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Complex Numbers

Scalar Multiplication of Complex Numbers

What is Scalar Multiplication?

Scalar multiplication involves multiplying a complex number by a real number (a scalar).

Visualization of Scalar Multiplication

Notice how the vector

z = 1+2i

changes when multiplied by the scalar

c=2

and .

Scalar Multiplication of Complex NumbersExplore scalar multiplication effects on complex vectors with interactive visualizations showing stretching, shrinking, and direction reversal.

On this page

What is Scalar Multiplication?
Mathematical Definition
- Calculation Examples
Visualization Examples

Comments

c=-0.5

From the visualization above, we can see:

Multiplying $z$ by a scalar $c > 1$ (like $2$ ) will stretch the vector $z$ in the same direction.
Multiplying $z$ by a scalar $0 < c < 1$ will shrink the vector $z$ in the same direction.
Multiplying $z$ by a scalar $c < 0$ (like $-0.5$ ) will reverse the direction of the vector $z$ (by $180^\circ$ ) and change its length according to the value of .

Mathematical Definition

If $z = x + iy$ is a complex number and $c$ is a scalar (a real number), then their scalar multiplication is:

cz = c(x + iy) = (cx) + i(cy)

This means we simply multiply the scalar $c$ by the real part ( $x$ ) and the imaginary part ( $y$ ) separately.

Calculation Examples

If $z = -3 + 4i$ and $c = 3$ , then:

cz = 3(-3 + 4i) = 3(-3) + i(3 \times 4) = -9 + 12i

If $z = 5 - i$ and $c = -2$ , then:

cz = -2(5 - i) = -2(5) + i(-2 \times -1) = -10 + 2i

Visualization Examples

Let's look at a few more examples to clarify the effect of scalar multiplication.

Scalar Greater than One

Multiplication by Scalar

c = 1.5

Vector

z = -2 + i

is stretched in the same direction when multiplied by

c = 1.5

, becoming

1.5z = -3 + 1.5i

.

Scalar between Zero and One

Multiplication by Scalar

c = 0.75

Vector

z = 3 - 2i

is shrunk in the same direction when multiplied by

c = 0.75

, becoming

0.75z = 2.25 - 1.5i

.

Negative Scalar

Multiplication by $-1$ yields the additive inverse (negative) of the complex number.

Multiplication by Scalar

c = -1

(Additive Inverse)

Vector

z = -1 - 3i

reverses direction (

180^\circ

) when multiplied by

c = -1

, becoming

-z = 1 + 3i

.

Properties of Principal Argument of Complex Numbers

|c|