Problem 63 - Algebra for College Students

Question

Find each of the products and express the answers in the standard form of a complex number. $$(7 i)(-6 i)$$

Step-by-Step Solution

Answer

The product is 42.

Step 1: Identify the Components

We need to find the product of \((7i)(-6i)\). Identify the components involved in the multiplication. We have two imaginary numbers: \(7i\) and \(-6i\).

Step 2: Multiply the Coefficients

Multiply the coefficients of the imaginary terms. This is the multiplication of real numbers:\[7 imes -6 = -42\]

Step 3: Multiply the Imaginary Components

When multiplying imaginary units \(i\), use the property \(i^2 = -1\). Thus:\(i imes i = i^2\) = -1.

Step 4: Determine the Product

Substitute the previously calculated values into the expression:\((-42) imes (i^2) = -42 imes (-1)\).

Step 5: Simplify and Write in Standard Form

Compute the results from step 4 to simplify the expression:\[-42 imes (-1) = 42\]Thus, the product in standard form (a + bi) of a complex number is:\(42 + 0i\).

Key Concepts

Imaginary Numbers

Imaginary numbers form the basis of complex numbers and are crucial for extending the idea of numbers beyond real numbers. The core element of imaginary numbers is the unit "i," defined such that \(i^2 = -1\). This property of "i" allows imaginary numbers to express roots of negative numbers, a concept not possible with real numbers alone.
One can think of an imaginary number as forming part of a plane, where the real numbers lay along the horizontal axis (like a number line) and the imaginary numbers lay along the vertical axis.

• An example of a simple imaginary number is \(3i\), where "3" is the coefficient.
• In the context of our exercise, \(7i\) and \(-6i\) are multiplied together, involving two imaginary numbers.
• When multiplying imaginary numbers, the property \(i^2 = -1\) becomes central, as seen in further steps of the exercise.

By understanding imaginary numbers, one can transition more comfortably into operations involving complex numbers.

Complex Multiplication

Multiplying complex numbers involves distributing terms similar to polynomials, but with attention to the unique properties of the imaginary unit \(i\). In complex multiplication, each component of the first complex number multiplies with each component of the second one.
In our case, we're focusing specifically on multiplying imaginary components as seen with \((7i)(-6i)\). Here's how it works:

• Multiply the coefficients: \(7 \times -6 = -42\).
• Multiply the imaginary units: \(i \times i = i^2\). Using the property of imaginary numbers, we replace \(i^2\) with \(-1\).
• Combine these products to simplify: \(-42 \times i^2 = -42 \times (-1)\).
• The result is \(42\), demonstrating how the imaginary units affect the sign and magnitude of the product.

Understanding complex multiplication is essential for solving equations involving complex numbers and can be expanded to include full complex numbers with both real and imaginary parts.

Standard Form of Complex Numbers

The standard form of complex numbers is expressed as \(a + bi\), where "a" is the real part and "bi" is the imaginary part. This notation is key to understanding and performing operations with complex numbers.
In our exercise, after performing the necessary multiplications and simplifications, the product was expressed in this standard form: \(42 + 0i\). Notice that:

• The real part "a" is 42.
• The imaginary part "bi" is 0i, indicating there's no remaining imaginary component.
• Complex numbers like \(42 + 0i\) are essentially real numbers, exemplifying how real numbers are special cases of complex numbers.

The structure of \(a + bi\) aids in clarity, ensuring that you can easily identify and separate real from imaginary contributions in complex numbers. This systematic approach is essential in both basic and advanced arithmetic with complex numbers.