Difference between revisions of "Complex number"
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| − | The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. | + | The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. |
| − | We | + | ==Derivation== |
| + | 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.) | ||
| − | As you can see, complex numbers enable us to remove the restriction of <math>x\ge 0</math> from the [[domain]] of <math>f(x)=\sqrt{x}</math> | + | ==Formal Definition== |
| + | 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> is the [[imaginary unit]]. 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, since <math>a = a + 0i</math>. | ||
| + | |||
| + | ==Parts== | ||
| + | Every complex number <math> z </math> has a ''[[real part]]'' denoted <math>\Re(z)</math> or <math>\mathrm{Re}(z)</math> and an ''[[imaginary part]]'' denoted <math> \Im(z)</math> or <math> \mathrm{Im}(z)</math>. Note that the imaginary part of a complex number is real: for example, <math>\Im(3 + 4i) = 4</math>. So, if <math>z\in \mathbb C</math>, we can write <math>z=\mathrm{Re}(z)+i\mathrm{Im}(z)</math>. (<math>z</math> and <math>w</math> are traditionally used in place of <math>x</math> and <math>y</math> as [[variable]]s when dealing with complex numbers, while <math>x</math> and <math>y</math> (and frequently also <math>a</math> and <math>b</math>) 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 <math>x\ge 0</math> from the [[domain]] of the [[function]] <math>f(x)=\sqrt{x}</math> (although some additional considerations are necessary). | ||
== Operations == | == Operations == | ||
| − | + | 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 <math>i^2 = -1</math>. 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 <math>\overline{z}</math>. | |
| − | + | ||
| − | + | 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 <math>|z|</math>. | |
| − | + | 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 <math>\arg(z)</math>. | |
| − | If <math>z=a+bi</math> and | + | === Examples === |
| + | |||
| + | If <math>z=a+bi</math> and <math>w = c + di</math>, | ||
* <math>\mathrm{Re}(z)=a</math>,<math>\mathrm{Im}(z)=b</math> | * <math>\mathrm{Re}(z)=a</math>,<math>\mathrm{Im}(z)=b</math> | ||
* <math>|z|=\sqrt{a^2+b^2}</math> | * <math>|z|=\sqrt{a^2+b^2}</math> | ||
| − | * <math>\ | + | * <math>\overline{z}=a-bi</math> |
| − | |||
* <math>z+w=(a+c)+(b+d)i</math> | * <math>z+w=(a+c)+(b+d)i</math> | ||
* <math>z-w=(a-c)+(b-d)i</math> | * <math>z-w=(a-c)+(b-d)i</math> | ||
| + | |||
| + | ==Alternate Forms== | ||
| + | |||
| + | In addition to the standard form <math>a+bi</math>, complex numbers can be expressed in two other forms. | ||
| + | |||
| + | The trigonometric form of a complex number is denoted by <math>r(\cos \theta + i \sin \theta)</math>, where <math>r</math> equals the magnitude of the complex number and <math>\theta</math> (in radians) is the argument of the complex number. | ||
| + | |||
| + | The exponential form of a complex number is denoted by <math>re^{i \theta}</math>, where <math>r</math> equals the magnitude of the complex number and <math>\theta</math> (in radians) is the argument of the complex number. | ||
== Topics == | == Topics == | ||
* [[Complex plane]] | * [[Complex plane]] | ||
| − | * [[ | + | * [[De Moivre's Theorem]] |
* [[Exponential form]] | * [[Exponential form]] | ||
* [[Roots of unity]] | * [[Roots of unity]] | ||
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== Problems == | == Problems == | ||
| − | + | ===Introductory=== | |
| − | + | *[[2007 AMC 12A Problems/Problem 18]] | |
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| + | ===Intermediate=== | ||
| + | *[[1984 AIME Problems/Problem 8|1984 AIME Problem 8]] | ||
| + | *[[1985 AIME Problems/Problem 3|1985 AIME Problem 3]] | ||
| + | *[[1988 AIME Problems/Problem 11|1988 AIME Problem 11]] | ||
| + | *[[1989 AIME Problems/Problem 14|1989 AIME Problem 14]] | ||
| + | *[[1990 AIME Problems/Problem 10|1990 AIME Problem 10]] | ||
| + | *[[1992 AIME Problems/Problem 10|1992 AIME Problem 10]] | ||
| + | *[[1994 AIME Problems/Problem 8|1994 AIME Problem 8]] | ||
| + | *[[1994 AIME Problems/Problem 13|1994 AIME Problem 13]] | ||
| + | *[[1995 AIME Problems/Problem 5|1995 AIME Problem 5]] | ||
| + | *[[1996 AIME Problems/Problem 11|1996 AIME Problem 11]] | ||
| + | *[[1997 AIME Problems/Problem 11|1997 AIME Problem 11]] | ||
| + | *[[1997 AIME Problems/Problem 14|1997 AIME Problem 14]] | ||
| + | *[[1998 AIME Problems/Problem 13|1998 AIME Problem 13]] | ||
| + | *[[1999 AIME Problems/Problem 9|1999 AIME Problem 9]] | ||
| + | *[[2000 AIME II Problems/Problem 9|2000 AIME II Problem 9]] | ||
| + | *[[2002 AIME I Problems/Problem 12|2002 AIME I Problem 12]] | ||
| + | *[[2004 AIME I Problems/Problem 13|2004 AIME I Problem 13]] | ||
| + | *[[2005 AIME II Problems/Problem 9|2005 AIME II Problem 9]] | ||
| + | *[[2009 AIME I Problems/Problem 2|2009 AIME I Problem 2]] | ||
| + | *[[2011 AIME II Problems/Problem 8|2011 AIME II Problem 8]] | ||
| + | |||
| + | ===Olympiad=== | ||
== See also == | == See also == | ||
| Line 58: | Line 79: | ||
* [[Trigonometry]] | * [[Trigonometry]] | ||
* [[Real numbers]] | * [[Real numbers]] | ||
| − | [[ | + | * [[Imaginary unit]] |
| + | |||
| + | [[Category:Definition]] | ||
| + | [[Category:Complex numbers]] | ||
Latest revision as of 14:36, 10 December 2023
The complex numbers arise when we try to solve equations such as 
.
Contents
Derivation
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.)
Formal Definition
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 
.
Parts
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).
Operations
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 
.
Examples
If 
and 
,
,
,
Alternate Forms
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.
Topics
Problems
Introductory
Intermediate
- 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
