Problem 3 - Intermediate Algebra

Question

Label each statement true or false. The real part of the complex number \(6 i\) is 0 .

Step-by-Step Solution

Answer

True

Step 1: Understanding the Complex Number

A complex number is usually written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In the given problem, the complex number is \(6i\). It is important to identify the real and imaginary parts of this number.

Step 2: Identifying the Real Part

In the complex number \(6i\), it is expressed as \(0 + 6i\). This format shows us that the real part \(a\) is 0, while the imaginary part is 6. The given statement asks if the real part is 0.

Step 3: Verifying the Statement

The statement 'The real part of the complex number \(6i\) is 0' is directly derived from the form \(0 + 6i\). Since the real part \(a = 0\), the statement is true.

Key Concepts

Real Part of a Complex Number

A complex number is typically expressed in the form \(a + bi\), where \(a\) is known as the real part, and \(b\) is the coefficient of \(i\), the imaginary part. In this expression, \(a\) is a real number that represents the horizontal position on the complex plane.
Understanding the real part is essential as it provides the 'real reality' of the complex number; it tells us the extent of the number along the real axis.

• For example, in a complex number like \(7 + 3i\), the real part is 7.
• If we consider \(0 + 5i\), the real part is 0.
• Even for a purely imaginary number like \(6i\), the format is \(0 + 6i\), hence the real part is 0.

The real part helps simplify complex equations and is essential in operations like addition and multiplication of complex numbers. It's this part that aligns closely with the numbers we are used to seeing in day-to-day arithmetic.

Imaginary Part of a Complex Number

The imaginary part of a complex number involves the imaginary unit \(i\), where \(i\) is defined as \(\sqrt{-1}\). In the standard complex number form \(a + bi\), \(b\) represents the imaginary part. This part defines how far along the imaginary axis the number extends.

• For instance, in the complex number \(3 + 4i\), the imaginary part is 4.
• In \(0 + 9i\), the imaginary part is 9.

The imaginary part indicates directions on the axis that is orthogonal, or perpendicular, to the real number line. It's essential in fields like electrical engineering, where both real and imaginary components are necessary for defining impedance (resistance in AC circuits).
Complex operations can often revolve around manipulating both real and imaginary parts. This allows mathematicians and engineers to solve equations that would be impossible using only real numbers.

Number Classification in Algebra

Number classification begins with the clear distinction between real numbers and complex numbers. Within algebra, numbers are classified to better understand their unique properties and how they interact when combined. Here’s a simplified breakdown:

• Real Numbers: These are numbers that can be found on the number line, including integers, rational numbers, and irrational numbers. Examples are -3, 0, 1.5, and \(\sqrt{2}\).
• Imaginary Numbers: Numbers that, when squared, give a negative result. These are generally represented as multiples of \(i\).


• Complex Numbers: Combinations of real and imaginary numbers; they have both a real component and an imaginary component, such as \(2 + 7i\).

Recognizing and using these classifications is crucial as it allows you to apply the properties that each type of number has. Real numbers fit into many common operations, while imaginary numbers give flexibility for equations that real numbers alone cannot address.
Complex numbers, which are a class of their own, combine these properties to expand the realm of solutions further, making them invaluable in higher mathematics and various applications across different sciences.