Type a command or search the notes. Tab cycles completions. Up and down recall history when blank. Escape dismisses, then leaves the prompt.
MATH
[1251]

Definitions from Complex arithmetic

All definitions from Complex arithmetic.

Back to Complex arithmetic

Complex number

A complex number is a number of the form

z=a+bi,a,b∈R,i2=−1.z=a+bi,\qquad a,b\in\mathbb R,\qquad i^2=-1.

The expression a+bia+bi is the Cartesian form of zz. Its real part is Re⁡(z)=a\operatorname{Re}(z)=a and its imaginary part is Im⁡(z)=b\operatorname{Im}(z)=b.

Every real number is a complex number with imaginary part zero. Equality is componentwise:

a+bi=c+di⟺a=c and b=d.a+bi=c+di\quad\Longleftrightarrow\quad a=c\text{ and }b=d.

When simplifying powers of ii, use the four-term cycle

i0=1,i1=i,i2=−1,i3=−i,i^0=1,\qquad i^1=i,\qquad i^2=-1,\qquad i^3=-i,

which then repeats because i4=1i^4=1.

  1. [1]:Argand plane
  2. [2]:Complex conjugate
  3. [3]:Complex modulus
  4. [4]:Add, multiply and divide complex numbers
  5. 8 more

Argand plane

The Argand plane represents z=a+biz=a+bi by the point (a,b)(a,b), with the real axis horizontal and the imaginary axis vertical.

Under this identification, addition of complex numbers is vector addition, and z−wz-w is the displacement from ww to zz. So ∣z−w∣|z-w| will measure the distance between the two points. The picture also makes modulus and argument geometric quantities in polar form.

The point represents the number but does not replace its algebra: complex multiplication still combines both components according to i2=−1i^2=-1.

  1. [1]:Complex number
  2. [2]:Read loci in the Argand plane

Complex conjugate

If z=a+biz=a+bi, its complex conjugate is

z‾=a−bi.\overline z=a-bi.

Conjugation changes the sign of the imaginary part and leaves the real part unchanged, so it reflects zz in the real axis. The product of zz and its conjugate is real:

zz‾=(a+bi)(a−bi)=a2+b2.z\overline z=(a+bi)(a-bi)=a^2+b^2.
  1. [1]:Complex number
  2. [2]:Rules for conjugation
  3. [3]:Conjugate roots for real coefficients
  4. [4]:A complex conjugate pair of roots

Complex modulus

The modulus of z=a+biz=a+bi is

∣z∣=a2+b2.|z|=\sqrt{a^2+b^2}.

In the Argand plane, ∣z∣|z| is the distance from zz to the origin. Division and polar form both use the identity zz‾=∣z∣2z\overline z=|z|^2.

  1. [1]:Complex number
  2. [2]:Argument
  3. [3]:Polar form
  4. [4]:Modulus laws
  5. [5]:Polar form
  6. 2 more

Field

A field is a non-empty set equipped with addition and multiplication satisfying closure, associativity and commutativity for both operations, distinct additive and multiplicative identities 0≠10\ne1, additive inverses, multiplicative inverses for non-zero elements and the two distributive laws.

In a field we can add, subtract, multiply and divide by any non-zero element without leaving the set. The real and complex numbers are fields. The natural numbers fail the additive inverse axiom: 1∈N1\in\mathbb N but −1∉N-1\notin\mathbb N. The integers fail the multiplicative inverse axiom: 2∈Z2\in\mathbb Z, but 2−1=1/2∉Z2^{-1}=1/2\notin\mathbb Z.

  1. [1]:Vector spaces

On this page

Complex numberArgand planeComplex conjugateComplex modulusField