Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ (5-6 i)+(4+2 i) $$

Step-by-Step Solution

Answer

The sum is \(9 - 4i\).

Step 1: Identify the Real Parts

Look at the complex numbers given: \((5-6i)\) and \((4+2i)\). Identify the real parts, which are 5 and 4, respectively.

Step 2: Identify the Imaginary Parts

Next, identify the imaginary parts of the complex numbers. For \((5-6i)\), the imaginary part is -6, and for \((4+2i)\), it is 2.

Step 3: Add the Real Parts

Add the real parts of the complex numbers: \(5 + 4 = 9\).

Step 4: Add the Imaginary Parts

Add the imaginary parts: \(-6 + 2 = -4\).

Step 5: Combine the Results

Combine the results from Step 3 and Step 4 to form the new complex number: \(9 - 4i\).

Key Concepts

Understanding Complex Number Addition

In the world of complex numbers, addition is a primary operation that combines two complex numbers into a new one. A complex number typically has the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. Adding complex numbers involves adding their corresponding real parts separately from their imaginary parts.

Consider two complex numbers: \((5 - 6i)\) and \((4 + 2i)\).

• First, identify the real components (\(5\) and \(4\)).
• Next, identify the imaginary components (\(-6i\) and \(2i\)).

Add these components as you would in arithmetic:

• Real parts: \(5 + 4 = 9\)
• Imaginary parts: \(-6i + 2i = -4i\)

Combine these results, and you have your new complex number: \(9 - 4i\).

This method of combining real and imaginary parts separately ensures that complex numbers remain in their standard form, simplifying calculations and comparisons.

Real and Imaginary Parts Explained

Every complex number is made up of two distinct parts: the real part and the imaginary part. Understanding these parts is crucial when working with complex numbers.

For example, in the complex number \(5 - 6i\):

• The 15 2 refers to the real part. It operates just like a regular number on the number line.
• The 1-6i 2 signifies the imaginary part. The letter 1i 2 here represents the square root of -1, a fundamental unit in complex numbers.

Similarly, in the complex number \(4 + 2i\):

• 14 2 is the real part.
• 12i 2 is the imaginary part.

Recognizing these parts allows us to perform operations like addition or multiplication effectively. When plotting on a spectral diagram or Argand plane, these components will place the number accurately in a two-dimensional space.

Mastering Complex Arithmetic

Complex arithmetic is an extension of traditional arithmetic that works with both real and imaginary numbers. It opens up a realm of mathematics necessary for fields like engineering, physics, and signal processing.

Key operations in complex arithmetic include:

• **Addition**: As observed earlier, complex addition involves combining like parts 6 real with real, imaginary with imaginary.
• **Subtraction**: This operation mirrors addition but instead involves "subtracting the imaginary and real parts of the complex numbers.
• **Multiplication**: Involves using distributive properties and substituting 1i^2 = -1 2 to simplify results.
• **Division**: Often performed by multiplying by the conjugate to remove the imaginary unit from the denominator.

Complex arithmetic allows calculations that accommodate directions and magnitudes, contributing to a deeper understanding of functions in mathematics.

This arithmetic, through its combination of real and imaginary components, facilitates the processing of signals and can model oscillations and waves, thereby expanding our computational abilities.