Trigonometry: The Complex Number System
This is a free lesson from our course in Trigonometry
 
   
This lesson is all about complex numbers, their operations and their properties. Complex numbers are introduced with the help of some examples and relations to rectangular coordinates. A complex number is any number of the form x + yi, where x and y are real numbers and i is the imaginary unit i.e. i = -1. We can pretty much treat 'i' as a variable in an algebraic expression and all algebraic rules are still to be followed in operations involving a complex number. In effect, the complex number system is an extension of the real number system. All real numbers are complex numbers with y = 0 in the expression (x + yi). Note that all real numbers are complex numbers, but all complex numbers are not real. (More text below video...)
<h2> Trigonometry - The Complex Number System</h2> <p> Complex numbers, operations and properties, equation, standard form, expressions, video, coordinate, trigonometry help, positive, graph, example, solution, add, subtract, standard form, conjugate, complex number system, real number, rectangular form of complex numbers, imaginary unit, numerator, denominator, x axis, y axis, complex plane, associative law, commutative law, distributive law, practice questions, quizzes</p> <p> Description" content="A complex number is any number of the form x + yi, where a and b are real numbers and i is the imaginary unit i.e. i = sqrt -1 . The conjugate of a complex number x + iy is a complex number equal to x -iy.</p>
Other useful lessons:
Polar Form of Complex Numbers
Multiplication of Complex Numbers

(Continued from above) The conjugate of a complex number x + iy is a complex number equal to x - iy. In a real number system, equation x2 = -1 has no solution because the square of a number is positive, but in the complex number system there is a solution to the equation x2 = -1, called the imaginary unit denoted by i, and is defined as i2 = -1.

A complex number can also be graphed on the complex plane, which consists of a real axis 'x' and an imaginary axis 'y'. For example, the complex number 3 + 4i has coordinates (3, 4) on the complex plane.
Complex numbers are added, subtracted, and multiplied by applying the associative, commutative and distributive laws, together with i = -1. To add or subtract complex numbers, add the real parts together and the imaginary parts together. For example, to convert the expression (3 + 2i)/(10 - i) into standard form, multiply its numerator and denominator by the conjugate of the denominator (10 - i) i.e., 10 + i, and then simplify the resulting expression to get (28/101) + (23/101)i.

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