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Complex Number

Addition of Complex Numbers

Addition of Two Complex Numbers

How do you add two complex numbers?

Suppose we have two complex numbers:

z1=x1+iy1z_1 = x_1 + iy_1z1​=x1​+iy1​
z2=x2+iy2z_2 = x_2 + iy_2z2​=x2​+iy2​

To add them (z1+z2z_1 + z_2z1​+z2​), simply add the real parts together and the imaginary parts together.

z1+z2=(x1+x2)+i(y1+y2)z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2)z1​+z2​=(x1​+x2​)+i(y1​+y2​)

Addition Example

Let z1=2+3iz_1 = 2 + 3iz1​=2+3i and z2=1−iz_2 = 1 - iz2​=1−i.

  • The real part of z1z_1z1​ is 2, the real part of z2z_2z2​ is 1.
  • The imaginary part of z1z_1z1​ is 3, the imaginary part of z2z_2z2​ is -1.

Then their sum is:

z1+z2=(2+1)+i(3+(−1))=3+i(2)=3+2iz_1 + z_2 = (2 + 1) + i(3 + (-1)) = 3 + i(2) = 3 + 2iz1​+z2​=(2+1)+i(3+(−1))=3+i(2)=3+2i

Visualization of Addition

Using the parallelogram rule, the addition of complex numbers can be viewed geometrically on the complex plane. If we represent z1z_1z1​ and z2z_2z2​ as vectors (arrows) from the origin (0,0), then their sum, z1+z2z_1 + z_2z1​+z2​, is the diagonal vector of the parallelogram formed by z1z_1z1​ and z2z_2z2​.

Geometric Addition of Complex Numbers
Visualization of the sum z1=2+3iz_1 = 2+3iz1​=2+3i and z2=1−iz_2 = 1-iz2​=1−i using the parallelogram rule.

Related Operations

Besides addition, other operations work similarly:

Scalar Multiplication

Multiplying a complex number z=x+iyz = x + iyz=x+iy by a real number (scalar) ccc is straightforward. Just multiply ccc into both the real and imaginary parts.

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

Geometrically, this scales the vector zzz by a factor of ccc. If ccc is negative, the vector's direction is reversed.

Negative of a Complex Number

The negative of z=x+iyz = x + iyz=x+iy is −z-z−z. This is the same as scalar multiplication by c=−1c = -1c=−1.

−z=−(x+iy)=−x+i(−y)=−x−iy-z = -(x + iy) = -x + i(-y) = -x - iy−z=−(x+iy)=−x+i(−y)=−x−iy

Geometrically, −z-z−z is a vector with the same length as zzz but pointing in the opposite direction (180 degrees rotation).

Subtraction of Two Complex Numbers

Subtracting z2z_2z2​ from z1z_1z1​ (z1−z2z_1 - z_2z1​−z2​) is the same as adding z1z_1z1​ to the negative of z2z_2z2​ (z1+(−z2)z_1 + (-z_2)z1​+(−z2​)).

z1−z2=z1+(−z2)=(x1+(−x2))+i(y1+(−y2))=(x1−x2)+i(y1−y2)z_1 - z_2 = z_1 + (-z_2) = (x_1 + (-x_2)) + i(y_1 + (-y_2)) = (x_1 - x_2) + i(y_1 - y_2)z1​−z2​=z1​+(−z2​)=(x1​+(−x2​))+i(y1​+(−y2​))=(x1​−x2​)+i(y1​−y2​)

So, subtract the real parts and subtract the imaginary parts.

Geometrically, z1−z2z_1 - z_2z1​−z2​ is the vector from the tip of z2z_2z2​ to the tip of z1z_1z1​.

