Problem 13

Question

In $3-18,$ write each number in terms of $i$ $$ -\sqrt{-51} $$

Step-by-Step Solution

Verified

Answer

The expression $-\sqrt{-51}$ is equivalent to $-i\sqrt{51}$.

1Step 1: Understand the Meaning of Imaginary Numbers

The imaginary unit, denoted by $i$, is defined as $i = \sqrt{-1}$. Therefore, when an expression involves the square root of a negative number, it can be rewritten in terms of $i$.

2Step 2: Recognize the Negative Square Root

The given expression is $-\sqrt{-51}$. The key part here is $\sqrt{-51}$, which implies the square root of a negative number, meaning we will have an imaginary component.

3Step 3: Express the Square Root in Terms of i

Using the property that $i = \sqrt{-1}$, we rewrite the expression inside the square root. Therefore, $\sqrt{-51} = \sqrt{-1 \times 51} = \sqrt{-1} \times \sqrt{51} = i\sqrt{51}$.

4Step 4: Apply the Negative Sign

The original expression had a negative sign outside the square root. Thus, we must include this when writing the expression in terms of $i$. Therefore, $-\sqrt{-51} = -i\sqrt{51}$.

Key Concepts

Square Root of Negative NumbersImaginary Unit $i$Rewriting Expressions with $i$

Square Root of Negative Numbers

When we encounter a negative number under a square root, it signals something special. The square root of a negative number cannot be handled in the same way as positive numbers. In typical real-number arithmetic, the square root of a negative number is not defined. However, that's where imaginary numbers come into play. To deal with these scenarios, mathematics introduces the concept of the imaginary unit, represented as $i$, which is defined as $i = \sqrt{-1}$.

Thus, whenever you see a square root with a negative number, you now know it involves $i$. This helps us calculate and visualize these complex numbers in an understandable way, even though we can't find a real-number solution for them.

Imaginary Unit $i$

The imaginary unit, represented by $i$, is crucial in understanding complex numbers. By definition, $i = \sqrt{-1}$. This means that $i$ is used to express the square roots of negative numbers in a form that's manageable and useful for further computations.

Let's consider this point through a basic property: since $i = \sqrt{-1}$, squaring $i$ results in $i^2 = -1$. This shows how multiplying by $i$ 'removes' the square root, but switches the polarity from negative to positive in the context of imaginary calculations.

In summary, the imaginary unit $i$ is an indispensable part of complex numbers, allowing us to work with square roots of negative numbers, expanding our number system beyond real numbers.

Rewriting Expressions with $i$

Rewriting expressions with $i$ makes handling negative square roots seamless and improves mathematical precision. Suppose you have an expression with a negative square root, like $-\sqrt{-51}$. The steps to rewrite it using $i$ involve a few logical ideas:

First, recognize that the expression $\sqrt{-51}$ needs $i$ since it is a negative root.
Next, break it down into $\sqrt{-1 \times 51}$, which is equivalent to $\sqrt{-1} \times \sqrt{51}$.
Replace $\sqrt{-1}$ with $i$, so it becomes $i\sqrt{51}$.
Finally, apply any outside signs from the original problem. Here, the expression $-\sqrt{-51}$ is adjusted to $-i\sqrt{51}$.

This process of rewriting with $i$ helps us treat and visualize complex numbers correctly, merging the imaginary and real parts seamlessly.

Problem 13