Problem 17 - Algebra 2

Question

Simplify. $$ (8+6 i)-(2+3 i) $$

Step-by-Step Solution

Answer

The expression simplifies to \(6 + 3i\).

Step 1: Distribute the Negative Sign

In the expression \[(8 + 6i) - (2 + 3i),\]we need to distribute the negative sign across the second group of terms. This changes \[-(2 + 3i)\]to \[-2 - 3i.\] The expression becomes:\[8 + 6i - 2 - 3i.\]

Step 2: Group Similar Terms

Now, arrange and group the like terms (real and imaginary parts) together:\[(8 - 2) + (6i - 3i).\]

Step 3: Simplify Real Parts

Solve the real part: \[8 - 2 = 6.\]

Step 4: Simplify Imaginary Parts

Solve the imaginary part: \[6i - 3i = 3i.\]

Step 5: Write the Simplified Expression

Combine the simplified real and imaginary parts to find the final simplified form of the expression: \[6 + 3i.\]

Key Concepts

Imaginary Unit

One of the fundamental components of complex numbers is the imaginary unit, often denoted by the symbol \(i\). The imaginary unit is unique because it is defined as \(i^2 = -1\). In mathematics, this allows for a more comprehensive number system that can solve equations which real numbers alone cannot address.
The imaginary unit \(i\) forms the backbone of imaginary numbers and plays a crucial role in complex numbers, which combine real and imaginary parts.

• It helps in performing operations that involve square roots of negative numbers.
• When multiplying complex numbers, the property \(i^2 = -1\) assists in simplifying expressions.

For example, in the given exercise, the term \(3i\) in the expression represents an imaginary part. Understanding \(i\) is key to working comfortably with complex numbers and recognizing how imaginary components combine or simplify.

Real Part

Complex numbers are composed of two parts: the real part and the imaginary part. The real part is the component of the complex number that does not involve the imaginary unit \(i\).
In our original expression \((8 + 6i) - (2 + 3i)\), the real parts are \(8\) and \(-2\). To simplify the expression, we first identify and group the real parts separately from the imaginary components.

• In the expression \((8 + 6i) - (2 + 3i)\), the real part of the result is obtained by subtracting \(2\) from \(8\), giving us \(6\).

The real part in a complex number refers to the horizontal axis if we were to graph complex numbers on a plane. This graphing approach often helps students better visualize how both real and imaginary parts interact.

Imaginary Part

The imaginary part of a complex number is what gives it its unique traits, being the term that includes the imaginary unit \(i\).
In the expression \((8 + 6i) - (2 + 3i)\), the imaginary parts are \(6i\) and \(-3i\). By grouping these terms, we can simplify them by basic arithmetic involving \(i\).

• The imaginary part is found by subtracting \(3i\) from \(6i\), resulting in \(3i\).

Imaginary parts cannot be processed in isolation like real numbers, but they are crucial to formulating a complete solution alongside real components.
Understanding and manipulating them alongside real parts allow us to address a broader class of problems within mathematics, especially in fields involving waves and oscillations, like electrical engineering and physics.