Problem 18 - Algebra for College Students

Question

Add or subtract as indicated. $$(-4-12 i)+(-3+16 i)$$

Step-by-Step Solution

Answer

The result is \(-7 + 4i\).

Step 1: Identify the Real and Imaginary Parts

The expression \((-4 - 12i) + (-3 + 16i)\) consists of real numbers and imaginary parts. Identify the real numbers, \(-4\) and \(-3\), then identify the imaginary components, \(-12i\) and \(16i\).

Step 2: Add the Real Parts

Combine the real parts: \(-4 + (-3) = -7\).

Step 3: Add the Imaginary Parts

Combine the imaginary parts: \(-12i + 16i\). Calculate as \(-12 + 16 = 4\) to get \(4i\).

Step 4: Write the Result as a Complex Number

The result of adding the real and imaginary parts is \(-7 + 4i\).

Key Concepts

Real Numbers

Real numbers are a core part of complex numbers. Think of them as the numbers you use every day. They include both positive and negative integers, fractions, and even decimals, without including any imaginary units like 'i'.

In the context of complex numbers, the real number is the part that does not have an 'i' next to it. For example, in a complex number like \(-4-12i\), the real number is \(-4\). It represents a position on the number line that runs horizontally. Real numbers help give meaning to the numeric quantity of the complex number without including an imaginary component.

Imaginary Numbers

Imaginary numbers might sound complicated, but they're actually pretty simple once you understand the basics. Imaginary numbers are defined by the unit 'i', where \(i^2 = -1\).

This means that imaginary numbers are essentially square roots of negative numbers, something that isn't possible with real numbers alone. In the expression \(-4 - 12i\), the \(-12i\) is the imaginary component. It tells us how the complex number behaves in the vertical direction on a complex plane.

Imaginary numbers work alongside real numbers to form complex numbers. They allow us to solve equations that have no real solutions and are essential in expressing quantities in many fields, such as physics and engineering.

Addition of Complex Numbers

Adding complex numbers becomes straightforward once you know how to handle the real and imaginary components separately. Here's the step-by-step approach:

• **Identify the parts**: Break the complex numbers into their real and imaginary parts. For instance, \((-4-12i) + (-3+16i)\) has real parts \(-4\) and \(-3\), and imaginary parts \(-12i\) and \(16i\).
• **Combine real parts**: Add the real numbers together. Here, \(-4 + (-3) = -7\). These are combined as they would be in any normal addition of real numbers.
• **Combine imaginary parts**: Add the imaginary parts next. Keep the 'i' aside and just add the numerical coefficients: \(-12 + 16 = 4\), which leads to a result of \(4i\).
• **Form the new complex number**: After combining, the complete complex number is now \(-7 + 4i\).

Addition doesn't change the complex nature; it simply combines like terms for an elegant solution, keeping the form \(a + bi\), where \(a\) is real and \(b\) is the imaginary coefficient.