Difference between revisions of "Complex number"
| Line 1: | Line 1: | ||
| − | + | The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. 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, <math> i </math>, such that <math> i = \sqrt{-1} </math>. If we add this new number to the reals, we will have solutions to <math> x^2 = -1 </math>. It turns out that in the system that results from this addition, we are not only able to find the solutions of <math> x^2 = -1 </math> 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 <math> a + bi </math> where <math> a,b\in \mathbb{R} </math> and <math> i = \sqrt{-1} </math>. The set of complex numbers is denoted by <math>\mathbb{C}</math>. The set of complex numbers contains the set <math>\mathbb{R}</math> of the [[real number]]s, but is much larger. Every complex number has a '''real part''', denoted by <math>\Re</math>, or simply <math>\mathrm{Re}</math>, and an '''imaginary part''', denoted by <math>\Im</math>, or simply <math>\mathrm{Im}</math>. So, if <math>z\in \mathbb C</math>, we can write <math>z=\mathrm{Re}(z)+i\mathrm{Im}(z)</math>, where <math>i</math> is the [[imaginary unit]]. | We are now ready for a more formal definition. A complex number is a number of the form <math> a + bi </math> where <math> a,b\in \mathbb{R} </math> and <math> i = \sqrt{-1} </math>. The set of complex numbers is denoted by <math>\mathbb{C}</math>. The set of complex numbers contains the set <math>\mathbb{R}</math> of the [[real number]]s, but is much larger. Every complex number has a '''real part''', denoted by <math>\Re</math>, or simply <math>\mathrm{Re}</math>, and an '''imaginary part''', denoted by <math>\Im</math>, or simply <math>\mathrm{Im}</math>. So, if <math>z\in \mathbb C</math>, we can write <math>z=\mathrm{Re}(z)+i\mathrm{Im}(z)</math>, where <math>i</math> is the [[imaginary unit]]. | ||
Revision as of 10:57, 8 July 2006
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 . The set of complex numbers is denoted by 
. The set of complex numbers contains the set 
of the real numbers, but is much larger. Every complex number has a real part, denoted by 
, or simply 
, and an imaginary part, denoted by 
, or simply 
. So, if 
, we can write 
, where is the imaginary unit.
As you can see, complex numbers enable us to remove the restriction of 
for the domain of 
.
The letters 
and 
are usually used to denote complex numbers.
Operations
- Addition
- Subtraction
- Multiplication
- Division
- Absolute value/Modulus/Magnitude (denoted by
). This is the distance from the origin to the complex number in the complex plane.
). This is the distance from the origin to the complex number in the Simple Example
If 
and w = c+di,
,
,
,
,
Topics
Problems
- AIME
