1 i i 5 i i 2 -1 i 6 -1 i 3 -i i 7 -i i 4 1 i 8 1 And so on. The powers of i repeat with every fourth power. 10 THE QUADRATIC FORMULA In the complex number system the solutions of the quadratic equation ax2 bx c 0 where a b and c are real numbers and a 0 are given by the formula 11 CHARACTER OF THE SOLUTIONS OF A QUADRATIC EQUATION In the complex number system consider a quadratic equation ax2 bx c 0 with real coefficients. 1. If b2 - 4ac gt 0 there are two unequal real solutions. 2. If b2 - 4ac 0 there is a repeated real solution a double root. 3. If b2 - 4ac lt 0 the equation has two complex solutions that are not real. These solutions are conjugates of each other.
PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.
You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!
For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!