Example of Combined Operations

Suppose we have:

z1=2+12iz_1 = 2 + \frac{1}{2}iz1​=2+21​i
z2=−3+2iz_2 = -3 + \sqrt{2}iz2​=−3+2​i

Let's calculate some operations:

  1. 2z12z_12z1​ (Scalar Multiplication):

    2z1=2(2+12i)=2(2)+i(2×12)=4+i2z_1 = 2(2 + \frac{1}{2}i) = 2(2) + i(2 \times \frac{1}{2}) = 4 + i2z1​=2(2+21​i)=2(2)+i(2×21​)=4+i

  2. z1+3z2z_1 + 3z_2z1​+3z2​ (Addition and Scalar Multiplication):

    z1+3z2=(2+12i)+3(−3+2i)z_1 + 3z_2 = (2 + \frac{1}{2}i) + 3(-3 + \sqrt{2}i)z1​+3z2​=(2+21​i)+3(−3+2​i)
    =(2+12i)+(3(−3)+i(32))= (2 + \frac{1}{2}i) + (3(-3) + i(3\sqrt{2}))=(2+21​i)+(3(−3)+i(32​))
    =(2+12i)+(−9+32i)= (2 + \frac{1}{2}i) + (-9 + 3\sqrt{2}i)=(2+21​i)+(−9+32​i)
    =(2−9)+i(12+32)= (2 - 9) + i(\frac{1}{2} + 3\sqrt{2})=(2−9)+i(21​+32​)
    =−7+i(12+32)= -7 + i(\frac{1}{2} + 3\sqrt{2})=−7+i(21​+32​)

  3. 2z1−z22z_1 - z_22z1​−z2​ (Subtraction and Scalar Multiplication):

    2z1−z2=(4+i)−(−3+2i)2z_1 - z_2 = (4 + i) - (-3 + \sqrt{2}i)2z1​−z2​=(4+i)−(−3+2​i)
    =(4−(−3))+i(1−2)= (4 - (-3)) + i(1 - \sqrt{2})=(4−(−3))+i(1−2​)
    =(4+3)+i(1−2)= (4 + 3) + i(1 - \sqrt{2})=(4+3)+i(1−2​)
    =7+i(1−2)= 7 + i(1 - \sqrt{2})=7+i(1−2​)

Exercise

If z1=1+2iz_1 = 1 + 2iz1​=1+2i and z2=3−iz_2 = 3 - iz2​=3−i. Determine:

  1. z1+z2z_1 + z_2z1​+z2​
  2. z1−z2z_1 - z_2z1​−z2​
  3. If z3=z1+z2z_3 = z_1 + z_2z3​=z1​+z2​, draw z1z_1z1​, z2z_2z2​, and z3z_3z3​ on the complex plane.

Answer Key

  1. z1+z2=(1+3)+i(2+(−1))=4+iz_1 + z_2 = (1+3) + i(2+(-1)) = 4 + iz1​+z2​=(1+3)+i(2+(−1))=4+i
  2. z1−z2=(1−3)+i(2−(−1))=−2+i(3)=−2+3iz_1 - z_2 = (1-3) + i(2-(-1)) = -2 + i(3) = -2 + 3iz1​−z2​=(1−3)+i(2−(−1))=−2+i(3)=−2+3i

  3. Visualization of z1z_1z1​, z2z_2z2​, and z3=z1+z2z_3 = z_1 + z_2z3​=z1​+z2​ on the complex plane using the parallelogram rule:

    Addition of Complex Numbers
    Visualization of z1z_1z1​, z2z_2z2​, and z3=z1+z2z_3 = z_1 + z_2z3​=z1​+z2​ on the complex plane using the parallelogram rule.
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Properties of Addition of Complex Numbers

  • Addition of Complex NumbersLearn how to add complex numbers step-by-step with geometric visualization. Master real and imaginary parts addition using parallelogram rule and examples.
On this page
  • Addition of Two Complex Numbers
    • Addition Example
    • Visualization of Addition
  • Related Operations
    • Scalar Multiplication
    • Negative of a Complex Number
    • Subtraction of Two Complex Numbers
    • Example of Combined Operations
  • Exercise
    • Answer Key
  • Comments
  • Report
  • Source code