Problem 13 - Introductory Algebra for College Students

Question

Express each number in terms of i. $$10+\sqrt{-3}$$

Step-by-Step Solution

Answer

The expression \(10+\sqrt{-3}\) expressed in terms of i is \(10 + i \sqrt{3}\)

Step 1: Identify the square root of a negative number

In this expression, the square root of a negative number is \(\sqrt{-3}\).

Step 2: Rewrite the square root of a negative number using i

To rewrite this in terms of i, we can consider that \(\sqrt{-3} = \sqrt{-1 * 3} = \sqrt{-1} * \sqrt{3}\). As \(i = \sqrt{-1}\), we rewrite the above expression as \(i * \sqrt{3}\).

Step 3: Finalize expression in terms of i

Substitute \(i * \sqrt{3}\) back into the original expression to get the final answer which is: \(10 + i \sqrt{3}\).

Key Concepts

Imaginary Unit

In the world of complex numbers, the imaginary unit is a crucial concept. This term is represented by the symbol \( i \), which is defined as \( i = \sqrt{-1} \). It is used to describe the square root of negative numbers, which do not have solutions within the set of real numbers. In other words, whenever you encounter the square root of a negative number, \( i \) comes into play.
Here’s a quick guide on how to use the imaginary unit:

• \( i^2 = -1 \) — This equality is fundamental, as it helps to simplify powers of \( i \).
• In expressions like \( \sqrt{-a} \), you can write \( \sqrt{-a} = i\sqrt{a} \), making it easier to work through problems involving negative square roots.
• When you multiply two imaginary numbers, their components follow the same rules of arithmetic, except \( i^2 \) becomes \(-1\).

By using these guidelines, understanding and manipulating expressions with imaginary numbers becomes straightforward.

Square Root

The square root operation is seeking a number which, when multiplied by itself, results in the original number. For positive numbers, this operation is intuitive. However, things get interesting with negative numbers.
With negative numbers, the traditional arithmetic operations don't apply as they do with positive numbers. Take \( \sqrt{-3} \), for instance. To manage this, we need the concept of the imaginary unit \( i \). So, \( \sqrt{-3} \) can be expressed as \( \sqrt{-1 \times 3} \), and split into \( \sqrt{-1} \times \sqrt{3} \).
Here’s a quick breakdown:

• Simplifying Negative Square Roots: The expression \( \sqrt{-a} \) can be rewritten using \( i \), leading to \( i\sqrt{a} \).
• Operations with Square Roots: When combined with algebraic operations, expressions like \( i \sqrt{a} \) can be easily managed.

Using \( i \), we effectively reinterpret the square root of negative numbers, expanding the limits of number operations beyond the real number scope.

Negative Numbers

Negative numbers are an extension of our typical counting numbers, but they represent values less than zero. When dealing with negative numbers, especially in the context of square roots, we step into the realm of complex numbers.
A negative sign inside a square root presents a problem because there are no real numbers whose squares are negative. For instance, \( \sqrt{-3} \) cannot be resolved using real numbers alone. Here’s where the imaginary unit \( i \) comes into the picture, making operations with negative numbers more tractable.
Consider:

• Expressing Negative Square Roots: Use \( i \) to convert \( \sqrt{-a} \) into \( i\sqrt{a} \). This simplifies operations and enables further calculations.
• Operations with Negatives: When negative numbers are under a square root, employing \( i \) strategically allows for full mathematical operations.

Negative numbers, combined with the magic of \( i \), allow us to explore beyond real numbers into the complex number plane.