Problem 34 - Algebra A Combined Function

Question

Multiply. Write your answers in the form \(a+b i\). $$ -6 i \cdot 4 i $$

Step-by-Step Solution

Answer

The answer is \(24\).

Step 1: Recognize the Expression

The expression given is the multiplication of two imaginary numbers, \(-6i\) and \(4i\). We need to multiply these to express the answer in the form \(a + bi\).

Step 2: Apply the Multiplication

Multiply the coefficients and the imaginary units: \((-6) \times 4 = -24\) and \(i \times i = i^2\). Combine these to get \(-24i^2\).

Step 3: Substitute the Value of \(i^2\)

Recall that \(i^2 = -1\). Substitute \(-1\) for \(i^2\) in the expression \(-24i^2\), which becomes \(-24(-1)\).

Step 4: Simplify the Expression

Simplify the expression \(-24(-1)\) to get \(24\). This is a real number, so it can be written in the form \(24 + 0i\).

Key Concepts

Complex Numbers: Understanding the Basics

Complex numbers are numbers that have a real part and an imaginary part. These numbers are written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) represents the imaginary part. This is the foundation for dealing with more advanced mathematical concepts.

• Real numbers are numbers without an imaginary component (e.g., 5, -3, 7.2).
• Imaginary numbers arise when dealing with square roots of negative numbers.
• Complex numbers, therefore, combine both real and imaginary components.

To visualize, imagine the real number line extended into two dimensions. The horizontal axis represents real numbers, while the vertical axis represents imaginary numbers. A complex number can be thought of as a point located on this plane, determined by its real and imaginary components. This concept is not only crucial in pure mathematics but also in engineering and physics, where complex numbers can represent different phenomena.

Algebra: Multiplying Complex Numbers

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. When it comes to complex numbers, algebra helps us handle operations, such as addition and multiplication.To multiply two complex numbers, like \(-6i\) and \(4i\), we follow certain steps. Each step utilises basic algebraic laws.

• First, multiply the numerical coefficients: \((-6) \times 4 = -24\).
• Then, multiply the imaginary units: \(i \times i = i^2\).

Combining these results gives us \(-24i^2\). Remember, when dealing with any operation involving complex numbers, it's vital to adhere to algebraic principles to ensure accurate calculations.

Imaginary Unit: The Core of Imaginary Numbers

The imaginary unit is denoted by \(i\) and is the cornerstone of imaginary and complex numbers. By definition, \(i\) is the square root of \(-1\), which is a non-real number.

• This means \(i^2 = -1\), a critical identity in complex number multiplication.
• Imaginary units allow for the expression of numbers that are not possible with real numbers alone.
• For example, the operation \(-24i^2\) simplifies by substituting \(-1\) for \(i^2\), resulting in \(24\).

Understanding the imaginary unit is essential when working with complex numbers. It enables us to transform seemingly impossible problems into solvable equations, widening the scope of what can be calculated in mathematics.