Question

Solve for all possible values of the real numbers \(x\) and \(y\) in the following equations. $$x+i y=0$$

Short answer

The solutions are \(x = 0\) and \(y = 0\).

Step-by-step solution

Understand the given equation

The equation given is \(x + i y = 0\), where \(i\) is the imaginary unit such that \(i^2 = -1\).

Equate real and imaginary parts

For the equation to hold, both the real part (\(x\)) and imaginary part (\(i y\)) of the left side must equal the real part (0) and imaginary part (0) on the right side. Thus, the real part \(x\) must equal 0 and \(i y\) must also equal 0.

Solve for the real part

From the real part, we get the equation \(x = 0\).

Solve for the imaginary part

The imaginary part gives us \(i y = 0\). To satisfy this, \(y\) must also be 0, because \(i\) is a non-zero constant.

Write the final solution

Combining both results, the real number solutions for the equations \(x + i y = 0\) are \(x = 0\) and \(y = 0\).

Key concepts

Real and Imaginary Parts

In complex numbers, each number is generally written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The term \(a\) is called the real part, while \(bi\) is the imaginary part. Understanding how to separate and work with these two parts is crucial when solving complex equations.
For instance, if we take a complex number like \(3 + 4i\), the real part is \(3\) and the imaginary part is \(4i\).

In the given exercise, we have the equation \(x + i y = 0\). Here, \(x\) is the real part and \(i y\) is the imaginary part. Both sides of the equation must balance individually for the equation to hold. This means we need to solve separately for the real part and the imaginary part of the equation.

Solving Equations

Solving complex equations involves breaking them down into their real and imaginary components. Let's revisit the step-by-step solution:

• Step 1: Understand the Equation
  We start with the equation \(x + i y = 0\).
• Step 2: Equate Real and Imaginary Parts
  For the equation to be true, both real and imaginary parts on the left must equal those on the right. Since the right side is \(0\), its real and imaginary parts are both \(0\).
• Step 3: Solve for the Real Part
  Set the real part \(x\) equal to \(0\): \(x = 0\).
• Step 4: Solve for the Imaginary Part
  For the imaginary part, set \(i y\) equal to \(0\). Since \(i\) is not zero, \(y\) must be \(0\).
• Step 5: Write the Final Solution
  Combining the solutions from both parts, we find that the complete solution is \(x = 0\) and \(y = 0\).

These steps illustrate solving equations involving complex numbers by treating their real and imaginary components separately.

Imaginary Unit

The imaginary unit \(i\) is fundamental in complex numbers. By definition, \(i\) satisfies the equation \(i^2 = -1\). This unique property allows us to define and work with numbers that extend beyond the real number line into what we call the complex plane.

When solving equations, recognizing the role of \(i\) helps in separating and balancing both the real and imaginary parts. For example, in our exercise with \(x + i y = 0\), the imaginary unit informs us that \(i y\) must simplify to zero for the equation to hold, which forces \(y\) to be zero since \(i\) itself is non-zero.

Here are a few key points about \(i\):

• \(i\) represents the square root of \(-1\).
• Multiplying an imaginary unit by itself gives \(-1\) (i.e., \(i^2 = -1\)).
• The imaginary unit allows for complex solutions and extends traditional algebra into complex algebra.

Understanding \(i\) and its properties is crucial when working with complex numbers and solving related equations.