Problem 15 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write each number in terms of \(i\) $$ 5+\sqrt{-5} $$

Step-by-Step Solution

Answer

The expression is written as \( 5 + i \sqrt{5} \).

Step 1: Identify the Imaginary Part

First, identify the part of the expression that involves the square root of a negative number, which is \ \( \sqrt{-5} \). In general, square root expressions of negative numbers are rewritten using the imaginary unit \ \( i \), where \ \( i = \sqrt{-1} \).

Step 2: Express the Imaginary Part in Terms of i

Express the square root of the negative number in terms of \ \( i \). Since \ \( i = \sqrt{-1} \), we rewrite \ \( \sqrt{-5} \) as \ \( \sqrt{-1} \times \sqrt{5} = i \sqrt{5} \).

Step 3: Combine Real and Imaginary Parts

Combine the real number \ \( 5 \) with the rewritten imaginary part \ \( i \sqrt{5} \) to express the entire expression in standard form for complex numbers. The expression becomes \ \( 5 + i \sqrt{5} \).

Key Concepts

Understanding Complex Numbers

Complex numbers might seem complicated at first, but are just an extension of the familiar real numbers. They help us manage expressions involving square roots of negative numbers. The standard form of a complex number is written as \( a + bi \), where \( a \) and \( b \) are real numbers. In this format:

• \( a \) is the real part of the complex number.
• \( bi \) is the imaginary part, where \( i \) is the imaginary unit.

Complex numbers are incredibly useful in both mathematics and science. They open doors to mathematical solutions for equations that don't have real number answers. For example, quadratic equations sometimes don't cross the x-axis, which means they have no real roots; but if we employ complex numbers, we find those elusive solutions. In essence, complex numbers enable us to express calculations that go 'beyond' the real number limitations.

The Role of the Imaginary Unit

Imaginary unit \( i \) is the backbone of complex numbers. It is defined as \( i = \sqrt{-1} \). This definition might seem strange, but it is a conceptual tool rather than a number in the traditional sense. When you multiply \( i \) by itself, you get:

• \( i^2 = -1 \)

This property is fundamental because it allows us to work with square roots of negative numbers. Here’s how you might see the imaginary unit in action:

• \( \sqrt{-4} = \sqrt{-1} \times \sqrt{4} = i \times 2 = 2i \)

The introduction of \( i \) doesn't make mathematics any more difficult—it's a tool to make certain operations possible. Remember, the idea is not to visualize \( i \) but to use it skillfully in mathematical expressions.

Rewriting Square Roots of Negative Numbers

Square roots of negative numbers can be tricky because we traditionally cannot find a real number whose square is negative. This is where the concept of the imaginary unit \( i \) comes into play. To handle something like \( \sqrt{-5} \), we use the following steps:

• First, recognize it as \( \sqrt{-1} \times \sqrt{5} \).
• Then, substitute \( i \) for \( \sqrt{-1} \).
• This gets you \( i \sqrt{5} \), a neat form that expresses the root in terms of the imaginary unit.

Through these steps, we've transformed a difficult concept so it can be worked with conventionally alongside real numbers. It simplifies complex expressions and allows us to perform algebraic manipulations and solve equations that incorporate roots of negative numbers.