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Chapter 1: Problem 68

Name the sets of numbers to which each number belongs. \(\sqrt{7}\)

Short Answer

Expert verified
\( \sqrt{7} \) is an irrational and real number.

Step by step solution

01

- Identify the Number Type

To determine the set of numbers that \( \sqrt{7} \) belongs to, start by recognizing what \( \sqrt{7} \) represents. The number \( \sqrt{7} \) is a non-perfect square root, meaning that it cannot be expressed as a ratio of two integers, so it is not a rational number. Instead, it is an irrational number because it has a non-repeating, non-terminating decimal expansion.
02

- Identify the Larger Set of Numbers

Although \( \sqrt{7} \) is irrational, it also belongs to the set of real numbers. Real numbers include both rational and irrational numbers, encompassing all numbers that can be found on the number line. Since \( \sqrt{7} \) is a value that can be plotted on the number line, it is a member of the real numbers set.

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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 fascinating because they cannot be written as a simple fraction with integers. Instead, they have decimal expansions that go on forever without repeating. Think of a number like \(\sqrt{7}\): when you try to calculate its square root, you get a messy decimal, approximately 2.645751311... and it keeps going without a pattern. That's what makes it irrational. Here are some key points about irrational numbers:
  • They cannot be expressed as a fraction of two integers.
  • The decimal does not repeat or terminate.
  • Examples include \(\pi\) (pi) and \(e\), the base of the natural logarithm.
Understanding irrational numbers helps us comprehend just how diverse the number line truly is.
Real Numbers
Real numbers are like the all-inclusive club on the number line. Every point you see there is a real number. They bring together rational numbers, like integers and fractions, with irrational numbers. Real numbers cover both the nice, neat fractions like \(\frac{1}{2}\), and the forever-unfinished decimals like \( \sqrt{7} \). Here's what to know about real numbers:
  • They include both rational and irrational numbers.
  • They can be negative, positive, or zero.
  • Any real number can be represented as a point on the number line.
If you visualize the number line, real numbers extend endlessly in both directions. They're vital for understanding almost all number-related topics, and they're the basis of much of mathematics.
Number Sets
Number sets are like categories where we place numbers based on their properties. They help us organize and better understand different types of numbers.Here's a clear hierarchy of common number sets:
  • Natural Numbers (\(\mathbb{N}\)): Counting numbers starting from 1, 2, 3, and so on.
  • Whole Numbers: Like natural numbers, but include 0.
  • Integers (\(\mathbb{Z}\)): All whole numbers plus their negative counterparts.
  • Rational Numbers (\(\mathbb{Q}\)): Numbers that can be expressed as a fraction of integers where the denominator is not zero.
  • Irrational Numbers: Numbers that cannot be written as fractions of integers and have non-repeating decimals.
  • Real Numbers (\(\mathbb{R}\)): All rational and irrational numbers.
Understanding these categories is crucial for navigating the different types of numbers you'll encounter in math. Each set builds on previous ones, creating a comprehensive framework for numerical classification.

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