Problem 5 - Precalculus

Question

Determine the real numbers \(c\) and \(d\) such that $$ 8-3 i=2 c+d i $$

Step-by-Step Solution

Answer

c = 4, d = -3.

Step 1: Understand the Equation

We are given the equation \( 8 - 3i = 2c + di \). Our goal is to find the real numbers \( c \) and \( d \) that satisfy this equation.

Step 2: Equate the Real Parts

Identify the real parts of both sides of the equation. On the left side, the real part is 8, and on the right side, the real part is \( 2c \). Therefore, we equate them: \[ 8 = 2c \]

Step 3: Solve for c

From the equation \( 8 = 2c \), divide both sides by 2 to find \( c \). Thus, \[ c = \frac{8}{2} = 4 \]

Step 4: Equate the Imaginary Parts

Identify the imaginary parts of both sides of the equation. On the left side, the imaginary part is \(-3\), and on the right side, the imaginary part is \( d \). Therefore, we equate them: \[ -3 = d \]

Step 5: Conclusion

The values for \( c \) and \( d \) that satisfy the given equation are \( c = 4 \) and \( d = -3 \).

Key Concepts

Equating Real and Imaginary Parts

When working with equations involving complex numbers, it's essential to understand the concept of equating parts. Complex numbers have both a real part and an imaginary part. For example, in the complex number \(8 - 3i\), 8 is the real part, and \(-3i\) is the imaginary part.
When we have an equation involving complex numbers, like \(8 - 3i = 2c + di\), we can separate it into two equations. The real parts on either side are equal, just as the imaginary parts must be equal.
This means we'll create one equation from the real components: \(8 = 2c\). Then, we extract another equation from the imaginary parts: \(-3 = d\). This way of breaking down the complex number equation helps isolate the variables, making it easier to solve for the unknowns.
So, by equating the real and imaginary parts, we maintain equality in the equation, facilitating solutions even in a domain involving complex numbers.

Solving Equations

Solving equations is a fundamental skill in mathematics. It involves finding the value of unknown variables that satisfy a given equation.
When solving the equation \(8 = 2c\) for the variable \(c\), the process involves isolating \(c\) on one side of the equation. This can be done by performing arithmetic operations.
For instance, dividing both sides of \(8 = 2c\) by 2 results in \(c = 4\). A similar method applies to the equation \(-3 = d\). Here, \(d\) is already isolated, making it evident that \(d = -3\).
The key to solving equations is to perform operations that simplify the expressions, ensuring the operations are reversible, which retains the balance of the equality. It's like peeling layers off until you reveal the unknown variable.

Real Numbers

In mathematics, real numbers include all the numbers that exist on the number line. This includes positive numbers, negative numbers, and zero.
Real numbers are versatile and appear in various contexts, from simple arithmetic to complex number calculations. These numbers do not have imaginary components.
In the context of our problem, we found that \(c = 4\) and \(d = -3\) are both real numbers. Real numbers like \(4\) and \(-3\) can be easily located on a standard number line, unlike imaginary numbers, which have a different set of properties.
Understanding real numbers helps in distinguishing them from complex numbers. Knowing that complex numbers have a structure \(a + bi\), where \(a\) and \(b\) are real numbers, provides a foundation for recognizing and handling real numbers within various mathematical operations.