Problem 15 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ (2-5 i)+(3+4 i) $$

Step-by-Step Solution

Answer

The expression evaluates to \(5 - i\).

Step 1: Identify the Terms

The given expression is \((2 - 5i) + (3 + 4i)\). Here, we have two complex numbers: \(2 - 5i\) and \(3 + 4i\). Each complex number is in the form \(a + bi\), where \(a\) and \(b\) are real numbers.

Step 2: Add the Real Parts

Add the real parts of both complex numbers. The real part of \(2 - 5i\) is \(2\), and the real part of \(3 + 4i\) is \(3\). So, the sum of the real parts is \(2 + 3 = 5\).

Step 3: Add the Imaginary Parts

Add the imaginary parts of both complex numbers. The imaginary part of \(2 - 5i\) is \(-5i\), and the imaginary part of \(3 + 4i\) is \(4i\). So, the sum of the imaginary parts is \(-5i + 4i = -1i\).

Step 4: Write the Result

Combine the results from Steps 2 and 3 to write the final expression in the form \(a + bi\). The result is \(5 - 1i\), or simply \(5 - i\).

Key Concepts

Adding Complex Numbers

When you add complex numbers, it might seem more complicated than simple arithmetic. However, it simply involves combining like terms. Complex numbers are expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Here, you follow a process similar to adding polynomials.

When adding complex numbers such as \((2 - 5i) + (3 + 4i)\), you need to work with each part separately. This means:

• Add the real parts together (\(2 + 3\)).
• Add the imaginary parts together (\(-5i + 4i\)).

After calculating each part, combine them to form the sum. This method will always guide you to correctly add any pair of complex numbers. The end result is a new complex number, simplified in the form \(a + bi\). Thus, our final result is \(5 - i\).

Real Part

The real part of a complex number is straightforward to identify. In the form \(a + bi\), the real part is \(a\). This is the non-imaginary component, which is treated like any normal real number in operations. Consider the numbers \(2 - 5i\) and \(3 + 4i\):

• For \(2 - 5i\), the real part is \(2\).
• For \(3 + 4i\), the real part is \(3\).

When calculating the sum of the real parts, you perform standard arithmetic. Add these values together: \(2 + 3 = 5\).

It's essential to handle the real parts separately from the imaginary parts because they are different kinds of numbers. Mixing them up can lead to mistakes. This separation ensures clarity and accuracy in your calculations.

Imaginary Part

The imaginary part of a complex number is the component that includes "\(i\)", the imaginary unit. The imaginary unit \(i\) is defined as \(\sqrt{-1}\), which represents a number that, when squared, equals \(-1\).

In our expression \(2 - 5i\) and \(3 + 4i\), the imaginary parts are:

• \(-5i\) from \(2 - 5i\)
• \(4i\) from \(3 + 4i\)

To find the resultant imaginary part when adding, simply sum these components: \(-5i + 4i = -1i\) or \(-i\). The imaginary parts should never be combined with real parts during addition, as they represent different dimensions.

Understanding how to handle the imaginary part helps ensure you perform operations on complex numbers accurately, leading to a correct and simplified result expressed as \(a + bi\). When you're adding complex numbers, remember this separation of parts, so your final answer maintains meaning and structural integrity.