Problem 10 - Algebra and Trigonometry

Question

Find the real and imaginary parts of the complex number. $$ 2-\sqrt{-5} $$

Step-by-Step Solution

Answer

Real part: 2; Imaginary part: \(-\sqrt{5}\)

Step 1: Rewrite the Complex Number

First, recognize that the expression \(2 - \sqrt{-5}\) can be rewritten in terms of the imaginary unit \(i\), where \(i = \sqrt{-1}\). We know \(\sqrt{-5} = \sqrt{5} \cdot \sqrt{-1} = \sqrt{5}i\). Therefore, the complex number becomes \(2 - \sqrt{5}i\).

Step 2: Identify the Real Part

The real part of any complex number in the form \(a + bi\) is the value \(a\). In the expression \(2 - \sqrt{5}i\), the term without the \(i\) is the real part, which is \(2\).

Step 3: Identify the Imaginary Part

The imaginary part of a complex number \(a + bi\) is the value \(b\), which is the coefficient of the \(i\). In \(2 - \sqrt{5}i\), the coefficient of \(i\) is \(-\sqrt{5}\), so the imaginary part is \(-\sqrt{5}\).

Key Concepts

Real Part

A complex number is typically in the form \(a + bi\), where \(a\) and \(b\) are real numbers. The real part refers to the \(a\) component, that is, the number that isn't attached to the imaginary unit \(i\).
The real part represents the "real" value of the complex number, as it lies on the horizontal axis of the complex plane.

• It is found by looking at the constant term in the complex number.
• In the expression \(2 - \sqrt{5}i\), the real part is \(2\).

Understanding the real part is straightforward once you recognize it as the value without the imaginary unit, \(i\).
This concept is crucial in fields like engineering and physics, where complex numbers model real-world phenomena. Always observe for the number that stands alone as your real part.

Imaginary Part

The imaginary part of a complex number, expressed as \(a + bi\), refers to the component attached to the imaginary unit \(i\). This part is a real number itself, and in equations, it's represented as \(b\). The imaginary part is crucial because:

• It determines the "height" or vertical position of the complex number on the complex plane.
• In \(2 - \sqrt{5}i\), the imaginary part is \(-\sqrt{5}\).

Despite being named "imaginary," this part has significant practical applications.
It is essential for computations in electronics and signal processing where signals are modulated and demodulated using imaginary parts. Understanding how to identify and use the imaginary part enhances your grasp of complex number operations.

Imaginary Unit

The imaginary unit, defined as \(i\), is the foundation of all operations involving complex numbers. It is specifically defined as \(i = \sqrt{-1}\). This definition is what allows us to take square roots of negative numbers and work with them consistently. Here's why \(i\) is important:

• It enables the expression of numbers that result from taking square roots of negative values, like \(\sqrt{-5}\).
• Using \(i\), \(\sqrt{-5}\) becomes \(\sqrt{5}i\), simplifying complex number expressions.

Imaginary units are vital in extending real number operations to a more general system, allowing for solutions to equations that don't have real solutions. In engineering, for instance, \(i\) is used extensively to represent and calculate alternating currents and waves in circles. Understanding \(i\) is essential to working with complex numbers efficiently.