ScientificCalcul

Number Systems

What Are Complex Numbers?

A complex number has the form a+bi, where a and b are real and i^2=-1. Real numbers are the special case b=0.

Reviewed

A complex number is a point as well as an expression

Plot z = a + bi at (a, b). Its conjugate reflects it across the real axis, and its modulus is the distance from the origin.

Complex plane for three plus two i The real axis is horizontal, the imaginary axis is vertical, and the point three plus two i is connected to the origin. realimaginary 3 + 2i32i
For z = 3 + 2i, the point is (3, 2), the conjugate is 3 - 2i, and |z| = sqrt(3^2 + 2^2) = sqrt(13).

Real and imaginary components

The real part of z=a+bi is a and the imaginary part is b. Two complex numbers are equal only when both corresponding parts are equal.

Complex numbers form a plane rather than a line: the horizontal axis represents real parts and the vertical axis represents imaginary parts.

Formulaz=a+bi; Re(z)=a; Im(z)=b

Arithmetic and conjugates

Addition combines matching components. Multiplication distributes normally and replaces i^2 with -1. The conjugate a-bi reflects a point across the real axis.

Multiplying a complex number by its conjugate gives the real value a^2+b^2, which makes complex division possible.

Formula(a+bi)(a-bi)=a^2+b^2

Worked example 1

Multiply (2+3i)(4-i)

Distribute and combine real and imaginary terms.

  1. 8-2i+12i-3i^2
  2. i^2=-1, so -3i^2=3
  3. real=11, imaginary=10i

Result: 11+10i

Modulus and polar form

The modulus |z| is the distance from the origin. The argument is an angle locating the number's direction, with quadrant handled by an atan2-type function.

Polar form separates magnitude and angle, making multiplication and division geometric: magnitudes multiply or divide while arguments add or subtract.

Formula|z|=sqrt(a^2+b^2); z=r(cos(theta)+i sin(theta))

Worked example 2

Find the modulus of -3+4i

Use the distance formula.

  1. |-3+4i|=sqrt((-3)^2+4^2)
  2. sqrt(9+16)=sqrt25

Result: |-3+4i|=5

Questions about complex numbers

Are imaginary numbers less real than real numbers?

They are well-defined mathematical numbers; the name is historical. Complex arithmetic is internally consistent and widely applied.

Is every real number complex?

Yes. A real number a can be written a+0i.

Sources and further reading