Problem 11 - Precalculus Mathematics for Calculus

Question

Find the real and imaginary parts of the complex number. $$-\frac{2}{3} i$$

Step-by-Step Solution

Answer

Real part: 0, Imaginary part: -2/3.

Step 1: Understand the Format of a Complex Number

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

Step 2: Identify the Given Complex Number

The complex number given in the problem is \( -\frac{2}{3} i \). This is already in the form of \( a + bi \), where \( a = 0 \) and \( b = -\frac{2}{3} \).

Step 3: Extract the Real Part

Identify the real part of the complex number. Since the complex number is \( -\frac{2}{3} i \), there is no real number added to the imaginary part \( bi \). Thus, the real part \( a \) is \( 0 \).

Step 4: Extract the Imaginary Part

Identify the imaginary part of the complex number. In \( -\frac{2}{3} i \), the imaginary part \( b \) is \( -\frac{2}{3} \), which multiplies the imaginary unit \( i \).

Key Concepts

Real Part of Complex Numbers

A complex number consists of two parts: the real part and the imaginary part. The real part of a complex number is any number without the imaginary unit, represented by \( i \).
The standard format of a complex number is \( a + bi \), where \( a \) is the real part. In many cases, the real part is immediately obvious, such as in the number 3 + 4i, where 3 is the real part.
However, sometimes only the imaginary part is present, as in the original exercise, where the number is \(-\frac{2}{3} i\). Here, the real part is simply 0 because there is no number before the imaginary part or the imaginary unit.

Imaginary Part of Complex Numbers

The imaginary part of a complex number involves the coefficient that multiplies the imaginary unit \( i \). It adds a new dimension to numbers, allowing for the representation of numbers that cannot appear on the traditional number line.
Written in the form \( a + bi \), the imaginary part is \( b \). In the complex number \(-\frac{2}{3} i\), the imaginary part is \(-\frac{2}{3}\). This means that the entire value of the complex number is concentrated in the imaginary dimension, with no real component present.

• In expressions like \(-\frac{2}{3} i\), it's crucial to recognize that \( b \) is \(-\frac{2}{3}\).
• The negative sign indicates the direction on the imaginary axis.

Imaginary Unit 'i'

The imaginary unit \( i \) is fundamental in defining complex numbers. By convention, \( i \) is defined as the square root of \(-1\), represented as \( i^2 = -1 \).
This property allows for the extension of real numbers to complex numbers, providing solutions to equations like \( x^2 + 1 = 0 \), which have no solution in the realm of real numbers alone.

• The imaginary unit \( i \) plays an essential role in distinguishing imaginary parts from real parts.
• When you multiply a real number by \( i \), it transitions the number into the imaginary dimension.

For example, in the expression \(-\frac{2}{3} i\), \(-\frac{2}{3}\) represents the coefficient, while \( i \) places this quantity in the imaginary realm, far from our everyday real numbers.