Simplifying Complex Number Expressions Into A + Bi Form

In the realm of mathematics, complex numbers play a pivotal role, extending the familiar number system to include the imaginary unit, denoted as 'i', where i² = -1. These numbers, expressed in the form a + bi, where 'a' and 'b' are real numbers, find applications in various fields, from electrical engineering to quantum mechanics. In this article, we will delve into the intricacies of simplifying a given complex number expression into its standard a + bi form, providing a comprehensive explanation that caters to both beginners and seasoned math enthusiasts.

Consider the Expression: Simplifying Complex Number Operations

Our journey begins with the expression:

$$(6-4i)(6-4i) - 3(7-11i)$$

This expression combines complex number multiplication and subtraction, requiring us to apply the fundamental rules of complex number arithmetic. To simplify this expression and express it in the form a + bi, we will embark on a step-by-step approach, meticulously explaining each operation.

Step 1: Expanding the Product of Complex Numbers

The first step involves expanding the product of the complex number (6 - 4i) with itself. We can achieve this by employing the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that each term in the first complex number is multiplied by each term in the second complex number.

Applying the FOIL method, we get:

$$(6 - 4i)(6 - 4i) = 6 * 6 + 6 * (-4i) + (-4i) * 6 + (-4i) * (-4i)$$

Simplifying each term, we obtain:

$$36 - 24i - 24i + 16i^2$$

Remembering that i² = -1, we can substitute this value into the expression:

$$36 - 24i - 24i + 16(-1) = 36 - 48i - 16$$

Combining the real terms (36 and -16), we arrive at:

$$20 - 48i$$

This is the simplified form of the product (6 - 4i)(6 - 4i).

Step 2: Distributing the Scalar Multiplication

Next, we need to address the term -3(7 - 11i). This involves distributing the scalar -3 to both the real and imaginary parts of the complex number (7 - 11i).

Multiplying -3 by 7 and -11i, we get:

$$-3(7 - 11i) = -3 * 7 + (-3) * (-11i) = -21 + 33i$$

This step simplifies the second part of our original expression.

Step 3: Combining the Simplified Terms

Now that we have simplified both parts of the expression, we can combine them:

$$(6 - 4i)(6 - 4i) - 3(7 - 11i) = (20 - 48i) + (-21 + 33i)$$

To combine these complex numbers, we add the real parts together and the imaginary parts together:

$$(20 - 21) + (-48i + 33i)$$

This simplifies to:

$$-1 - 15i$$

Step 4: Expressing the Result in a + bi Form

The expression is now in the form a + bi, where a = -1 and b = -15. Therefore, the final simplified form of the given expression is:

$$-1 - 15i$$

Expressing Complex Numbers in a + bi Form: A Deep Dive

The a + bi form is the standard way to represent complex numbers, providing a clear separation between the real (a) and imaginary (b) components. Understanding this form is crucial for performing various operations with complex numbers, including addition, subtraction, multiplication, and division.

The Significance of the Imaginary Unit 'i'

The imaginary unit 'i' is the cornerstone of complex numbers. It is defined as the square root of -1, a concept that expands the number system beyond real numbers. The introduction of 'i' allows us to solve equations that have no solutions within the realm of real numbers, such as x² + 1 = 0.

Visualizing Complex Numbers on the Complex Plane

Complex numbers can be visualized on a complex plane, which is similar to the Cartesian plane but with the horizontal axis representing the real part (a) and the vertical axis representing the imaginary part (b). A complex number a + bi can be plotted as a point (a, b) on this plane. This graphical representation provides a geometric interpretation of complex numbers and their operations.

Operations with Complex Numbers in a + bi Form

Addition and Subtraction

Adding and subtracting complex numbers in a + bi form is straightforward. We simply add or subtract the real parts and the imaginary parts separately:

$$(a + bi) + (c + di) = (a + c) + (b + d)i$$

$$(a + bi) - (c + di) = (a - c) + (b - d)i$$

Multiplication

As demonstrated in the initial expression, multiplying complex numbers involves using the distributive property (FOIL method) and remembering that i² = -1:

$$(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$$

Division

Dividing complex numbers requires a slightly more intricate process. We multiply both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of a + bi is a - bi. This eliminates the imaginary part from the denominator, allowing us to express the result in a + bi form:

$$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$$

Applications of Complex Numbers

Complex numbers are not merely abstract mathematical concepts; they have practical applications in various fields:

• Electrical Engineering: Complex numbers are used to analyze alternating current (AC) circuits, where impedance (resistance to AC current) is a complex quantity.
• Quantum Mechanics: Complex numbers are fundamental in quantum mechanics, where wave functions that describe the behavior of particles are complex-valued.
• Fluid Dynamics: Complex numbers are used to represent fluid flow and analyze aerodynamic forces.
• Signal Processing: Complex numbers are used in signal processing to analyze and manipulate signals, such as audio and images.

Conclusion: Mastering Complex Number Simplification

Simplifying complex number expressions and expressing them in the standard a + bi form is a fundamental skill in mathematics. By understanding the rules of complex number arithmetic and practicing various examples, you can confidently tackle more complex problems involving these fascinating numbers. Complex numbers unlock a world of mathematical possibilities and have significant applications in various scientific and engineering disciplines. Mastering these concepts can open doors to deeper understanding and problem-solving capabilities in mathematics and its related fields.

In this article, we meticulously simplified a given expression, demonstrating the step-by-step process of expanding products, distributing scalars, and combining terms. We also delved into the significance of the a + bi form, the imaginary unit 'i', and the various operations that can be performed with complex numbers. With this knowledge, you are well-equipped to explore the world of complex numbers and their applications further.