Multiplication of Complex Numbers

Definition

Multiplication of complex numbers is a process that combines two complex numbers to produce another complex number. This operation can be understood algebraically, where two complex numbers are multiplied using the distributive property, as well as geometrically, where multiplication corresponds to the rotation and scaling of points in the complex plane. The result of this operation reveals both the magnitude and direction associated with the product of the two original numbers.

5 Must Know Facts For Your Next Test

1. When multiplying two complex numbers, say $$z_1 = a + bi$$ and $$z_2 = c + di$$, the product is given by $$z_1 z_2 = (ac - bd) + (ad + bc)i$$.
2. Geometrically, multiplication by a complex number represents a rotation by the argument (angle) of that number and scaling by its magnitude.
3. The multiplication of complex numbers is commutative; that is, $$z_1 z_2 = z_2 z_1$$ for any two complex numbers $$z_1$$ and $$z_2$$.
4. The modulus of the product of two complex numbers is equal to the product of their moduli: $$|z_1 z_2| = |z_1| |z_2|$$.
5. Multiplying by a pure imaginary number can lead to a 90-degree rotation in the complex plane, which visually represents the effect of multiplication.

Review Questions

• How does multiplication of complex numbers relate to their geometric representation in the complex plane?
  • Multiplication of complex numbers can be visualized geometrically as both rotation and scaling. When multiplying two complex numbers, the angle (argument) of each number adds together while their magnitudes (moduli) multiply. This means that if you visualize one number as a point in the complex plane, multiplying it by another complex number rotates that point around the origin and changes its distance from the origin based on the magnitude of both numbers.
• Explain how you would use polar form to multiply two complex numbers and why it simplifies the process.
  • To multiply two complex numbers using polar form, you first express each number as $$r_1( ext{cos} \theta_1 + i\text{sin} \theta_1)$$ and $$r_2( ext{cos} \theta_2 + i\text{sin} \theta_2)$$. The product then becomes $$r_1 r_2 [ ext{cos} (\theta_1 + \theta_2) + i \text{sin} (\theta_1 + \theta_2)]$$. This approach simplifies multiplication because it reduces the need for expanding terms and makes it straightforward to combine angles and magnitudes.
• Analyze how understanding multiplication in terms of geometric transformations can enhance your problem-solving skills in complex analysis.
  • Understanding multiplication as a geometric transformation allows for deeper insights into how complex functions behave. It enables you to predict how shapes will transform when subjected to multiplication by different complex numbers, enhancing intuition about function mapping in the complex plane. This perspective also aids in visualizing concepts like analytic functions and conformal mappings, as it connects algebraic operations with their geometric interpretations, making it easier to conceptualize solutions to more complicated problems.

Related terms

Complex Conjugate:

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part, which helps in simplifying expressions and performing operations like division.

Polar Form:

The polar form of a complex number expresses it in terms of its magnitude and angle, making it easier to perform multiplication and division using trigonometric functions.

Magnitude:

The magnitude of a complex number is its distance from the origin in the complex plane, calculated using the formula $$|z| = \sqrt{a^2 + b^2}$$ for a complex number $$z = a + bi$$.