Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (5-2 i)^{2} $$

Step-by-Step Solution

Answer

The expression is \(21 - 20i\).

Step 1: Identify the Expression

The expression given is \((5 - 2i)^{2}\). We are required to expand this expression and rewrite it in the standard form \(a + bi\), where \(a\) and \(b\) are real numbers.

Step 2: Use the Binomial Theorem

Using the binomial theorem, recall that \((x-y)^2 = x^2 - 2xy + y^2\). Apply this formula with \(x = 5\) and \(y = 2i\).

Step 3: Square the Real Part

Calculate \(5^2 = 25\).

Step 4: Calculate Twice the Product

Compute \(-2 \cdot 5 \cdot 2i = -20i\).

Step 5: Square the Imaginary Part

Calculate \((2i)^2 = 4i^2\), and since \(i^2 = -1\), this becomes \(-4\).

Step 6: Combine Like Terms

Add the results from the previous steps: \(25 - 20i - 4 = 21 - 20i\).

Step 7: Write in Standard Form

The expression \((5 - 2i)^2\) can be written in the form \(a + bi\) as \(21 - 20i\), where \(a = 21\) and \(b = -20\).

Key Concepts

The Binomial Theorem

The binomial theorem is a powerful algebraic tool used to expand expressions of the form \((x + y)^n\) for any positive integer \(n\). In the context of complex numbers, it allows us to simplify expressions by breaking them down into easier-to-manage parts.
If you consider the expression \((5 - 2i)^2\), the binomial theorem comes into play.

• First, note that \(x = 5\) and \(y = -2i\).
• Apply the theorem: \((x-y)^2 = x^2 - 2xy + y^2\).
• This results in the expression \(5^2 - 2(5)(2i) + (2i)^2\).

This step-by-step approach, guided by the binomial theorem, helps simplify complex numbers into a form that's easier to understand and manipulate.

Imaginary Unit

The imaginary unit, represented as \(i\), is a fundamental concept in mathematics and plays a key role in complex numbers. It is defined by the property \(i^2 = -1\).
This unique feature allows mathematicians and engineers to work with quantities that involve the square root of negative numbers, something not possible within the real number system.

• When multiplying by \(i\), such as in \(2i\), you create an imaginary number.
• When squaring an imaginary number like \((2i)^2\), you multiply \(2i \times 2i\), resulting in \(4i^2\).
• Given \(i^2 = -1\), \(4i^2\) simplifies to \(-4\).

Understanding the imaginary unit \(i\) is crucial for manipulating and simplifying complex numbers effectively.

Complex Conjugate

Complex conjugates are closely related to complex numbers and are crucial for many mathematical operations. For a complex number \(z = a + bi\), its complex conjugate is \(\overline{z} = a - bi\).
The complex conjugate has several important properties:

• When you multiply a complex number by its conjugate, the result is always a real number. For example, \((5 - 2i)(5 + 2i) = 5^2 - (2i)^2 = 25 - (-4) = 29\).
• Complex conjugates are helpful in division involving complex numbers, ensuring the denominator becomes a real number.
• This makes computations with complex numbers in standard form \(a + bi\) much simpler.

While the specific exercise doesn't require finding a complex conjugate, understanding this concept is essential for comprehensively grasping complex numbers.

Real and Imaginary Parts

Every complex number \(z\) can be expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Understanding these components will help you work effectively with complex numbers.
Here's how you identify each part:

• The real part \(a\) is the component not associated with \(i\). In the expression \(21 - 20i\), \(21\) is the real part.
• The imaginary part \(b\) is the coefficient of \(i\). In this example, \(-20\) is the imaginary part.
• A complex number in standard form is written as \(a + bi\), facilitating operations such as addition, subtraction, and multiplication.

By clearly identifying the real and imaginary parts, applying mathematical operations becomes more straightforward.