Complex Number Operations Calculator
Enter the real and imaginary parts for two complex numbers (Z1 and Z2) to perform various operations.
Complex Number 1 (Z1)
Complex Number 2 (Z2)
Results:
Z1 + Z2:
Z1 – Z2:
Z1 * Z2:
Z1 / Z2:
Modulus of Z1 (|Z1|):
Argument of Z1 (arg(Z1)):
Modulus of Z2 (|Z2|):
Argument of Z2 (arg(Z2)):
Understanding Complex Numbers and Their Operations
Complex numbers are an extension of real numbers, incorporating an imaginary unit denoted by 'i', where i² = -1. A complex number is typically expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. They are fundamental in various fields, including electrical engineering, quantum mechanics, signal processing, and fluid dynamics.
Components of a Complex Number
- Real Part (a): The component of the complex number that does not involve 'i'.
- Imaginary Part (b): The coefficient of 'i' in the complex number.
For example, in the complex number 3 + 4i, 3 is the real part and 4 is the imaginary part.
Basic Operations with Complex Numbers
1. Addition
To add two complex numbers, Z1 = a + bi and Z2 = c + di, you simply add their real parts and their imaginary parts separately:
Z1 + Z2 = (a + c) + (b + d)i
Example: If Z1 = 3 + 4i and Z2 = 1 – 2i, then Z1 + Z2 = (3 + 1) + (4 – 2)i = 4 + 2i.
2. Subtraction
Similar to addition, to subtract Z2 from Z1, you subtract their real parts and their imaginary parts separately:
Z1 - Z2 = (a - c) + (b - d)i
Example: If Z1 = 3 + 4i and Z2 = 1 – 2i, then Z1 – Z2 = (3 – 1) + (4 – (-2))i = 2 + 6i.
3. Multiplication
Multiplying two complex numbers Z1 = a + bi and Z2 = c + di involves using the distributive property, remembering that i² = -1:
Z1 * Z2 = (a + bi)(c + di) = ac + adi + bci + bdi² = ac + adi + bci - bd = (ac - bd) + (ad + bc)i
Example: If Z1 = 3 + 4i and Z2 = 1 – 2i, then Z1 * Z2 = (3*1 – 4*(-2)) + (3*(-2) + 4*1)i = (3 + 8) + (-6 + 4)i = 11 – 2i.
4. Division
Dividing complex numbers Z1 = a + bi by Z2 = c + di (where Z2 ≠ 0) requires multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of c + di is c - di.
Z1 / Z2 = (a + bi) / (c + di) = [(a + bi)(c - di)] / [(c + di)(c - di)]
The denominator becomes c² + d² (a real number). The numerator is calculated as a multiplication:
Numerator = (ac + bd) + (bc - ad)i
So, Z1 / Z2 = [(ac + bd) / (c² + d²)] + [(bc - ad) / (c² + d²)]i
Example: If Z1 = 3 + 4i and Z2 = 1 – 2i:
- Conjugate of Z2 is 1 + 2i.
- Denominator = (1² + (-2)²) = 1 + 4 = 5.
- Numerator Real Part = (3*1 + 4*(-2)) = 3 – 8 = -5.
- Numerator Imaginary Part = (4*1 – 3*(-2)) = 4 + 6 = 10.
- Z1 / Z2 = (-5 / 5) + (10 / 5)i = -1 + 2i.
5. Modulus (Magnitude)
The modulus (or magnitude) of a complex number Z = a + bi is its distance from the origin in the complex plane. It is calculated using the Pythagorean theorem:
|Z| = sqrt(a² + b²)
Example: If Z1 = 3 + 4i, then |Z1| = sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5.
6. Argument (Phase)
The argument of a complex number Z = a + bi is the angle (in radians or degrees) that the line connecting the origin to the point (a, b) makes with the positive real axis in the complex plane. It is typically found using the arctangent function:
arg(Z) = atan2(b, a) (using the two-argument arctangent function to correctly handle quadrants)
Example: If Z1 = 3 + 4i, then arg(Z1) = atan2(4, 3) ≈ 0.927 radians or ≈ 53.13 degrees.