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Complex Number

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+2iz = 1+2iz=1+2i changes when multiplied by the scalar c=2c=2c=2 and c=−0.5c=-0.5c=−0.5.

From the visualization above, we can see:

  • Multiplying zzz by a scalar c>1c > 1c>1 (like 2) will stretch the vector zzz in the same direction.
  • Multiplying zzz by a scalar 0<c<10 < c < 10<c<1 will shrink the vector zzz in the same direction.
  • Multiplying zzz by a scalar c<0c < 0c<0 (like -0.5) will reverse the direction of the vector zzz (by 180 degrees) and change its length according to the value of ∣c∣|c|∣c∣.

Mathematical Definition

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

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

This means we simply multiply the scalar ccc by the real part (xxx) and the imaginary part (yyy) separately.

Calculation Examples

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

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

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

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

Visualization Examples

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

Positive Scalar (> 1)

Multiplication by Scalar c=1.5c = 1.5c=1.5
Vector z=−2+iz = -2 + iz=−2+i is stretched in the same direction when multiplied by c=1.5c = 1.5c=1.5, becoming 1.5z=−3+1.5i1.5z = -3 + 1.5i1.5z=−3+1.5i.

Positive Scalar (0 < c < 1)

Multiplication by Scalar c=0.75c = 0.75c=0.75
Vector z=3−2iz = 3 - 2iz=3−2i is shrunk in the same direction when multiplied by c=0.75c = 0.75c=0.75, becoming 0.75z=2.25−1.5i0.75z = 2.25 - 1.5i0.75z=2.25−1.5i.

Negative Scalar (c = -1)

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

Multiplication by Scalar c=−1c = -1c=−1 (Additive Inverse)
Vector z=−1−3iz = -1 - 3iz=−1−3i reverses direction (180°) when multiplied by c=−1c = -1c=−1, becoming −z=1+3i-z = 1 + 3i−z=1+3i.
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Multiplication of Complex Numbers

  • Scalar Multiplication of Complex NumbersExplore scalar multiplication effects on complex vectors with interactive 3D visualizations showing stretching, shrinking, and direction reversal.
On this page
  • What is Scalar Multiplication?
  • Mathematical Definition
    • Calculation Examples
  • Visualization Examples
    • Positive Scalar (> 1)
    • Positive Scalar (0 < c < 1)
    • Negative Scalar (c = -1)
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