Problem 12 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The result is \(6 - i\).

Step 1: Identify Components of Complex Numbers

The expression given is \((2+5i) + (4-6i)\). Each complex number has a real part and an imaginary part. For \(2+5i\), the real part is 2 and the imaginary part is 5. For \(4-6i\), the real part is 4 and the imaginary part is -6.

Step 2: Add the Real Parts

To simplify the expression, first add the real parts of the complex numbers. The real parts are 2 and 4.Calculate: \(2 + 4 = 6\).

Step 3: Add the Imaginary Parts

Next, add the imaginary parts of the complex numbers. The imaginary parts are 5 and -6.Calculate:\(5 + (-6) = -1\).

Step 4: Combine Real and Imaginary Sums

Now, combine the sums of the real parts and imaginary parts to get the result in the form \(a + bi\).The expression becomes:\(6 - i\).

Key Concepts

Real Part

In complex numbers, the real part is the component which does not involve the imaginary unit, denoted as \(i\). When you look at a complex number such as \(2 + 5i\), the real part is the number \(2\). This is the same component you would encounter in regular algebra with real numbers.
To find the real part of a complex number, you simply identify the coefficient that stands alone without the imaginary unit \(i\). This is an essential step whenever working with complex numbers, as it guides you in operations like addition, subtraction, and many more.
Understanding the real part is critical for the correct categorization and manipulation of complex numbers in both mathematical and applied contexts.

Imaginary Part

The imaginary part of a complex number involves the multiplication factor of the imaginary unit \(i\) — a unique element that represents the square root of negative one. In the example \(2 + 5i\), the imaginary part is \(5\). It signifies the component opposite to what we defined as the real part.
Recognizing and isolating the imaginary part of a complex number is crucial, especially in problems requiring operations on complex numbers like addition or subtraction. By identifying the factor paired with \(i\), you ensure precision in computations and solutions. Imaginary parts are not just abstract concepts; they have practical applications in fields like engineering and quantum physics.

Adding Complex Numbers

When adding complex numbers, it's vital to manage the real and imaginary parts separately. Let's consider two complex numbers: \(2 + 5i\) and \(4 - 6i\). To add them, you need to simply combine the respective parts:

• Real Parts: Add the real parts together. For \(2 + 5i\) and \(4 - 6i\), that is \(2 + 4\), resulting in \(6\).
• Imaginary Parts: Likewise, add the imaginary parts. For our numbers, that is \(5 + (-6)\), resulting in \(-1\).

The final step is to express these combined results as a single complex number, \(6 - i\). This systematic approach ensures accuracy when performing operations with complex numbers and helps keep your calculations error-free.