Question

Find the real numbers \(a\) and \(b\) such that the equation is true. $$a+b i=7+12 i$$

Short answer

The real number \(a\) is 7 and the imaginary number \(b\) is 12.

Step-by-step solution

Comparing Real Parts

In the given equation, the real parts are \(a\) from \(a+bi\) and 7 from \(7 + 12i\). For the equation to hold true, the real parts of both sides must be equal. Therefore we equate \(a = 7\).

Comparing Imaginary Parts

Similarly, the imaginary parts are \(b\) from \(a+bi\) and 12 from \(7 + 12i\). For the equation to hold true, we must have the imaginary parts of both sides being equal. Therefore we equate \(b = 12\).

Key concepts

Understanding the Real Part in Complex Numbers

When working with complex numbers, each is expressed in the form of \(a + bi\), where \(a\) represents the real part and \(bi\) represents the imaginary part. The real part is the component without the imaginary unit \(i\). In any complex number, the real part tells us its position along the horizontal axis of the complex plane. If you think of a complex number on a graph, the real part is simply how far left or right the number is from the origin.
The key point when dealing with equations involving complex numbers is that the real parts must match on both sides of the equation. If you have \(a + bi = c + di\), then \(a = c\). This concept is crucial because it allows us to separately solve for the real component in equations. In our exercise, with \(a + bi = 7 + 12i\), it is clear from the equation that the real part of our number, denoted by \(a\), must equal 7.

Deciphering the Imaginary Part in Complex Numbers

The imaginary part in a complex number is the component that includes the imaginary unit \(i\), noted as \(bi\) in \(a + bi\). This part contributes to the number's position along the vertical axis of the complex plane, indicating its 'imaginary' distance from the origin.
Although it might seem complex initially, understanding the imaginary part is straightforward once you realize it behaves almost independently from the real part. In the equation \(a + bi = 7 + 12i\), we see that the imaginary part of \(a+bi\) is \(b\), and this must match the imaginary part of \(7 + 12i\), which is 12. Hence, it becomes evident that \(b = 12\).

• Always treat the real and imaginary parts as separate when solving equations.
• Remember that the imaginary part defines the position along the vertical axis.

The Art of Equation Solving with Complex Numbers

Solving equations involving complex numbers requires confidence, especially in separating the real and imaginary parts. This process is often referred to as 'equation solving with complex numbers', and it involves equating the respective real and imaginary parts on both sides of an equation.
In our problem, given the complex equation \(a + bi = 7 + 12i\), we approach it by first comparing the real parts and then the imaginary parts. This method simplifies the process into manageable steps, ensuring that each component is tackled independently.
The steps generally follow this pattern:

• Identify the real parts on both sides of the equation and set them equal. Solve for the unknown real component.
• Identify the imaginary parts and equate them too. Solve for the imaginary component.

This structured approach might seem repetitive, but its success lies in ensuring each part of the complex number is properly accounted for, leading to accurate solutions. Ultimately, it confirms our results, such as \(a = 7\) and \(b = 12\) in the exercise, aligning with the simple yet powerful concept that complex numbers are indeed comprised of independent real and imaginary parts.