Problem 65

Question

Find each of the products and express the answers in the standard form of a complex number. $$(3 i)(2-5 i)$$

Step-by-Step Solution

Verified

Answer

The product is $15 + 6i$.

1Step 1: Expand the Expression

The product is \[(3i)(2 - 5i)\]To find this product, apply the distributive property: \[3i \cdot 2 + 3i \cdot (-5i)\]

2Step 2: Calculate Each Term

Compute each term separately:- For the first term:\[3i \cdot 2 = 6i\]- For the second term:\[3i \cdot (-5i) = -15i^2\]

3Step 3: Substitute and Simplify Using $i^2 = -1$

Given that $i^2 = -1$, substitute and simplify:\[6i - 15i^2 = 6i - 15(-1) = 6i + 15\]

4Step 4: Express in Standard Form

Rearrange to express the complex number in standard form, which is $a + bi$. Thus, the final expression is:\[15 + 6i\]

Key Concepts

The Distributive Property with Complex NumbersUnderstanding the Imaginary Unit $i$Expressing Complex Numbers in Standard Form

The Distributive Property with Complex Numbers

The distributive property is a fundamental algebraic principle that allows us to multiply a single term by two or more terms inside a parenthesis. It can be expressed as:

For real numbers, it appears as: \[a(b + c) = ab + ac\].

When working with complex numbers, we apply the distributive property in the same way. In the given exercise, the expression $(3i)(2 - 5i)$ is distributed as follows:

Multiply $3i$ by 2 to get $6i$.
Multiply $3i$ by $-5i$ to get $-15i^2$.

This shows how each term in the parenthesis interacts with the term outside, a crucial step for solving problems involving complex numbers. Understanding and applying the distributive property correctly ensures that you can break down and simplify expressions effectively.

Understanding the Imaginary Unit $i$

The imaginary unit, denoted as $i$, is a unique number in mathematics with the defining property that $i^2 = -1$. This means when you multiply the imaginary unit by itself, you get $-1$.

This concept extends the real number system into the complex number system.
It's essential for operations involving complex numbers.

In our exercise, calculating $3i \cdot (-5i)$ yields $-15i^2$, which simplifies to 15 due to the equivalence $i^2 = -1$. Replacing $i^2$ with $-1$ is a critical step, turning the product from an imaginary number into a real number, thereby simplifying the expression. Without understanding $i^2$, working with complex numbers would be significantly more challenging.

Expressing Complex Numbers in Standard Form

The standard form for expressing complex numbers is $a + bi$, where $a$ is the real part and $bi$ is the imaginary part.

This format helps to clearly identify and separate the real and imaginary components.

In the exercise, after applying the distributive property and understanding the role of the imaginary unit $i$, the expression $6i + 15$ is rearranged into standard form as $15 + 6i$.

$a = 15$ represents the real part.
$b = 6$ represents the coefficient of the imaginary part.

By writing complex numbers in this standard form, it becomes easier to conduct additional operations, such as addition, subtraction, or comparison with other complex numbers. This standard notation also provides clarity and consistency in mathematical communication.

Problem 65

Problem 66