Problem 39 - Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i.\) $$ (6-i)+(9+3 i) $$

Step-by-Step Solution

Answer

The result is \(15 + 2i\).

Step 1: Identify the Real Parts

We begin by identifying the real parts of each complex number. In the expression \((6-i) + (9+3i)\), the real parts are 6 and 9.

Step 2: Add the Real Parts

Add the real parts identified in Step 1: \(6 + 9 = 15\). This is the real part of the resulting complex number.

Step 3: Identify the Imaginary Parts

Next, identify the imaginary parts from each complex number. In \((6-i) + (9+3i)\), the imaginary parts are -1 and +3.

Step 4: Add the Imaginary Parts

Add the imaginary parts: \(-1 + 3 = 2\). This is the imaginary part of the resulting complex number.

Step 5: Combine Real and Imaginary Parts

Now combine the sums from Step 2 and Step 4 to form \(15 + 2i\), which is the final answer in the form \(a + bi\).

Key Concepts

Addition of Complex Numbers

Complex numbers are fascinating as they bridge the gap between algebra and geometry, incorporating both real and imaginary components. When asked to add two complex numbers, such as (6-i) + (9+3i), the task is straightforward.
We handle the real and imaginary parts separately. Here's how the process works:

• Extract the Real Components: From 6 - i and 9 + 3i, we identify the real parts: '6' from the first and '9' from the second.
• Add the Real Components: Just like regular numbers, add these real parts together: 6 + 9 = 15.
• Extract the Imaginary Components: Recognize the imaginary parts: '-i' (or '-1i') and '3i'.
• Add the Imaginary Components: Sum the imaginary parts: -1 + 3 = 2. This results in the new imaginary part 2i.

Finally, combining the results from both real and imaginary calculations, we get 15 + 2i. Such simple steps help us easily solve addition problems involving complex numbers.
Remembering to handle the components separately prevents mistakes and allows us to confidently work with more complex equations in the future.

Imaginary Numbers

Imaginary numbers often seem, well, imaginary when first introduced, but they have practical uses in fields ranging from engineering to physics. At its heart, an imaginary number is a number that gives a negative result when squared.
This key property revolves around the imaginary unit, denoted as 'i', where i is defined such that i² = -1.
In complex numbers, an imaginary number pairs with a real number for more nuanced equations.
Examples:

• The imaginary part of 6-i is '-i', equivalent to '-1i'.
• In 9+3i, the imaginary part is ' 3i'.

When adding or manipulating these numbers, treat these imaginary parts independently, just like the real numbers.
It's essential to understand that even though they're termed "imaginary" due to their initial abstraction, these numbers have very real applications in modern technology, making them an exciting area of mathematics to explore further.

Real Numbers

Real numbers are the numbers we've known and loved since primary school. They include all the integers, fractions, and decimals we use every day.
When we deal with complex numbers like 6-i or 9+3i, the real component is quite straightforward as it is just the regular number part of the expression.
The real number is what is added to or subtracted from the imaginary part to build a complex number.
Let's illustrate with examples:

• In 6-i, '6' is the real part.
• In 9+3i, '9' is the real part.

When combining complex numbers, we focus separately on these real parts to ensure calculations remain clear and correct.
Ultimately, recognizing and working with the real number part of complex numbers helps maintain simplicity and accuracy in mathematical equations, making it an indispensable skill in higher-level math.