Express (1-i/21)^2 In Standard Form A Comprehensive Guide

by BRAINLY IN FTUNILA 58 views
Iklan Headers

Before diving into the solution, let's recap complex numbers and their standard form. In mathematics, a complex number is a number that can be expressed in the form a + bi, where:

  • a is the real part
  • b is the imaginary part
  • i is the imaginary unit, defined as the square root of -1 (i² = -1)

The standard form of a complex number is a + bi, where a and b are real numbers. This form clearly separates the real and imaginary components, making it easier to perform arithmetic operations and visualize complex numbers on the complex plane (also known as the Argand diagram).

Complex numbers extend the concept of real numbers by including an imaginary component. They are essential in various fields, including mathematics, physics, engineering, and computer science. Understanding how to manipulate and express complex numbers in standard form is crucial for solving problems involving these numbers.

Why is the Standard Form Important?

Expressing complex numbers in standard form offers several advantages:

  1. Clarity and Uniqueness: The standard form a + bi provides a clear and unambiguous representation of a complex number. Each complex number has a unique representation in standard form, making comparisons and algebraic manipulations straightforward.
  2. Arithmetic Operations: Adding, subtracting, multiplying, and dividing complex numbers are greatly simplified when they are in standard form. The real and imaginary parts can be treated separately, making calculations more manageable.
  3. Geometric Interpretation: Complex numbers in standard form can be easily plotted on the complex plane, where the real part a corresponds to the horizontal axis and the imaginary part b corresponds to the vertical axis. This geometric representation provides a visual understanding of complex number operations and relationships.
  4. Applications in Various Fields: Many mathematical and scientific applications rely on the standard form of complex numbers. For instance, in electrical engineering, complex numbers are used to represent alternating currents and impedances. In quantum mechanics, they are fundamental to describing wave functions and quantum states.

In summary, the standard form of complex numbers is a fundamental concept that simplifies their understanding and use in various mathematical and scientific contexts. Converting a complex number to its standard form is often a crucial first step in solving problems involving complex numbers. Now, let's apply this understanding to the given expression.

Our objective is to express the complex number (1 - i/21)² in the standard form a + bi. This involves expanding the expression, simplifying it, and separating the real and imaginary parts. This problem will demonstrate the application of algebraic principles to complex numbers, emphasizing the significance of the imaginary unit i and its properties.

The given expression (1 - i/21)² is a complex number raised to the power of 2. To express this in standard form, we need to perform the following steps:

  1. Expand the expression: Use the binomial expansion or the FOIL (First, Outer, Inner, Last) method to expand the square of the binomial.
  2. Simplify the expression: Combine like terms and simplify any powers of i. Remember that i² = -1.
  3. Express in standard form: Separate the real and imaginary parts to express the result in the form a + bi, where a and b are real numbers.

By following these steps, we will transform the given expression into its standard complex number form. This process will reinforce our understanding of complex number arithmetic and algebraic manipulation. The result will clearly show the real and imaginary components of the original expression.

Let's break down the solution step-by-step:

Step 1: Expand the expression

We begin by expanding the expression (1 - i/21)² using the binomial expansion formula (A - B)² = A² - 2AB + B², where A = 1 and B = i/21:

(1 - i/21)² = 1² - 2(1)(i/21) + (i/21)²

This expansion sets the stage for simplifying the expression by applying the properties of complex numbers. The next step involves calculating each term in the expanded expression.

Step 2: Simplify the expression

Now, we simplify each term:

  • 1² = 1
  • 2(1)(i/21) = 2i/21
  • (i/21)² = i²/21² = i²/441

Substituting these values back into the expanded expression, we get:

1 - 2i/21 + i²/441

Recall that i² = -1. Substituting this into the expression, we get:

1 - 2i/21 + (-1)/441

Step 3: Combine real parts

Now, we combine the real parts of the expression:

1 - 1/441 = (441 - 1)/441 = 440/441

Our expression now looks like this:

440/441 - 2i/21

This step simplifies the expression by combining the real number components, making it easier to identify the real part of the complex number.

To express the result in the standard form a + bi, we need to identify the real part (a) and the imaginary part (b). From the simplified expression:

440/441 - 2i/21

We can see that:

  • The real part, a, is 440/441.
  • The imaginary part, b, is -2/21.

Therefore, we can write the complex number in standard form as:

440/441 + (-2/21)i

This is the standard form a + bi, where a = 440/441 and b = -2/21.

In conclusion, the standard form of the complex number (1 - i/21)² is:

440/441 - (2/21)i

This result clearly shows the real part (440/441) and the imaginary part (-2/21) of the complex number. Expressing complex numbers in standard form simplifies their manipulation and interpretation in various mathematical and practical contexts. This step-by-step solution demonstrates the process of expanding, simplifying, and converting a complex expression into its standard form, highlighting the fundamental properties of complex numbers and their arithmetic.

This problem demonstrates several key concepts in complex number manipulation:

  1. Binomial Expansion: The ability to expand binomial expressions is crucial for simplifying complex numbers raised to a power. In this case, we used the formula (A - B)² = A² - 2AB + B².
  2. Properties of the Imaginary Unit: Understanding that i² = -1 is essential for simplifying expressions involving the imaginary unit. This property allows us to eliminate i² terms and express the complex number in standard form.
  3. Standard Form (a + bi): Expressing complex numbers in standard form makes it easier to identify the real and imaginary parts, perform arithmetic operations, and visualize the number on the complex plane.
  4. Combining Real and Imaginary Parts: Simplifying complex expressions involves combining like terms, which means grouping the real parts and the imaginary parts separately.
  5. Fractions and Simplification: Working with complex numbers often involves fractions, so proficiency in fraction arithmetic is necessary for simplifying expressions.

By mastering these concepts, you can confidently manipulate complex numbers and solve a wide range of problems in mathematics, physics, engineering, and other fields. Practice is key to developing fluency in complex number arithmetic and algebra.

To further solidify your understanding of complex numbers and their manipulation, consider trying the following exercises:

  1. Express the following complex numbers in standard form:
    • (2 + 3i)²
    • (1 - i/5)³
    • (3 + 4i)/(1 - i)
  2. Simplify the following expressions:
    • (i⁵ + i¹⁰ + i¹⁵)
    • (1 + i)⁴
    • (1/i)
  3. Solve the following equations for complex numbers:
    • z² + 4 = 0
    • (z + 1)² = -9
    • 2z + 3i = 5 - i

Working through these problems will help you develop a deeper understanding of complex number arithmetic, algebraic manipulation, and problem-solving techniques. Remember to focus on the fundamental principles and properties of complex numbers, and don't hesitate to review the concepts and examples discussed in this article.

By consistently practicing and applying your knowledge, you can become proficient in working with complex numbers and unlock their power in various mathematical and scientific applications.