Problem 25 - Algebra 2

Question

Find the additive inverse of each number. $$ 5-3 i $$

Step-by-Step Solution

Answer

The additive inverse of the number \( 5 - 3i \) is \( -5 + 3i \).

Step 1: Identify the Additive Inverse of a Real Number

The additive inverse of any number a is -a, which when added to a, results in zero.

Step 2: Identify the Additive Inverse of an Imaginary Number

Similarly, the additive inverse of any imaginary number bi is -bi, which when added to bi, results in zero.

Step 3: Combine the Additive Inverses

For a complex number a + bi, its additive inverse is -a - bi. Therefore, the additive inverse of the given number 5 - 3i is -5 + 3i.

Key Concepts

Understanding Complex Numbers

Complex numbers form a vast and fascinating field of mathematics. They consist of two parts: a real part and an imaginary part. In a complex number like \( a + bi \), \( a \) represents the real part and \( bi \) represents the imaginary part where \( i \) is the square root of \( -1 \). Complex numbers are crucial in various fields, including engineering, physics, and computer science, as they can describe oscillations, rotations, and many other phenomena that can't be explained with just real numbers.

When encountering complex numbers in algebra, it is essential to understand their structure so that you can perform operations like addition, subtraction, and finding the additive inverse, which refers to the number that when added to the original number, will result in zero. For example, for the complex number \( 5 - 3i \), the additive inverse is \( -5 + 3i \). This fundamental concept helps to solve equations and understand the symmetrical nature of the complex plane.

Imaginary Numbers and their Role

Imaginary numbers are often seen as the more mysterious part of complex numbers, yet they are no less important. An imaginary number is a value that gives a negative number when squared. The most common imaginary unit is \( i \), defined as \( i^2 = -1 \). This might seem abstract, but it's a crucial construct that helps us solve equations that have no real solutions, like \( x^2 + 1 = 0 \).

It's also worth noting that imaginary numbers are not 'imaginary' in the sense of being unreal or made-up; they are a genuine concept with practical applications. In the context of additive inverses, for an imaginary number \( bi \), its additive inverse is \( -bi \), as this will zero out the imaginary part when added to the original number. So if you're dealing with the imaginary part \( -3i \) of a complex number, its additive inverse would be \( 3i \), simplifying the concept.

Algebraic Operations with Complex Numbers

Algebraic operations with complex numbers are an extension of the operations we perform with real numbers. For addition and subtraction, we combine the real parts and the imaginary parts separately. The process is straightforward: treat the real components as you would in standard arithmetic, and apply the same principles to the imaginary components.

To illustrate, let's add two complex numbers, \( a + bi \) and \( c + di \). The sum would be \((a + c) + (b + d)i\). Subtraction works similarly, and when finding the additive inverse, we effectively 'negate' both the real and imaginary parts, resulting in a new complex number that, if added to the original, will yield zero, maintaining the balance of the complex number system. Hence, algebraic operations with complex numbers take into account both the real and the imaginary elements, respecting the distinct properties that define complex numbers as a whole.