Problem 55 - Beginning and Intermediate Algebra

Question

Tell whether each statement is true or false. 0 is a real number.

Step-by-Step Solution

Answer

True, 0 is a real number.

Step 1: Understanding Real Numbers

Real numbers include all the numbers that can be found on the number line. This includes both rational numbers (like integers and fractions) and irrational numbers (like √2 and π).

Step 2: Identify the Category of 0

The number 0 is an integer, making it a rational number. Since real numbers encompass both rational and irrational numbers, 0 is also considered a real number.

Step 3: Conclusion on 0's Status

Since 0 is a rational number and all rational numbers are real numbers, we conclude that 0 is indeed a real number.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction or a ratio of two integers. Specifically, a rational number is any number that can be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \).
It's important to remember that integers, fractions, and finite decimals are all considered rational numbers.

• For example, \( \frac{1}{2} \) is a rational number because it is a fraction.
• The number 5 is also a rational number because it can be expressed as \( \frac{5}{1} \).
• Even repeating decimals like 0.333... are rational, as they can be converted into the fraction \( \frac{1}{3} \).
• Zero is a unique rational number because it can be expressed as \( \frac{0}{1} \) or any \( \frac{0}{b} \) where \( b eq 0 \).

Rational numbers are a subset of real numbers, which means every rational number can also be located on the number line. They are one of the simplest forms of numbers used in everyday mathematics.

Integers

Integers are whole numbers that can be positive, negative, or zero. These numbers do not have any decimal or fractional parts. Essentially, integers are the basic counting numbers, their negatives, and zero.

• Examples include -3, -2, -1, 0, 1, 2, 3, and so on.
• Zero is particularly interesting because it serves as the neutral point between positive and negative integers.
• While integers are whole numbers, not all whole numbers are integers, especially when they include fractions or decimals.

All integers are rational numbers because each integer \( n \) can be represented as \( \frac{n}{1} \). This means they can also easily be placed on a number line. Integers form an important foundational set in mathematics, used in everything from counting objects to performing algebraic operations.

Number Line

The number line is a visual representation of numbers, evenly spaced along a straight horizontal line. This tool is used to illustrate the position, order, and value of numbers, making concepts in arithmetic and algebra more tangible.

• It typically includes both positive and negative numbers, with zero at the center.
• Numbers to the right of zero are positive, while numbers to the left are negative.
• It is an essential tool for showing operations like addition, subtraction, and illustrating inequalities.
• Real numbers, including both rational and irrational numbers, can be located on the number line.

Since all real numbers can be placed on the number line, it serves as a great way to compare and contrast numbers, providing a clear picture of how numbers relate to one another. The number line helps clarify that numbers like 0, despite their neutral value, are very much part of the real number system. This makes the concept of "real numbers" more accessible and functional in mathematics.