How do I multiply two complex numbers like (4 + 3i) and (12 + 2i)?

Multiply each term using the distributive property: (4)(12) + (4)(2i) + (3i)(12) + (3i)(2i), then simplify, remembering that i^2 = -1.

What is the step-by-step process to find the product of (4 + 3i) and (12 + 2i)?

First, multiply: 4×12 = 48, 4×2i = 8i, 3i×12 = 36i, 3i×2i = 6i^2. Since i^2 = -1, replace 6i^2 with -6. Then, combine like terms: 48 - 6 + (8i + 36i) = 42 + 44i.

What is the final simplified form of (4 + 3i) (12 + 2i)?

The simplified product is 42 + 44i.

Why do we replace i^2 with -1 when multiplying complex numbers?

Because i is the imaginary unit defined by i^2 = -1, which helps simplify expressions involving imaginary numbers.

What Is The Product Of ( 4 + 3 I ) And ( 12 2 I )

Q: What is the product of (4 + 3i) and (12 + 2i)?

The product is (48 + 8i + 36i + 6i^2), which simplifies to (48 + 44i - 6) because i^2 = -1, resulting in 42 + 44i.

What Is The Product Of ( 4 + 3 I ) And ( 12 + 2 I )

When dealing with complex numbers, one of the fundamental operations is multiplication. The product of two complex numbers involves combining their real and imaginary parts following specific algebraic rules. Understanding how to multiply complex numbers like (4 + 3i) and (12 + 2i) is essential not only in pure mathematics but also in fields like engineering, physics, and computer science, where complex numbers are used to describe oscillations, waveforms, and signal processing.

In this article, we will explore how to find the product of the complex numbers (4 + 3i) and (12 + 2i), breaking down each step in detail. We will discuss the underlying algebraic principles, demonstrate the calculation process, and explain the significance of the result in various contexts. This comprehensive guide aims to provide clarity for students, educators, and professionals working with complex numbers.

Understanding Complex Numbers

What Are Complex Numbers?

Complex numbers extend the real number system to include imaginary numbers. A complex number has two parts:

Real part: A real number representing the "horizontal" component.

Imaginary part: A real number multiplied by the imaginary unit, denoted as i.

A complex number is generally written as:

\[ z = a + bi \]

where:

\( a \) is the real part.

\( b \) is the imaginary coefficient.

\( i \) is the imaginary unit, satisfying \( i^2 = -1 \).

For example, in \( 4 + 3i \):

Real part: 4

Imaginary part: 3

Similarly, in \( 12 + 2i \):

Real part: 12

Imaginary part: 2

The Importance of Multiplying Complex Numbers

Complex number multiplication is crucial in several applications:

Signal processing: Representing and manipulating wave signals.

Control systems: Analyzing system stability.

Electromagnetism: Describing oscillating fields.

Mathematics: Studying polynomial roots and complex analysis.

Understanding the product of two complex numbers helps in transforming and simplifying complex expressions, solving equations, and interpreting physical phenomena.

Step-by-Step Calculation of the Product

Given the two complex numbers:

\[ (4 + 3i) \quad \text{and} \quad (12 + 2i) \]

we want to compute:

\[ (4 + 3i) \times (12 + 2i) \]

Using the Distributive Property (FOIL Method)

The multiplication of two binomials follows the distributive property, often remembered by the FOIL acronym:

F: First terms

O: Outer terms

I: Inner terms

L: Last terms

Applying FOIL to the complex numbers:

\[ (4 + 3i)(12 + 2i) = (4 \times 12) + (4 \times 2i) + (3i \times 12) + (3i \times 2i) \]

Calculating each term:

\( 4 \times 12 = 48 \)

\( 4 \times 2i = 8i \)

\( 3i \times 12 = 36i \)

\( 3i \times 2i = 6i^2 \)

Now, combine these results:

\[ 48 + 8i + 36i + 6i^2 \]

Next, combine like terms:

\[ 48 + (8i + 36i) + 6i^2 \]
\[ 48 + 44i + 6i^2 \]

Recall that \( i^2 = -1 \), so:

\[ 6i^2 = 6 \times (-1) = -6 \]

Substitute back:

\[ 48 + 44i - 6 \]

Finally, combine the real parts:

\[ (48 - 6) + 44i = 42 + 44i \]

Result:

\[ (4 + 3i)(12 + 2i) = \boxed{42 + 44i} \]

Interpreting the Result

The product of the two complex numbers is \( 42 + 44i \), which is itself a complex number with:

Real part: 42

Imaginary part: 44

This result can be visualized in the complex plane as a point located at coordinates (42, 44).

