Multiplication of complex numbers

Definition

Multiplication of complex numbers involves combining two complex numbers to produce another complex number, using the distributive property and the fact that the imaginary unit $i$ satisfies $i^2 = -1$. This operation can be performed in rectangular form or in trigonometric form, providing different insights into the behavior and properties of complex numbers. Understanding this multiplication is crucial for further topics like polar coordinates and analyzing functions in the complex plane.

5 Must Know Facts For Your Next Test

1. When multiplying two complex numbers in rectangular form, such as $(a + bi)(c + di)$, you apply the distributive property, leading to $ac + adi + bci + bdi^2$, which simplifies to $(ac - bd) + (ad + bc)i$.
2. In trigonometric form, multiplying two complex numbers involves multiplying their magnitudes and adding their angles: if $z_1 = r_1(cos(\theta_1) + i sin(\theta_1)$ and $z_2 = r_2(cos(\theta_2) + i sin(\theta_2)$, then $z_1 z_2 = r_1 r_2(cos(\theta_1 + \theta_2) + i sin(\theta_1 + \theta_2))$.
3. The multiplication of complex numbers is commutative and associative, meaning that the order in which you multiply them does not affect the outcome.
4. Multiplying a complex number by its conjugate results in a real number: $(a + bi)(a - bi) = a^2 + b^2$, which is always non-negative.
5. In both rectangular and trigonometric forms, multiplying complex numbers reveals how their geometric representation on the complex plane combines rotation and scaling.

Review Questions

• How does the multiplication of complex numbers in rectangular form demonstrate the use of the distributive property?
  • Multiplying complex numbers in rectangular form requires applying the distributive property to each part. For example, when multiplying $(a + bi)(c + di)$, you first multiply each term in the first binomial by each term in the second. This results in four products: $ac$, $adi$, $bci$, and $bdi^2$. After simplifying by substituting $i^2$ with $-1$, you combine like terms to yield $(ac - bd) + (ad + bc)i$, showing how distribution works even with imaginary units.
• Explain how multiplying two complex numbers in trigonometric form differs from multiplying them in rectangular form.
  • When multiplying complex numbers in trigonometric form, you focus on their magnitudes and angles. The process simplifies multiplication by converting it into a matter of scalar multiplication and angle addition. For instance, if $z_1 = r_1(cos(\theta_1) + i sin(\theta_1)$ and $z_2 = r_2(cos(\theta_2) + i sin(\theta_2)$, then their product is $r_1 r_2(cos(\theta_1 + \theta_2) + i sin(\theta_1 + \theta_2))$. This method emphasizes geometric interpretation—scaling by magnitudes while rotating by angles—making calculations potentially easier compared to rectangular forms.
• Evaluate how the concept of multiplication of complex numbers integrates with other algebraic operations to enhance understanding of mathematical structures.
  • Multiplication of complex numbers interlinks with other operations like addition and division to create a comprehensive algebraic framework for understanding both real and imaginary components. It supports operations with functions in calculus and further emphasizes geometric interpretations through polar coordinates. By recognizing patterns such as commutativity and associativity alongside relationships like those between a number and its conjugate, students can appreciate how these operations build towards deeper concepts like roots of unity or transformations in the complex plane, enriching their overall grasp of mathematics.

Related terms

Complex Conjugate: The complex conjugate of a complex number is formed by changing the sign of its imaginary part, providing a useful tool for division and finding magnitudes.

Magnitude: The magnitude of a complex number, also known as the modulus, represents its distance from the origin in the complex plane and is calculated using the formula $$|a + bi| = \sqrt{a^2 + b^2}$$.

Trigonometric Form:

Trigonometric form expresses complex numbers using a magnitude and an angle, which simplifies multiplication and division through angle addition and subtraction.