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Chapter 1: Problem 43

Determine if the statement is true or false. a. \(6 \in \mathbb{R}\) b. \(6 \in \mathbb{C}\)

Short Answer

Expert verified
Both statements a and b are true.

Step by step solution

01

Understanding the sets

Set \( \mathbb{R} \) represents the set of all real numbers. Set \( \mathbb{C} \) represents the set of all complex numbers.
02

Analyzing statement a

Statement a: \(6 \in \mathbb{R} \). The number 6 is a real number since it can be represented on the number line. Thus, the statement is true.
03

Analyzing statement b

Statement b: \(6 \in \mathbb{C} \). The number 6 can also be considered a complex number, written as \(6 + 0i \), since complex numbers consist of a real part and an imaginary part. Hence, this statement is also true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real numbers
Real numbers, represented by the set \(\mathbb{R} \), include all the numbers you can find on a number line. This set contains both rational numbers (like \(1/2\) and \(4\)) and irrational numbers (like \(\sqrt{2}\) and \(\pi\). Real numbers can be positive or negative, and they cover many familiar types of numbers:
  • Whole numbers (e.g., 0, 1, 2,...)
  • Integers (e.g., -1, 0, 1,...)
  • Fractions (e.g., 1/2, 3/4)
  • Decimals (e.g., 2.5, -0.75)
If you can place a number on a number line without any imaginary component, it's a real number. In the given exercise, 6 is a real number because it fits neatly onto the number line, making the statement \(6 \in \mathbb{R}\) true.
Complex numbers
Complex numbers are numbers that have both real and imaginary parts. They are represented by the set \(\mathbb{C}\). A complex number is typically written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part (with \(i\) being the imaginary unit, \( i = \sqrt{-1}\)).
  • For example, \(3 + 4i\) is a complex number with a real part of 3 and an imaginary part of 4i.
  • A purely real number can also be seen as a complex number with an imaginary part of 0 (like \(6 + 0i\)).
This means that all real numbers are technically also complex numbers. Given this, the statement \(6 \in \mathbb{C}\) is true because 6 can be written as \(6 + 0i\).
Number sets
Number sets help us categorize numbers into different groups, each with its unique properties. This makes it easier to study and work with numbers. Some main number sets include:
  • Natural Numbers (\(\mathbb{N}\)
    These are counting numbers starting from 1, 2, 3, and so on. Sometimes, 0 is included depending on the context.
  • Whole Numbers (\(\mathbb{W}\)
    These are natural numbers plus 0. So it includes numbers like 0, 1, 2, 3,...
  • Integers (\(\mathbb{Z}\)
    These include all natural numbers, their negative counterparts, and 0. For example, -2, -1, 0, 1, 2,...
  • Rational Numbers (\(\mathbb{Q}\)
    These are numbers that can be expressed as fractions, such as 1/2, -3/4, and 5.
  • Irrational Numbers
    Numbers that cannot be written as simple fractions. Examples include \(\pi\) and \(\sqrt{3}\).
  • Real Numbers (\(\mathbb{R}\)
    This set encompasses all rational and irrational numbers, essentially covering anything that can appear on a number line.
  • Complex Numbers (\(\mathbb{C}\)
    These include both real numbers and numbers with imaginary parts.
Understanding these sets helps us see where numbers fit and how they relate to each other. In the exercise, recognizing that 6 fits into multiple sets (real numbers and complex numbers) makes answering the questions much easier.

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