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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>.  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.)
+
The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>.   
  
−
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>, but it is much larger. 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.)
+
==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.)
 +
 
 +
==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).
 
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).
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== Operations ==
 
== Operations ==
  
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* Addition
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Addition and subtraction of complex numbers are similar to doing the same operations to polynomials -- add the real parts then add the imaginary parts.
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* Subtraction
+
 
−
* Multiplication
+
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>.
−
* Division
+
 
−
* [[Absolute value]]/Modulus/Magnitude (denoted by <math>|z|</math>). This is the distance from the origin to the complex number in the [[complex plane]].
+
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>.
−
* [[Complex conjugate | Conjugation]]
 
−
* The [[argument]] function <math>\arg</math>
 
  
−
== Simple Example ==
+
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 <math>\displaystyle w = c + di</math>,
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=== 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>\mathrm{Re}(w)=c</math>,<math>\mathrm{Im}(w)=d</math>
+
* <math>\overline{z}=a-bi</math>
−
* <math>|w|=\sqrt{c^2+d^2}</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 ==
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* [[Exponential form]]
 
* [[Exponential form]]
 
* [[Roots of unity]]
 
* [[Roots of unity]]
−
* [[Geometry with complex numbers]]
 
  
 
== Problems ==
 
== Problems ==
−
*AIME
+
===Introductory===
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=392620#p392620 1984 #8]
+
*[[2007 AMC 12A Problems/Problem 18]]
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=421338#p421338 1985 #3]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=458040#p458040 1988 #11]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=436603#p436603  1989 #14]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=459508#p459508 1990 #10]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=430620#p430620 1992 #10]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=53847#p53847 1994 #8]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=394743#p394743 1994 #13]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=394478#p394478 1995 #5]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=394249#p394249 1996 #11]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=393654#p393654 1997 #11]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=393661#p393661 1997 #14]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=392484#p392484 1998 #13]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=392227#p392227 1999 #9]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=385894#p385894 2000 Alternate #9]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=378395#p378395 2002 #12]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=378129#p378129 2004 #13]
 
−
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=368277#p368277 2005 Alternate #9]
 
  
 +
===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 ==
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* [[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 $x^2 = -1$.

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, $i$, such that $i = \sqrt{-1}$. If we add this new number to the reals, we will have solutions to $x^2 = -1$. It turns out that in the system that results from this addition, we are not only able to find the solutions of $x^2 = -1$ 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 $a + bi$ where $a,b\in \mathbb{R}$ and $i = \sqrt{-1}$ is the imaginary unit. The set of complex numbers is denoted by $\mathbb{C}$. The set of complex numbers contains the set $\mathbb{R}$ of the real numbers, since $a = a + 0i$.

Parts

Every complex number $z$ has a real part denoted $\Re(z)$ or $\mathrm{Re}(z)$ and an imaginary part denoted $\Im(z)$ or $\mathrm{Im}(z)$. Note that the imaginary part of a complex number is real: for example, $\Im(3 + 4i) = 4$. So, if $z\in \mathbb C$, we can write $z=\mathrm{Re}(z)+i\mathrm{Im}(z)$. ($z$ and $w$ are traditionally used in place of $x$ and $y$ as variables when dealing with complex numbers, while $x$ and $y$ (and frequently also $a$ and $b$) 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 $x\ge 0$ from the domain of the function $f(x)=\sqrt{x}$ (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 $i^2 = -1$. 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 $\overline{z}$.

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 $|z|$.

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 $\arg(z)$.

Examples

If $z=a+bi$ and $w = c + di$,

  • $\mathrm{Re}(z)=a$,$\mathrm{Im}(z)=b$
  • $|z|=\sqrt{a^2+b^2}$
  • $\overline{z}=a-bi$
  • $z+w=(a+c)+(b+d)i$
  • $z-w=(a-c)+(b-d)i$

Alternate Forms

In addition to the standard form $a+bi$, complex numbers can be expressed in two other forms.

The trigonometric form of a complex number is denoted by $r(\cos \theta + i \sin \theta)$, where $r$ equals the magnitude of the complex number and $\theta$ (in radians) is the argument of the complex number.

The exponential form of a complex number is denoted by $re^{i \theta}$, where $r$ equals the magnitude of the complex number and $\theta$ (in radians) is the argument of the complex number.

Topics

Problems

Introductory

Intermediate

Olympiad

See also

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