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Chapter 0: Problem 6

In Exercises determine whether the real number is rational or irrational. $$\frac{22}{7}$$

Short Answer

Expert verified
The number \( \frac{22}{7} \) is rational.

Step by step solution

01

Understand the Mathematical Definitions

Recall that a rational number is a number that can be written as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are both integers and \( b \neq 0 \). An irrational number is a number that cannot be written in this way.
02

Check if the Number can be Expressed as a Fraction

The given number \( \frac{22}{7} \) is already in the form of a fraction where both 22 and 7 are integers, and 7 is not equal to zero.
03

Identify the Number

Since the given number can be expressed as a fraction, we can conclude that \( \frac{22}{7} \) is a rational number.

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Key Concepts

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

Irrational Numbers
Irrational numbers are a set of numbers that are quite unique in the world of mathematics. They are numbers that cannot be expressed as a simple fraction of two integers, which means they do not have a finite or repeating decimal representation. The decimal expansion of an irrational number goes on forever without repeating a pattern. A famous example of an irrational number is the square root of 2, often written as \( \sqrt{2} \), which equates to a never-ending decimal that cannot be expressed as a fraction.
  • They include numbers such as \( \pi \) (pi) and the natural logarithm base \( e \), both of which extend infinitely without repetition.
  • Irrational numbers are crucial in many mathematical fields such as geometry and calculus, providing solutions that rational numbers cannot.
Understanding irrational numbers helps in grasping the complete set of real numbers, showcasing the diversity within number types.
Integer
An integer is one of the simplest and most recognizable types of numbers in mathematics. Integers are the set of whole numbers that include the positive numbers, negative numbers, and zero. They do not have any fractional or decimal component.
  • Some examples include -3, 0, 1, and 42.
  • Integers have excellent properties such as closure under addition, subtraction, and multiplication.
  • They are also used to represent discrete quantities, which are invaluable in counting and ordering items in everyday life.
Unlike rational numbers, integers do not allow fractions or decimals, making them easier to work with when performing basic arithmetic operations.
Fraction
Fractions are a way to represent parts of a whole and are expressed as a ratio of two integers. They are written in the form \( \frac{a}{b} \), where \( a \) is called the numerator and \( b \) is the denominator. The denominator (\( b \)) specifies into how many equal parts the whole is divided, and the numerator (\( a \)) tells how many of those parts are being considered.
  • Fractions can be proper (numerator is less than the denominator) or improper (numerator is greater than or equal to the denominator).
  • A fraction like \( \frac{22}{7} \) is an example of a rational number, as it represents a division of two integers (22 and 7).
  • Fractions are versatile and used extensively in different mathematical contexts, from simple arithmetic to complex equations.
Understanding fractions is essential for a comprehensive knowledge of rational numbers, providing the flexibility to represent numbers in a form that decimal notation sometimes cannot.

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Most popular questions from this chapter

simplify the expression. $$ \frac{7 x^{2}}{x^{-3}} $$

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