Complex Numbers

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Definition

Complex numbers are numbers which take the form   a+bi  , where  a and  b are real numbers and  i = \sqrt{-1} .

Such numbers frequently appear in mathematical equations, even in those describing physical systems. Extending our notion of numbers to include complex numbers results in many astounding symmetries and relationships throughout mathematics.

Basic Properties

Complex numbers have two parts: a "real" part represented by  a and an "imaginary" part represented by  bi ; the  i factor in the imaginary part forces the two to be separate. The same operations that are used on real numbers, such as addition, subtraction, multiplication, and division, can be used on complex numbers. For example, two complex numbers are added by components, real added to real part and imaginary to imaginary part:

 (6-1i)+(3+0.5i) = 9 -0.5i\,

As another example, multiplying two complex numbers is carried out in the same way that we would multiply two real binomials:

 (3+2i) \times (1-3i) = 3-9i+2i+6 = 9-7i \,

Note that because each  i is the square root of  -1 , the product of two  i terms gives  -1 , so  2i \times (-3i) = 6 \,.

Visualizing the Complex Numbers

We traditionally visualize the real numbers, such as 2 and 0.5, as points on the number line. We can visualize real numbers this way because all real numbers can be identified by a single value. The real number 5 is unique, and has its own place on the number line. Because complex numbers have two parts, we can think of them as vectors contained in a plane. We call the plane which contains complex numbers the Argand Plane, or the Complex Plane. The y-axis represents the imaginary component of our complex number, and the x-axis the real component. The complex number  2+3i is shown below:

We visualize complex numbers in the same way we visualize an ordered pair on a plane.
We visualize complex numbers in the same way we visualize an ordered pair on a plane.
Additionally, complex numbers can be thought of as a rotation about the origin. Transforming a positive number to a negative number with the same absolute value can be thought of as multiplying it by  -1 . For example,  3*-1 rotates the number  3  180º about the origin.
Because  i is the square root of  -1 ,  3*i rotates  3  90º about the origin.
Thus two successive multiplications by  i is equivalent to multiplying by  -1 , or one 180-degree rotation about the origin:
Rotating 3 about the origin.
Rotating 3 about the origin.



We can thus speak of the magnitude of a complex number as the length of this vector, which is r = \sqrt{a^2+b^2}, as is readily shown by the Pythagorean Theorem.This vector idea leads to an important relation between trigonometry and the complex numbers. Euler's formula, which can be derived using Taylor Series, tells us that

 e^{i\theta} = \cos(\theta) +i\sin(\theta) \,.
This page's main image shows that
 \sin(\theta)=b/r \, and  \cos(\theta) = a/r \, .
Substituting gives
 e^{i\theta} = a/r +ib/r \,
or
 re^{i\theta} = a+bi \,.

Therefore all complex numbers of the form  a +bi can be expressed with an exponential function.

Relation to the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree   n has  n roots. Polynomials frequently have extra roots in addition to their real roots. For example,  2 and  -2 are the roots of the polynomial  x^2-4 . Since this is a polynomial of degree  2 , it should have two roots, and it does. But the polynomial  x^3-8 only has one real root,  2 . But by the Fundamental Theorem of Algebra, it must have three roots. The complex ones turn out to be  -1- \sqrt{-3} and  -1+ \sqrt{-3} or  -1-i \sqrt{3} and  -1+i \sqrt{3} . The Fundamental Theorem of Algebra could not have been proved without the use of complex numbers, because it is untrue if only real roots are considered.

References

Nahin, Paul J. An Imaginary Tale: The Story of [the Square Root of minus One]. Princeton, NJ: Princeton UP, 1998.

Strogatz, Steven. "Finding Your Roots." Opinionator Finding Your Roots Comments. N.p., n.d. Web. 20 Nov. 2012. <http://opinionator.blogs.nytimes.com/2010/03/07/finding-your-roots/>.

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