Question

For \(i=\sqrt{-1}\), which of the following complex numbers is equivalent to \(\left(10 i-4 i^2\right)-(7-3 i) ?\) A) \(-11+7 i\) B) \(-3+13 i\) C) \(3-13 i\) D) \(11-7 i\)

Short answer

B) \(-3 + 13i\)

Step-by-step solution

Rewrite the expression using \(i=\sqrt{-1}\).

The provided expression is \(\left(10i - 4i^2\right) - (7 - 3i)\). Since \(i^2 = -1\), replace all instances of \(i^2\) in the given expression: \[\left(10i - 4(-1)\right) - (7-3i).\]

Simplify the expression

Now, simplify the expression by doing the arithmetic operations as: \[\left(10i + 4\right) - (7-3i) = 4 + 10i - 7 + 3i.\]

Combine the real and imaginary parts

Combine the real parts (constant terms) and the imaginary parts (terms with \(i\)): \[(4-7) + (10i+3i) = -3 + 13i.\]

Compare the result with the given options

We have found that the simplified complex number is \(-3 + 13i\). Comparing this to the given options, we can see that this matches option B. Therefore, the correct answer is: B) \(-3 + 13i\).

Key concepts

Imaginary Unit

In the world of complex numbers, the imaginary unit is a fundamental building block. It is denoted by the symbol \(i\) and is defined as the square root of -1, expressed as \(i = \sqrt{-1}\). Now, why is this important?

Regular numbers, like 2 or -5, can't handle the square root of a negative number. The imaginary unit \(i\) expands our number system to work with these calculations. This concept allows mathematicians and engineers to solve equations that would otherwise be unsolvable with just real numbers.

When you square \(i\), you get \(i^2 = -1\). This property is key in simplifying expressions that involve complex numbers. While working with complex numbers, whenever you encounter \(i^2\), replace it with \(-1\) to simplify calculations. Doing this consistently will help prevent mistakes and streamline your problem-solving process.

Complex Plane

The complex plane is like a map for complex numbers. It helps us visualize them in a two-dimensional space. How do we do this?

Each complex number has two parts: a real part and an imaginary part. For instance, for the complex number \(-3 + 13i\), -3 is the real part and 13 is the imaginary part.

Here's how it works:

• The real part is plotted on the horizontal axis (often called the "real axis").
• The imaginary part is plotted on the vertical axis (often called the "imaginary axis").

By plotting these, you can visually compare the sizes of complex numbers, determine their distances from each other, and even understand operations like addition and multiplication geometrically. A complex number like \(-3 + 13i\) can be imagined as a point or a vector extending from the origin to the coordinates (-3, 13) on this plane.

Algebraic Expression Simplification

When working with complex numbers, simplifying algebraic expressions is crucial for finding solutions efficiently. Let's break it down:

1. **Use properties:** As seen in the exercise, you start by using the properties of the imaginary unit, like \(i^2 = -1\). This makes the expression easier to handle. Once all \(i^2\) terms are managed, it simplifies the problem a great deal.

2. **Distribute correctly:** Handle each part of the expression carefully. Distribute signs across the terms, remembering that subtraction affects both terms within the parenthesis.

3. **Combine like terms:** Group and simplify terms into real and imaginary parts. This makes the final answer clear. For instance, \(4 + 10i - 7 + 3i\) becomes \(-3 + 13i\), by combining the real parts: \(4 - 7\), and the imaginary parts: \(10i + 3i\).

By using these strategies, algebraic expressions involving complex numbers become manageable and often lead directly to the solution.