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Chapter 0: Problem 4

\(\mathbb{Z}=\\{\ldots,-3,-2,-1,0,1,2,3, \ldots\\}\) is called the set of _____.

Short Answer

Expert verified
integers

Step by step solution

01

- Identify the Given Set

Observe the given set: \(\mathbb{Z}=\{\ldots,-3,-2,-1,0,1,2,3, \ldots\}\) . This set includes both positive and negative integers.
02

- Recognize the Pattern and Structure

Notice the pattern of the elements in the set. It includes all positive integers, all negative integers, and zero, without any fractions or decimals.
03

- Define the Set

This specific structure of the set is known to represent the 'integers'.

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Key Concepts

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

Integers
The given set \(\mathbb{Z}=\{...,-3,-2,-1,0,1,2,3,...\} \)is known as the set of integers. Integers are whole numbers.
They can be positive, negative, or zero.In mathematical terms, integers are represented by the symbol \(\mathbb{Z}\).This symbol comes from the German word 'Zahlen,' which means 'numbers.'
Some key points to remember about integers are:
  • Integers do not include fractions or decimals.
  • They form an infinite set both in the positive and negative direction.
  • Integers are useful in various mathematical computations and real-world applications.

Understanding integers is crucial because they are foundational to many math concepts.
Positive and Negative Integers
Integers can be divided into positive and negative integers along with zero.Positive integers are represented as \(1, 2, 3, ...\)and are found to the right of zero on the number line.
These numbers are greater than zero. Negative integers are represented as \(...,-3,-2,-1\)and are found to the left of zero on the number line.
These numbers are less than zero.Here are some insights about positive and negative integers:
  • Positive integers: Examples include 5, 23, and 100.
  • Negative integers: Examples include -7, -45, and -3.
Zero is a unique integer as it is neither positive nor negative. It acts as a neutral point on the number line, separating positive and negative integers.
Whole Numbers
Whole numbers are a subset of integers.
They include all non-negative integers: \(0, 1, 2, 3, ...\)This means whole numbers include zero and all the positive integers but do not include any negative numbers.
Whole numbers can be useful when counting objects or describing quantities that cannot be divided into parts.
Key points to remember about whole numbers:
  • Whole numbers start from zero and go on to infinity.
  • They do not include fractions, decimals, or negative numbers.
  • Examples of whole numbers: 0, 15, 263.
When performing arithmetic operations, whole numbers maintain their identity by staying within the set of whole numbers unless subtracted resulting in a negative output.

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