Problem 32 - Algebra 2

Question

Simplify each expression. $$ 6-(8+3 i) $$

Step-by-Step Solution

Answer

-2 - 3i

Step 1: Identify the complex numbers

Recognize that 6 is a real number and that 8+3i is a complex number where 8 is the real part and 3i is the imaginary part.

Step 2: Simplify the expression

Subtract the real and imaginary parts separately to simplify: Real part: 6 - 8 = -2, Imaginary part: 0 - 3i = -3i.

Step 3: Combine the simplified parts

Write the simplified expression combining the real part and the imaginary part: -2 - 3i.

Key Concepts

Complex Numbers

When we talk about numbers, we're often referring to real numbers, like 1, 2, or 3. However, there is another set of numbers known as complex numbers that are crucial in the field of mathematics. A complex number is a combination of a real number and an imaginary number, expressed in the form of \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part with \( i \) being the square root of -1. Complex numbers are essential because they allow us to solve equations that have no solution within the system of real numbers, like \( x^2 + 1 = 0 \).

The beauty of complex numbers lies in their ability to extend our number system and provide completeness for algebra, which is analogous to how negative numbers were introduced to make subtraction universally applicable. They are widely used in various disciplines, including engineering, physics, and applied mathematics, particularly in the study of electrical circuits and waveforms.

Real and Imaginary Parts

Understanding the real and imaginary parts of complex numbers is critical to working with them effectively. Every complex number is made up of two parts: the real part, which could be any real number, and the imaginary part, which is any real number multiplied by \( i \). The real part, represented by \( a \) in the standard form \( a + bi \), can be thought of as the 'horizontal' component on a complex plane, while the imaginary part, represented by \( bi \), is akin to the 'vertical' component.

Imaginary numbers, including the imaginary unit \( i \) itself (which is essentially \( \sqrt{-1} \)), enable computations and solutions in scenarios where only real numbers fall short. For example, in the expression \( 8 + 3i \), 8 is the real part and \( 3i \) is the imaginary part. When dealing with complex numbers, it's important to keep these parts distinct, as they will be manipulated by algebraic operations.

Algebraic Operations

Performing algebraic operations on complex numbers follows similar rules as operations on real numbers, with extra attention given to the imaginary unit \( i \). The fundamental operations—addition, subtraction, multiplication, and division—can be performed on the real and imaginary parts separately. For instance, when simplifying the expression \( 6 - (8 + 3i) \), we deal with the real parts as one group and the imaginary parts as another. You subtract the real part of the second number from the real part of the first, yielding \( 6 - 8 = -2 \), and likewise with the imaginary parts: \( 0 - 3i = -3i \) as there is no explicit imaginary part in the number 6.

An important note is to always remember to square \( i \) as -1 whenever you're multiplying two imaginary parts. Algebraic operations with complex numbers expand our problem-solving toolkit and enable a coherent way to work through a wider variety of mathematical problems.