Problem 33 - Algebra A Combined Function

Question

Multiply. Write your answers in the form \(a+b i\). $$ -9 i \cdot 7 i $$

Step-by-Step Solution

Answer

The result is \(63 + 0i\), which simplifies to 63.

Step 1: Multiply the real coefficients

First, we focus on the coefficients of the imaginary numbers, which are -9 and 7. We multiply them as we would with regular numbers: \(-9 \times 7 = -63\).

Step 2: Multiply the imaginary units

Next, we multiply the imaginary units \(i\) together. Recall that \(i^2 = -1\). So multiplying \(i\) by \(i\) gives us \(i \times i = i^2 = -1\).

Step 3: Combine results from Step 1 and Step 2

Now, we take the result from Step 1, which is -63, and the result from multiplying the imaginary units: \(-63 \times (-1) = 63\).

Step 4: Express in the form \(a+bi\)

The product \(63\) is a real number, so we express it in the form \(a + bi\) as \(63 + 0i\).

Key Concepts

Imaginary Unit

The imaginary unit is a fundamental concept in complex numbers, represented by the letter \(i\). It is defined as the square root of '-1'. This concept might seem strange at first because there are no real numbers whose square is negative.
However, mathematicians introduced \(i\) to extend the real numbers and solve equations that otherwise would have no solution.

• Essentially, \(i = \sqrt{-1}\).
• As a crucial property, \(i^2 = -1\).

Understanding the imaginary unit is vital because it allows us to work with complex numbers. Having \(i\) makes it possible to solve polynomial equations and perform calculations in physics and engineering that involve more than just real numbers.

Multiplication of Complex Numbers

Multiplying complex numbers involves both the real and imaginary parts of the numbers. In complex numbers, a+b\(i\), \(a\) represents the real part and \(b\) is the coefficient of the imaginary unit \(i\). However, when multiplying pure imaginary numbers, such as \(-9i\) and \(7i\), you mainly focus on the coefficients and \(i\).
Here's how it works:

• Multiply the coefficients: For \(-9i \cdot 7i\), you multiply \(-9\) and \(7\) to get \(-63\).
• Multiply the imaginary parts: \(i \cdot i\), which equals \(i^2\).
• Replace \(i^2\) with \(-1\): Simplifying \(-63 \cdot i^2\) gives us \(-63 \times (-1) = 63\).

Always remember that \(i^2 = -1\), which flips the sign of the product, from negative to positive. This essential rule makes multiplying complex numbers straightforward once understood.

Expressing in a+bi form

Complex numbers are usually expressed in the form \(a+bi\), which clearly shows their real and imaginary components. This format makes it easier to perform arithmetic operations and understand the structure of the number.
When given a product of complex numbers, the ultimate goal is to express it in this standard form. In our exercise, after multiplication and simplification, the result was \(63\).
Although \(63\) is just a real number, it can be written in complex form as \(63 + 0i\). Here:

• \(a = 63\), representing the real part of the number.
• \(b = 0\), indicating that there is no imaginary component.

Being able to express results in the \(a+bi\) form ensures clarity in complex number arithmetic and naturally extends to adding or subtracting complex quantities.