Thinkwell's Precalculus video tutorials feature award-winning teacher Edward Burger, who has a unique ability to break down concepts and explain examples in ways that stick with students. Our Precalculus has hundreds of video tutorials and thousands of automatically graded exercises with step-by-step feedback, covering everything in the most popular textbooks, so you have all of the pre-calculus math help you need to prepare for Calculus. Precalculus | Thinkwell account icon arrow-left-long icon arrow-left icon arrow-right-long icon arrow-right icon bag-outline icon bag icon cart-outline icon cart icon chevron-left icon chevron-right icon cross-circle icon cross icon expand-less-solid icon expand-less icon expand-more-solid icon expand-more icon facebook-square icon facebook icon google-plus icon instagram icon kickstarter icon layout-collage icon layout-columns icon layout-grid icon layout-list icon link icon Lock icon mail icon menu icon minus-circle-outline icon minus-circle icon minus icon payment-american_express icon Artboard 1 payment-cirrus icon payment-diners_club icon payment-discover icon payment-google icon payment-interac icon payment-jcb icon payment-maestro icon payment-master icon payment-paypal icon payment-shopifypay payment-stripe icon payment-visa icon pinterest-circle icon pinterest icon play-circle-fill icon play-circle-outline icon plus-circle-outline icon plus-circle icon plus icon rss icon search icon Shopify logo shopify icon snapchat icon trip-advisor icon tumblr icon twitter icon vimeo icon vine icon yelp icon youtube iconĪ 2-in-1 value: Thinkwell's Precalculus combines concepts from College Algebra with Trigonometry for a comprehensive experience of precalculus mathematics. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the 106 109 + 10 109 i Separate the real and imaginary parts. 6 – 20 i + 30 i – 100 i 2 9 – 30 i + 30 i – 100 i 2 Multiply using the distributive property or the FOIL method. 2 + 10 i 3 + 10 i ⋅ 3 – 10 i 3 – 10 i Prepare to multiply the numerator and denominator by the complex conjugate of the denominator. 2 + 10 i 3 + 10 i Rewrite the denominator in standard form. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.Ģ + 10 i 10 i + 3 Substitute 10 i for x. Suppose we want to divide c + d i c + d i by a + b i, a + b i, where neither a a nor b b equals zero. Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another. Note that complex conjugates have a reciprocal relationship: The complex conjugate of a + b i a + b i is a − b i, a − b i, and the complex conjugate of a − b i a − b i is a + b i. In other words, the complex conjugate of a + b i a + b i is a − b i. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. Dividing Complex Numbersĭivision of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator.
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