Problem 7 - Algebra and Trigonometry

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 Complex Number Format

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

Step 2: Identify the Real Part

Look at the given complex number \(-\frac{2}{3} i \). The real part \( a \) corresponds to the number without the imaginary unit \( i \). Here, since there's no number beside an explicit real component (a real part is not present), the real part is 0.

Step 3: Identify the Imaginary Part

The imaginary part \( b \) of the complex number is the coefficient of \( i \). In \(-\frac{2}{3} i \), the coefficient of \( i \) is \(-\frac{2}{3}\). Therefore, the imaginary part is \(-\frac{2}{3}\).

Key Concepts

real part

The real part of a complex number refers to the component that does not include the imaginary unit \( i \). In the complex number format \( a + bi \), the real part is represented by \( a \). Think of it as the number's "normal" part that you'd usually encounter outside of complex numbers.

• The real part identifies the horizontal shift from the origin in the complex plane.
• If you see a complex number like \( 2 + 3i \), the real part is \( 2 \).
• If the number looks like \( -\frac{2}{3} i \) with no explicit part without \( i \), the real part is simply \( 0 \).

This distinction is crucial in separating what part of the number affects the real number line versus what affects the imaginary axis in the complex plane.

imaginary part

The imaginary part of a complex number is the part that includes the imaginary unit \( i \). In the format \( a + bi \), the imaginary part is represented by \( b \).

• This part determines the vertical shift from the origin in the complex plane.
• For the complex number \(-\frac{2}{3} i\), the imaginary part is \(-\frac{2}{3}\).
• It is important to remember that \( b \) (the imaginary part) is just a real number multiplying \( i \).

The imaginary part gives the number its unique properties in calculations involving complex algebra, influencing how numbers behave on the imaginary axis.

complex number format

Complex numbers are an essential concept in mathematics, typically expressed in the form \( a + bi \). Here, \( a \) is the real part, while \( b \) is the imaginary part linked with the imaginary unit \( i \).

• The structure \( a + bi \) makes it clear which component affects the real axis and which affects the imaginary axis.
• The imaginary unit \( i \) has a special property: \( i^2 = -1 \), which is crucial in many calculations.
• Complex numbers bridge gaps in algebraic equations where real solutions might not exist, like solving \( x^2 + 1 = 0 \).

Understanding complex number format is pivotal for exploring fields such as engineering and physics, where real-world phenomena are modeled using complex numbers to capture both magnitude and direction in a unified form.