The complex numbers arise when we try to solve equations such as .
We know (from the Trivial Inequality) that the square of a real number cannot be negative, so this equation has no solutions in the real numbers. However, it is possible to define a number, , such that . If we add this new number to the reals, we will have solutions to . It turns out that in the system that results from this addition, we are not only able to find the solutions of but we can now find all solutions to every polynomial. (See the Fundamental Theorem of Algebra for more details.)
We are now ready for a more formal definition. A complex number is a number of the form where and is the imaginary unit. The set of complex numbers is denoted by . The set of complex numbers contains the set of the real numbers, since .
Every complex number has a real part denoted or and an imaginary part denoted or . Note that the imaginary part of a complex number is real: for example, . So, if , we can write . ( and are traditionally used in place of and as variables when dealing with complex numbers, while and (and frequently also and ) are used to represent real values such as the real and imaginary parts of complex numbers. This mathematical convention is often broken when it is inconvenient, so be sure that you know what set variables are taken from when dealing with the complex numbers.)
As you can see, complex numbers enable us to remove the restriction of from the domain of the function (although some additional considerations are necessary).
Addition and subtraction of complex numbers are similar to doing the same operations to polynomials -- add the real parts then add the imaginary parts.
Multiplication is also similar to doing the same operations to polynomials -- use the distributive property and apply . For division, however, the denominator needs to be a real number; this is done so by multiplying the complex conjugate, where the sign of the imaginary part is swapped. The complex conjugated is denoted by .
The absolute value (or modulus or magnitude) of a complex number is the distance from the complex number to the origin. It is denoted by .
The argument of a complex number is the angle formed between the line drawn from the complex number to the origin and the positive real axis on the complex coordinate plane. It is denoted by .
If and ,
In addition to the standard form , complex numbers can be expressed in two other forms.
The trigonometric form of a complex number is denoted by , where equals the magnitude of the complex number and (in radians) is the argument of the complex number.
The exponential form of a complex number is denoted by , where equals the magnitude of the complex number and (in radians) is the argument of the complex number.
- 1984 AIME Problem 8
- 1985 AIME Problem 3
- 1988 AIME Problem 11
- 1989 AIME Problem 14
- 1990 AIME Problem 10
- 1992 AIME Problem 10
- 1994 AIME Problem 8
- 1994 AIME Problem 13
- 1995 AIME Problem 5
- 1996 AIME Problem 11
- 1997 AIME Problem 11
- 1997 AIME Problem 14
- 1998 AIME Problem 13
- 1999 AIME Problem 9
- 2000 AIME II Problem 9
- 2002 AIME I Problem 12
- 2004 AIME I Problem 13
- 2005 AIME II Problem 9
- 2009 AIME I Problem 2
- 2011 AIME II Problem 8