Problem 35 - Algebra A Combined Function

Question

Multiply. Write your answers in the form \(a+b i\). $$ -10 i \cdot-4 i $$

Step-by-Step Solution

Answer

The result is \(-40\), which is \(-40 + 0i\) in standard form.

Step 1: Identify the problem

We need to multiply two imaginary numbers: \(-10i\) and \(-4i\) and express the result in the form \(a + bi\).

Step 2: Multiply the coefficients

Multiply the coefficients of the imaginary numbers together. \(-10 \times -4 = 40\).

Step 3: Multiply the imaginary units

Multiply the imaginary units \(i\) together. \(i \times i = i^2\).

Step 4: Use the property of imaginary unit

Recall the property that \(i^2 = -1\). Apply this to the product from Step 3.

Step 5: Combine the results

Combine the results from Step 2 and Step 4: \(40 \times i^2 = 40 \times (-1) = -40\).

Step 6: Write in standard form

Since the result is \(-40\), it can be expressed in the form \(a + bi\) as \(-40 + 0i\).

Key Concepts

Understanding Imaginary Numbers

Imaginary numbers play a crucial role in mathematics, especially when dealing with complex numbers. They are built upon the unit 'i', which is defined as the square root of -1:

• This means that i multiplied by itself (i.e., \(i^2\)) gives -1.
• Imaginary numbers expand the real number system, allowing us to find solutions to equations like \(x^2 + 1 = 0\), which don’t have solutions within the real numbers alone.
• In this context, imaginary numbers help provide a comprehensive view of mathematical operations.

When working with imaginary numbers, like \(-10i\) and \(-4i\) in the exercise above, it is important to remember that they are similar to constants multiplied by the imaginary unit \(i\). Breaking down operations with imaginary numbers relies heavily on understanding their properties, especially \(i^2 = -1\), to find solutions and interpret results.

Multiplication of Complex Numbers

The multiplication of complex numbers might seem perplexing at first, but it follows straightforward arithmetic rules. When multiplying, treat the real and imaginary components separately:

• The general form of a complex number is \(a + bi\), where 'a' is the real part and 'bi' is the imaginary part.
• To multiply two complex numbers, apply the distributive property to each component.
• Using the FOIL method can simplify the expansion: First, Outer, Inner, Last terms.

In the example of \(-10i\cdot-4i\), we only have imaginary components, so the multiplication simplifies to:

• \(-10 \times -4 = 40\)
• The imaginary unit multiplication can be viewed as \(i \times i = i^2\).

Remember, \(i^2\) simplifies to \(-1\). Thus, the result becomes \(40 \times (-1) = -40\). By understanding these steps, the multiplication of two imaginary numbers can become an intuitive process.

The Standard Form of Complex Numbers

Expressing complex numbers in standard form is a fundamental and necessary skill in mathematics. The standard form is expressed as \(a + bi\), where 'a' represents the real part and 'b' the imaginary part.

• This structure helps clearly distinguish between the components and makes computation and manipulation straightforward.
• When writing complex numbers in standard form, use both real and imaginary coefficients for clarity and precision.
• A complex number where \(b = 0\) is a purely real number, while one where \(a = 0\) is purely imaginary.

From our exercise, the result of multiplying two imaginary numbers was \(-40\), which in complex form becomes \(-40 + 0i\). This highlights the key aspect of the standard form, even when the imaginary part is zero. Having this result underscores the importance of representing complex solutions comprehensively and helps to contrast and compare their real and imaginary components.