Complex number

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 $a = a + 0i$, but it is much larger.

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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