Additional Insights and Contexts

Magnitude and Phase of the Product

Understanding the magnitude (also called modulus) and phase (angle) of the product provides deeper insights into the nature of complex multiplication.

Magnitude (|z|) of a complex number \( z = a + bi \):

\[ |z| = \sqrt{a^2 + b^2} \]

Phase (θ):

\[ \theta = \arctan \left( \frac{b}{a} \right) \]

Applying this to the product \( 42 + 44i \):

Magnitude:

\[ |42 + 44i| = \sqrt{42^2 + 44^2} = \sqrt{1764 + 1936} = \sqrt{3700} \approx 60.83 \]

Phase:

\[ \theta = \arctan \left( \frac{44}{42} \right) \approx \arctan(1.0476) \approx 46.4^\circ \]

This indicates the product's vector in the complex plane has a length of approximately 60.83 units and is positioned at an angle of about 46.4 degrees from the positive real axis.

Multiplication in Polar Coordinates

Complex numbers can also be expressed in polar form:

\[ z = r (\cos \theta + i \sin \theta) \]

where \( r \) is the magnitude, and \( \theta \) is the argument (phase).

Multiplying two complex numbers in polar form involves:

Multiplying their magnitudes.

Adding their angles.

For the original numbers:

Convert to polar form:

\( 4 + 3i \):

\[ r_1 = \sqrt{4^2 + 3^2} = 5 \]
\[ \theta_1 = \arctan \left( \frac{3}{4} \right) \approx 36.87^\circ \]

\( 12 + 2i \):

\[ r_2 = \sqrt{12^2 + 2^2} = \sqrt{144 + 4} = \sqrt{148} \approx 12.17 \]
\[ \theta_2 = \arctan \left( \frac{2}{12} \right) \approx 9.59^\circ \]

Multiply magnitudes:

\[ r{product} = r1 \times r_2 \approx 5 \times 12.17 = 60.83 \]

Add angles:

\[ \theta{product} = \theta1 + \theta_2 \approx 36.87^\circ + 9.59^\circ = 46.46^\circ \]

This matches the earlier calculation of the magnitude and phase, confirming the consistency of complex multiplication across different representations.

Applications of Complex Number Multiplication

Understanding how to multiply complex numbers, such as (4 + 3i) and (12 + 2i), is vital across various disciplines:

Electrical Engineering: Analyzing AC circuits where impedance and phasors are represented as complex numbers.

Quantum Mechanics: Wave functions involve complex probability amplitudes; their interactions often involve multiplication.

Control Theory: Stability analysis relies on the roots of characteristic equations, which are complex numbers.

Signal Processing: Fourier transforms utilize complex multiplication to analyze frequency components.

Practical Example: Signal Modulation

Suppose an engineer is working with two signals represented by complex numbers. Multiplying these signals corresponds to combining their amplitudes and phases, which is essential in modulation techniques like amplitude modulation (AM). The calculation of their product allows for understanding how signals interact and combine in the frequency domain.

Conclusion

The product of the complex numbers (4 + 3i) and (12 + 2i) is \( 42 + 44i \). This calculation involves applying the distributive property, recognizing that \( i^2 = -1 \), and simplifying the expression accordingly. The process demonstrates the fundamental algebraic rules governing complex numbers and highlights the importance of both rectangular and polar forms.

Understanding complex multiplication is not only a mathematical skill but also a gateway to numerous practical applications in science and engineering. By mastering the step-by-step process outlined here, students and professionals can confidently work with complex numbers and harness their power in diverse fields.

Summary of the key steps:

Expand the product using FOIL.

Simplify each term, especially replacing \( i^2 \) with -1.

Combine like