Problem 15 - Algebra for College Students

Question

Add or subtract as indicated. $$(-7+3 i)-(5-2 i)$$

Step-by-Step Solution

Answer

The result is \(-12 + 5i\).

Step 1: Identify the Complex Numbers

The expression given for addition or subtraction is \[ (-7 + 3i) - (5 - 2i) \] which consists of two complex numbers: 1. \(-7 + 3i\)2. \(5 - 2i\).

Step 2: Distribute the Negative Sign

Subtracting the second complex number is equivalent to adding its additive inverse. Thus, distribute the negative sign to each term inside the second set of brackets:\[ -7 + 3i - 5 + 2i \]

Step 3: Group Like Terms

Combine the real parts and the imaginary parts separately. The real parts are \(-7\) and \(-5\), and the imaginary parts are \(3i\) and \(2i\):\[ (-7 - 5) + (3i + 2i) \]

Step 4: Simplify Each Group

Calculate each group:1. For the real parts: \(-7 - 5 = -12\)2. For the imaginary parts: \(3i + 2i = 5i\)Thus, the simplified expression is:\[-12 + 5i\]

Key Concepts

Addition and Subtraction of Complex Numbers

When working with complex numbers, addition and subtraction follow similar logic to combining like terms in algebra. A complex number is generally expressed in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part with \(i\) representing the imaginary unit. For example, in the problem \((-7 + 3i) - (5 - 2i)\), you have two complex numbers.

To subtract complex numbers, combine their respective parts separately. You first need to distribute the negative sign across the second complex number, resulting in the expression: \(-7 + 3i - 5 + 2i\). Here, the operation turns into simple addition of real and imaginary parts:

• Real parts: \(-7\) and \(-5\)
• Imaginary parts: \(3i\) and \(2i\)

Thus, you get the final result: \(-12 + 5i\). Complex addition and subtraction are critical in solving problems involving electrical engineering, quantum physics, and more, where complex numbers frequently represent alternating currents or waves.

Imaginary Numbers

Imaginary numbers form a fascinating part of mathematics, introducing the concept of imaginary units. The imaginary unit, denoted as \(i\), is specially defined by the property that \(i^2 = -1\). This means \(i\) is used to represent the square root of negative numbers, which is not possible using real numbers.

For example, when dealing with a simple expression like \(3i + 2i\), you are adding two imaginary numbers. This follows typical arithmetic rules and results in \(5i\), similar to adding elements with a common variable in algebra. While \(i\) itself cannot be visualized on the traditional number line, imaginary numbers are an essential component of the complex number system used to solve equations that have no real solutions otherwise.

Imaginary numbers find applications in many fields, such as engineering and physics, especially in dealing with signal processing and control systems.

Real and Imaginary Parts

Every complex number consists of two parts: a real and an imaginary part. The real part of a complex number \(a + bi\) is \(a\), and the imaginary part is \(b\). This duality forms the foundation of complex arithmetic.

In the expression \(-7 + 3i\):

• The real part is \(-7\)
• The imaginary part is \(3\)

In a subtraction problem like \((-7 + 3i) - (5 - 2i)\), you address the real and imaginary components separately to achieve clarity and coherence in calculations. This means computing the real parts \((-7 - 5)\) to get \(-12\), and for the imaginary parts \((3i + 2i)\) to get \(5i\).

Understanding the separation of these parts allows for more efficient calculations, especially when utilizing complex numbers within mathematical operations like calculus, discrete mathematics, or electrical circuits. This bifurcation helps in visualizing the components of complex numbers on a two-dimensional plane called the complex plane.