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...)
(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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-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.