Problem 12 - Introductory Algebra for College Students

Question

Express each number in terms of i. $$9+\sqrt{-4}$$

Step-by-Step Solution

Answer

Expressing the number in terms of \(i\), we get \(9 + 2i\).

Step 1: Identify the Imaginary Part

First, identify the part of the number that involves the square root of a negative number. In this case, that is \(\sqrt{-4}\).

Step 2: Express the Imaginary Part in Terms of i

Next, using the definition of \(i\), which is \(i = \sqrt{-1}\), express \(\sqrt{-4}\) in terms of \(i\). This gives \(2i\) because \(\sqrt{-4} = \sqrt{4} * \sqrt{-1} = 2i\).

Step 3: Combine the Real and Imaginary Parts

Lastly, combine the real part 9 with the imaginary part we calculated in Step 2, which gives the answer \(9 + 2i\).

Key Concepts

Imaginary Unit i

The concept of the imaginary unit, denoted as 'i', is fundamental in understanding complex numbers. It's defined as the square root of -1, which means that it satisfies the equation
\[ i^2 = -1 \].
Mathematicians created this concept to extend the real number system to include solutions for square roots of negative numbers, which are not possible within the realm of real numbers alone. This unit allows for the expression of numbers in a two-dimensional plane, known as the complex plane where real and imaginary numbers coexist. Think of 'i' as a new kind of number that follows its own unique rules, but when combined with real numbers, it forms a complex number.

Expressing \(\sqrt{-1}\) of Negative Numbers

When faced with the square root of a negative number, algebra introduces the concept of 'i' to express them in a meaningful way. As there's no real number whose square is negative, the imaginary unit enables the expression of these roots. For example, to express \(\sqrt{-4}\), we can split it into two parts: \(\sqrt{4}\) and \(\sqrt{-1}\). Since the square root of 4 is 2 and \(\sqrt{-1}\) is defined as 'i', we rewrite \(\sqrt{-4}\) as \(2i\). This step is crucial as it bridges the gap between what is real and what is complex, empowering students to handle a wider variety of algebraic problems.

Combining Real and Imaginary Parts

Complex numbers consist of two parts: a real part and an imaginary part. When combining these, we simply place them together, often expressed in the form 'a + bi' where 'a' is the real part, and 'b' is the coefficient of the imaginary part 'i'.
In our exercise, we have a real number, 9, and an imaginary number, 2i, derived from \(\sqrt{-4}\). The combination is done by addition, placing the real and imaginary parts side by side, resulting in \(9 + 2i\). This new complex number represents a point on the complex plane and holds all the properties of both real and imaginary numbers. Understanding how to combine these two seemingly different types of numbers unlocks a deeper understanding of algebraic operations and functions involving complex numbers.