Problem 16 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ (2+5 i)+(4-6 i) $$

Step-by-Step Solution

Answer

The result is \(6 - i\).

Step 1: Identifying the Terms

The given expression is \((2 + 5i) + (4 - 6i)\). Identify and separate the real and imaginary parts of each complex number: \(2 + 5i\) and \(4 - 6i\).

Step 2: Adding Real Parts

Add the real parts of the complex numbers: \(2\) from \(2+5i\) and \(4\) from \(4-6i\). This gives: \(2 + 4 = 6\).

Step 3: Adding Imaginary Parts

Add the imaginary parts of the complex numbers: \(5i\) from \(2+5i\) and \(-6i\) from \(4-6i\). This results in: \(5i + (-6i) = -1i\) or just \(-i\).

Step 4: Combine Results

Combine the results of Step 2 and Step 3 to write the final expression in the form \(a + bi\). The final result is \(6 - i\).

Key Concepts

Addition of Complex Numbers

Complex numbers are numbers that have two primary components: a real part and an imaginary part. To add two complex numbers, we need to add their corresponding real and imaginary parts separately. The basic form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\).

When you are given an expression like \((2 + 5i) + (4 - 6i)\), the addition involves two steps:

• First, add the real components of the complex numbers. In this case, add \(2\) and \(4\), resulting in \(6\).
• Then, add the imaginary components, which are \(5i\) and \(-6i\). This gives us \(-i\), simplifying from \(5i + (-6i)\).

Finally, combine these results to get the summed complex number \(6 - i\). By following these steps, you can efficiently add any pair of complex numbers, always ending up with a result in the standard form \(a + bi\).

Real Part

The real part of a complex number is the component that does not involve the imaginary unit \(i\). It is represented by the term \(a\) in the general form \(a + bi\). Focusing on the real part allows us to interpret complex numbers similarly to ordinary real numbers in standard arithmetic.

Considering the original expression \((2 + 5i) + (4 - 6i)\), the real parts of these numbers are \(2\) and \(4\), respectively. By adding these numbers together \(2 + 4\), we get \(6\), which serves as the real part of the resulting sum.

Understanding the real component of complex numbers is crucial because it provides the grounding in real number arithmetic within the context of complex numbers, helping to establish the framework upon which imaginary and complex number operations build.

Imaginary Part

The imaginary part of a complex number is the component that involves the imaginary unit \(i\). This part is represented by \(bi\) in the expression \(a + bi\), with \(b\) being a real number.

In the expression \((2 + 5i) + (4 - 6i)\), our focus on imaginary parts highlights \(5i\) and \(-6i\). When we perform the addition \(5i + (-6i)\), the calculation equates to \(-i\), demonstrating how imaginary parts interact directly with one another, just like the real parts.

Understanding how to work with the imaginary parts is vital to mastering complex arithmetic. It underscores how complex numbers extend the real number system while providing powerful tools for solving a variety of mathematical problems, particularly those involving phenomena that cannot be captured by real numbers alone.