Table of content

  1. Why Complex Numbers Are Needed
  2. Algebraic Form and Equality of Numbers
  3. Real and Purely Imaginary Numbers
  4. Powers of the Imaginary Unit
  5. Addition and Subtraction
  6. Multiplication and Conjugation
  7. Division
  8. A Complex Number as a Pair of Numbers

Why Complex Numbers Are Needed

Consider the equation:

\[ x^2+1=0. \]

Moving one to the other side gives:

\[ x^2=-1. \]

There are no solutions among the real numbers: the square of any real number is non-negative.

\[ x^2\geq0, \qquad x\in\mathbb R. \]

To solve such equations, the imaginary unit \(i\) is introduced, which is defined by:

\[ \boxed{i^2=-1}. \]

The equation \(x^2=-1\) now has two solutions:

\[ x=i, \qquad x=-i. \]

Therefore, over the complex numbers, the expression can be factored:

\[ x^2+1=(x-i)(x+i). \]

Algebraic Form and Equality of Numbers

Any complex number is written in the form:

\[ \boxed{z=x+iy}, \qquad x,y\in\mathbb R. \]

The number \(x\) is called the real part, and the number \(y\) is called the imaginary part of a complex number:

\[ x=\operatorname{Re}z, \qquad y=\operatorname{Im}z. \]

Thus, we can also write:

\[ z=\operatorname{Re}z+i\operatorname{Im}z. \]

Example

For the number \(z=3-2i\):

\[ \operatorname{Re}z=3, \qquad \operatorname{Im}z=-2. \]

The representation \(z=x+iy\) is unique. Therefore, two complex numbers are equal if and only if both their real and imaginary parts are equal:

\[ \boxed{ z_1=z_2 \iff \begin{cases} \operatorname{Re}z_1=\operatorname{Re}z_2,\\ \operatorname{Im}z_1=\operatorname{Im}z_2. \end{cases} } \]

Example

\[ x+iy=3-2i \]

means that

\[ x=3, \qquad y=-2. \]

Real and Purely Imaginary Numbers

If the imaginary part equals zero, the complex number is real:

\[ y=0, \qquad z=x+i0=x. \]

For example:

\[ 5=5+0i. \]

Therefore, real numbers are included in the set of complex numbers:

\[ \boxed{\mathbb R\subset\mathbb C}. \]

If the real part equals zero while the imaginary part does not, the number is called purely imaginary:

\[ x=0, \qquad y\ne0, \qquad z=iy. \]

For example, \(-3i\) is purely imaginary.

Zero has both parts equal to zero:

\[ 0=0+0i. \]

Powers of the Imaginary Unit

From the definition \(i^2=-1\), we obtain:

\[ i^0=1, \qquad i^1=i, \qquad i^2=-1, \qquad i^3=-i, \qquad i^4=1. \]

The values then repeat with a period of four:

\[ \boxed{1\longrightarrow i\longrightarrow-1\longrightarrow-i\longrightarrow1} \]
\[ \boxed{i^{n+4}=i^n}, \qquad n\in\mathbb N_0. \]

To calculate a large power of \(i\), find the remainder when its exponent is divided by \(4\).

Example

\[ 2026=4\cdot506+2. \]

Therefore:

\[ \boxed{i^{2026}=i^2=-1}. \]

Addition and Subtraction

Let:

\[ z_1=x_1+iy_1, \qquad z_2=x_2+iy_2. \]

When adding and subtracting, work with the real and imaginary parts separately:

\[ \boxed{ z_1+z_2=(x_1+x_2)+i(y_1+y_2) }. \]
\[ \boxed{ z_1-z_2=(x_1-x_2)+i(y_1-y_2) }. \]

Addition

\[ (1+2i)+(3-i) =(1+3)+(2i-i) =\boxed{4+i}. \]

Subtraction

\[ (4+3i)-(1+5i) =(4-1)+i(3-5) =\boxed{3-2i}. \]

Multiplication and Conjugation

To multiply complex numbers, expand the brackets using the usual rules of algebra and then use \(i^2=-1\):

\[ (x_1+iy_1)(x_2+iy_2) = x_1x_2+ix_1y_2+ix_2y_1+i^2y_1y_2. \]
\[ \boxed{ (x_1+iy_1)(x_2+iy_2) = (x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1) }. \]

Example

\[ (1+i)(2-i) = 2-i+2i-i^2 = \boxed{3+i}. \]

For a number \(z=x+iy\), its conjugate is the number in which the sign of the imaginary part is changed:

\[ \boxed{\overline{z}=x-iy}. \]

Examples

\[ \overline{1+i}=1-i, \qquad \overline{3-2i}=3+2i. \]

Conjugation has important properties:

\[ \overline{\overline{z}}=z, \qquad z+\overline{z}=2\operatorname{Re}z, \qquad z-\overline{z}=2i\operatorname{Im}z. \]
\[ \boxed{ z\overline{z} = (x+iy)(x-iy) = x^2+y^2 }. \]

The product \(z\overline{z}\) is always a non-negative real number. This property is useful for division.

Division

Consider the quotient:

\[ \frac{z_1}{z_2}, \qquad z_2\ne0. \]

To eliminate the imaginary part from the denominator, multiply both the numerator and denominator by the conjugate of the denominator:

\[ \frac{z_1}{z_2} = \frac{z_1\overline{z_2}} {z_2\overline{z_2}}. \]

If \(z_2=x_2+iy_2\), then:

\[ z_2\overline{z_2}=x_2^2+y_2^2>0, \]

because \(z_2\ne0\). Therefore, the denominator becomes real:

\[ \boxed{ \frac{x_1+iy_1}{x_2+iy_2} = \frac{x_1x_2+y_1y_2}{x_2^2+y_2^2} + i\frac{x_2y_1-x_1y_2}{x_2^2+y_2^2} }. \]

Example

\[ \frac{1+i}{1-i} \cdot \frac{1+i}{1+i} = \frac{(1+i)^2}{(1-i)(1+i)}. \]
\[ (1+i)^2=1+2i+i^2=2i, \qquad (1-i)(1+i)=1-i^2=2. \]
\[ \boxed{ \frac{1+i}{1-i}=i }. \]

A Complex Number as a Pair of Numbers

A complex number \(z=x+iy\) corresponds to an ordered pair of real numbers \((x,y)\):

\[ \boxed{ z=x+iy \longleftrightarrow (x,y) }. \]

The first coordinate is the real part, and the second is the imaginary part. For example:

\[ 1+i \longleftrightarrow (1,1), \qquad 3-2i \longleftrightarrow (3,-2). \]

This is why equality of complex numbers is checked by comparing their real and imaginary parts.

In Brief

  • The algebraic form is convenient for calculations.
  • Addition and subtraction are performed component by component.
  • For multiplication, use \(i^2=-1\).
  • For division, the conjugate of the denominator is useful.
  • The pair \((x,y)\) leads to a geometric representation of a number.

Next Topic

The Complex Plane, Modulus, and Argument

Next, we can move from the representation \(z=x+iy\) to a point \((x,y)\) on the complex plane, study the modulus of a number, and its argument.