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Introduction to Complex Numbers

Long ago, mathematicians believed quadratic equations such as x 2 + 3x + 6 = 0 , which were irreducible over the reals, to be unsolvable because their solution contained a negative radical. Not until the number ı was invented could they be solved. ı = . It is called the imaginary unit. A number of the form bı , where b is a real number, is called an imaginary number.

Using ı , an equation like x 2 + 3x + 8.5 = 0 can be solved using the quadratic equation.


x =    

x =    

x =    

x =    

x = - -2.5ı, - +2.5ı    

Because ı = , = ı . Any negative radical can be expressed as ı with a real coefficient.

The powers of ı are repeating. ı0 = 1,ı1 = 2 = - 1, andı3 = - ı . Then ı4 = 1 , and the cycle repeats itself.

A complex number is a number of the form a + bı , where a and b are real numbers. a is called the real part of the complex number, and bı is called the imaginary part of the complex number. a + bı is called the standard form of a complex number. An imaginary number is a complex number whose real part equals zero. Thus 0 is the only number that is both real and imaginary.

Complex numbers can be plotted on the complex plane. Let z = a + bı . The rectangular form of the complex number z is the ordered pair (a, b) , such that the first coordinate is the real part, and the second coordinate is the coefficient of the imaginary unit of the imaginary part. In rectangular form, complex numbers can be represented by ordered pairs of real numbers. Next, let the x-axis of the rectangular coordinate plane be called the real axis, and let the y-axis of the rectangular coordinate plane be called the imaginary axis. These two axes, the real and imaginary axes, form the complex plane, in which complex numbers in rectangular form (a, b) are plotted the same way as points are plotted using rectangular coordinates.

Figure %: The complex plane