Problem 11 - Introductory Algebra for College Students

Question

Express each number in terms of i. $$7+\sqrt{-16}$$

Step-by-Step Solution

Answer

The number is expressed as \(7 + 4i\) in terms of i.

Step 1: Identifying \(i\) in the expression

We start by identifying the complex number. The given number is 7 + \(\sqrt{-16}\), which can be expressed as a complex number in the form a + bi, where 'a' is the real part and 'bi' is the imaginary part.

Step 2: Expressing \(\sqrt{-16}\) as \(4i\)

Since i is defined such that \(i^2 = -1\), we can express \(\sqrt{-16}\) as \(4i\). This is because \(\sqrt{-16} = \sqrt{16}*\sqrt{-1} = 4*\sqrt{-1} = 4i\).

Step 3: Summing the real and imaginary parts

Now we just add the real part (7) to the imaginary part (4i) to get the complex number 7 + 4i.

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a mathematical concept used to handle square roots of negative numbers. It's a cornerstone of complex numbers.Complex numbers extend the idea of one-dimensional numbers to two dimensions by introducing \(i\), which satisfies the equation \(i^2 = -1\). This means that \(i\), when squared, gives \(-1\).

• The imaginary unit allows for mathematical operations with numbers previously deemed unsolvable, such as the square root of negative numbers.
• \( \sqrt{-1} \) can be rewritten as \(i\).

Understanding \(i\) is fundamental for branching into complex number arithmetic. In this scenario, \(\sqrt{-16}\) involves multiplying \(\sqrt{16}\), which is 4, by \(i\), resulting in \(4i\). This transformation of negative roots using the imaginary unit is crucial for operations within complex numbers.

Real and Imaginary Parts

Each complex number is composed of a real part and an imaginary part, formatted as \(a + bi\).Here, \(a\) represents the real component, and \(bi\) represents the imaginary component.A comfortable understanding of these parts helps in performing basic arithmetic and seeing how complex numbers interact.

• Real part: This is the number \(a\), which exists on the real number line alone, without the imaginary unit.
• Imaginary part: This is the number \(bi\), which includes the imaginary unit \(i\).

For example, in our problem, the real part is 7, and the imaginary part is \(4i\). Connecting these components gives us the full complex number, \(7 + 4i\). This construction of combining both parts allows for broader calculations and insights, especially in advanced mathematics applications.

Expressing in Terms of i

Expressing numbers in terms of \(i\) is simply a way to simplify and work with square roots of negative numbers, bringing them under the umbrella of complex numbers.To express a square root of a negative number using \(i\), you decompose it by separating the negative sign from its positive counterpart.

• Start with the expression involving a square root of a negative number, such as \( \sqrt{-16} \).
• Rewrite it by separating the number as \( \sqrt{16} \times \sqrt{-1} \).
• This becomes \( 4 \times i \), which is \( 4i \).

This approach, of expressing in terms of \(i\), not only resolves the complication of negative square roots but also forms the base structure of dealing with complex numbers like \(7 + 4i\). It’s a foundational skill required for success in many fields of mathematics and engineering.