Problem 12 - Algebra and Trigonometry

Question

Find the real and imaginary parts of the complex number. $$ i \sqrt{3} $$

Step-by-Step Solution

Answer

Real part: 0; Imaginary part: \( \sqrt{3} \).

Step 1: Identify the Complex Number Form

A complex number is generally represented in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part coefficient.

Step 2: Express \( i \sqrt{3} \) in Standard Form

The complex number given is \( i \sqrt{3} \). This can be expressed in standard form as \( 0 + i \sqrt{3} \), indicating that the real part \( a = 0 \) and the imaginary part \( b = \sqrt{3} \).

Step 3: Determine the Real Part

From the expression \( 0 + i \sqrt{3} \), it is evident that the real part of the complex number is 0.

Step 4: Determine the Imaginary Part

From the expression \( 0 + i \sqrt{3} \), we see that the coefficient of \( i \) is \( \sqrt{3} \), so the imaginary part is \( \sqrt{3} \).

Key Concepts

Real Part

In mathematics, when dealing with complex numbers, the real part is an essential component to understand. A complex number usually takes the form \( a + bi \), where \( a \) is specifically the real part. This component represents the horizontal component on the complex plane and is a real number. When looking at a complex number, the real part is the "non-imaginary" component. For instance, if you have a complex number \( 2 + 3i \), the real part is \( 2 \).
For our original exercise, the complex number presented was \( i \sqrt{3} \), which can be rethought of as \( 0 + i \sqrt{3} \). Here, the real part is \( 0 \) as there is no additional real number alongside the imaginary part. Understanding the real part helps clarify how complex numbers are situated in relation to the real number line. If a complex number has a real part of \( 0 \), like in our exercise, it is purely imaginary, meaning it solely lies on the imaginary axis.

Imaginary Part

The imaginary part of a complex number refers to the component that involves the imaginary unit \( i \). The imaginary unit \( i \) is defined as \( i^2 = -1 \). In the standard form of a complex number \( a + bi \), the term \( bi \) is the imaginary part where \( b \) is a real number that multiplies the imaginary unit. Here, \( b \) is the coefficient that determines the "strength" or "direction" of the imaginary part on the complex plane.
To put it simply, the imaginary part can be thought of as the vertical component. Going back to our example, the complex number \( i \sqrt{3} \) can be rewritten in standard form as \( 0 + i \sqrt{3} \), meaning the imaginary part is \( \sqrt{3} \). This shows how high up or down from the horizontal axis (real axis) the number sits. It is crucial to grasp this concept to fully handle operations and interpretations involving complex numbers.

Standard Form of Complex Numbers

The standard form of complex numbers is a concise way of expressing numbers that have both a real and imaginary part. It is articulated as \( a + bi \), where \( a \) represents the real part and \( b \) is the coefficient of the imaginary part. This form is beneficial because it clearly indicates both components, making it easier to perform arithmetic operations and visualize their positions on the complex plane.
In our exercise, the number \( i \sqrt{3} \) is expressed in this standard form as \( 0 + i \sqrt{3} \). This immediately tells us that the real part is \( 0 \) and the imaginary part is \( \sqrt{3} \).

• Real Part: Located on the horizontal axis; here, it is \( 0 \).
• Imaginary Part: Located on the vertical axis; it is \( \sqrt{3} \).

The standard form is powerful because it allows us to easily separate, manipulate, and analyze both real and imaginary components independently. This clarity is especially important when adding, subtracting, or multiplying complex numbers. It's a reliable guide for solving mathematical problems involving complex numbers efficiently.