Scalar Multiplication of Complex Numbers

Multiplying a Complex Vector by a Real Number

Scalar multiplication involves multiplying a complex number by a real number, called a scalar.

Visualization of Scalar Multiplication

Notice how the vector

z = 1+2i

changes when multiplied by the scalar

c=2

and

c=-0.5

The visualization shows how the scalar changes the vector:

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 $c = 0$ sends the vector to the origin, so $0z = 0$ .
For $z\neq0$ , multiplying by a scalar $c < 0$ (like $-0.5$ ) reverses the direction of the vector by $180^\circ$ and multiplies its length by $|c|$ .

Multiplying a Complex Number by a Scalar

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)

The formula multiplies the scalar $c$ by the real part $x$ and the imaginary part $y$ separately.

Scaling Complex Numbers by Real Scalars

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

\begin{aligned} cz &= 3(-3 + 4i) \\ &= 3(-3) + i(3 \times 4) \\ &= -9 + 12i \end{aligned}

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

\begin{aligned} cz &= -2(5 - i) \\ &= -2(5) + i(-2 \times -1) \\ &= -10 + 2i \end{aligned}

Stretching Shrinking and Reversing on the Complex Plane

A real scalar changes the length of the arrow and can also turn it around. The cases here separate the effects of stretching, shrinking, and reversing direction, so the size and the sign of the scalar each stay clear.

Scalar Greater than One

Here the factor is larger than one, so the arrow keeps its direction and its length grows by the factor.

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

Here the factor lies between zero and one, so the arrow keeps its direction and becomes shorter.

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

Multiplying a Complex Vector by a Real Number

Scalar multiplication involves multiplying a complex number by a real number, called a scalar.

Visualization of Scalar Multiplication

Notice how the vector

z = 1+2i

changes when multiplied by the scalar

c=2

and

c=-0.5

The visualization shows how the scalar changes the vector:

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 $c = 0$ sends the vector to the origin, so $0z = 0$ .
For $z\neq0$ , multiplying by a scalar $c < 0$ (like $-0.5$ ) reverses the direction of the vector by $180^\circ$ and multiplies its length by $|c|$ .

Multiplying a Complex Number by a Scalar

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)

The formula multiplies the scalar $c$ by the real part $x$ and the imaginary part $y$ separately.

Scaling Complex Numbers by Real Scalars

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

\begin{aligned} cz &= 3(-3 + 4i) \\ &= 3(-3) + i(3 \times 4) \\ &= -9 + 12i \end{aligned}

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

\begin{aligned} cz &= -2(5 - i) \\ &= -2(5) + i(-2 \times -1) \\ &= -10 + 2i \end{aligned}

Stretching Shrinking and Reversing on the Complex Plane

Scalar Greater than One

Here the factor is larger than one, so the arrow keeps its direction and its length grows by the factor.

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

Here the factor lies between zero and one, so the arrow keeps its direction and becomes shorter.

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