Problem 20 - Algebra A Combined Function

Question

Add or subtract as indicated. Write your answers in the form \(a+b i .\) $$ (2-4 i)-(2-i) $$

Step-by-Step Solution

Answer

The result is \(-3i\).

Step 1: Distribute the Negative Sign

To begin, distribute the negative sign in front of the second set of parentheses across the terms within it. This changes the expression from \( (2 - 4i) - (2 - i) \) to \( 2 - 4i - 2 + i \).

Step 2: Group Like Terms

Next, group the real parts and the imaginary parts of the expression separately. You have: \( (2 - 2) + (-4i + i) \).

Step 3: Combine the Real Parts

Combine the real numbers. The expression for the real parts is:\( 2 - 2 = 0 \).

Step 4: Combine the Imaginary Parts

Now, combine the imaginary numbers. The expression for the imaginary parts is:\( -4i + i = -3i \).

Step 5: Write the Final Result

Finally, write the answer in the form \(a+bi\). The real part is 0 and the imaginary part is \(-3i\), so the final answer is:\( 0 - 3i \), or simply \(-3i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are a fundamental part of complex numbers, represented as multiples of the imaginary unit, denoted by the letter \( i \). This concept may seem a bit abstract at first, but it's quite straightforward once you break it down.

The imaginary unit \( i \) is defined by the property that \( i^2 = -1 \). Hence, an imaginary number can be expressed in the form \( bi \), where \( b \) is a real number.

Here are a few important points about imaginary numbers:

• Imaginary numbers are not greater or less than real numbers, they are simply located at a different axis on the complex plane.
• When adding or subtracting imaginary numbers, you treat them like algebraic terms, combining like terms.

In our problem, we combined the imaginary parts \(-4i + i\) to get \(-3i\). Imaginary numbers make complex numbers rich and allow solutions to all polynomial equations through the Fundamental Theorem of Algebra.

Real Numbers

Real numbers are numbers that can be found on the number line. This includes all the positive numbers, negative numbers, and zero. They can be integer values, fractions, or decimals. Real numbers are fundamental in calculating and analyzing almost anything in mathematics and science.

In the context of complex numbers, real numbers are the component associated with the real axis of the complex plane.

A few key characteristics of real numbers include:

• They obey all the usual rules of arithmetic: addition, subtraction, multiplication, and division.
• They are capable of representing known quantities and are crucial when combining terms with real coefficients.

In the exercise, the real parts were \(2 - 2\), resulting in \(0\). This is standard when dealing with real numbers: combining two identical, but opposite, real values results in zero.

Algebraic Expressions

Algebraic expressions involve combining variables and constants into mathematical statements that represent numerical relationships. Within the context of complex numbers, they are used to capture both the real and imaginary parts of a number in a compact form.

Equipped with full power of operations like addition, subtraction, multiplication, and division, algebraic expressions enable us to solve mathematical problems efficiently. In this context, they help organize our approach to dealing with complex numbers.

Here are some things to remember:

• Always distribute operation signs, such as negatives, appropriately across terms.
• Group like terms together, such as real parts with real parts and imaginary parts with imaginary parts.
• Properly simplify expressions by performing all possible arithmetic operations on grouped terms.

Our exercise used algebraic approaches to handle expressions properly, first by distributing the negative, then by grouping terms. These methods ensure clarity and correctness in dealing with complex expressions like \(2 - 4i - 2 + i\), ultimately leading to \(-3i\).