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Chapter 1: Problem 2

The natural numbers, their additive inverses, and 0 form the set of _________.

Short Answer

Expert verified
The set of integers.

Step by step solution

01

Understand the given terms

Natural numbers are counting numbers starting from 1, such as 1, 2, 3, and so on. Additive inverses are the numbers that add up to zero when paired with another number. For example, the additive inverse of 3 is -3.
02

Incorporate zero into the set

The statement also includes zero, which is the neutral element for addition (i.e., adding zero to any number does not change the number).
03

Identify the set formed by these elements

The natural numbers, their additive inverses, and zero together form the set of integers. The set of integers includes all whole numbers and their negative counterparts, as well as zero.

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Key Concepts

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

Natural Numbers
Natural numbers are the foundation of counting. They start from 1 and continue infinitely in the sequence: 1, 2, 3, 4, and so on.
These numbers help us count objects, days, steps, and much more in our daily life.
Natural numbers are positive whole numbers and do not include zero or any negative numbers.
While natural numbers are primarily used for counting, they also form the basis of other mathematical sets and operations.
For instance, when we include zero and negative numbers, we expand our understanding beyond just counting.
Additive Inverses
An additive inverse is a number that, when added to a given number, results in zero.
For example, the additive inverse of 3 is -3, because \(3 + (-3) = 0\).
This property is crucial in understanding more complex mathematical concepts such as algebra.
Additive inverses allow us to solve equations and simplify expressions by balancing out positive and negative values.
Remember, every integer has an additive inverse:
  • The additive inverse of 5 is -5
  • The additive inverse of -7 is 7
  • The additive inverse of 0 is still 0
Zero
Zero is a unique number in mathematics. It is the neutral element in addition, meaning adding zero to any number does not change that number.
Think of zero as a placeholder or a starting point. For example, \(7 + 0 = 7\) and \(-5 + 0 = -5\).
Zero is neither positive nor negative, making it a special case in mathematics.
In many structures, zero serves as the baseline from which positive and negative numbers extend out.
Understanding zero fully is crucial because it plays a key role in various mathematical operations and sets.
For example, when combined with natural numbers and their additive inverses, zero helps form the set of integers.

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Most popular questions from this chapter

Find the average of each group of numbers. All integers between -15 and \(-10,\) including both -15 and -10

To find the average (mean) of a group of numbers, we add the numbers and then divide the sum by the number of terms added. Work Exercises 139-142 in order, to find the average of \(23,18,13,-4,\) and -8 How many numbers are in the group?

Write a numerical expression for each phrase, and simplify the expression. The difference of 4 and -8

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To find the average (mean) of a group of numbers, we add the numbers and then divide the sum by the number of terms added. Work Exercises 139-142 in order, to find the average of \(23,18,13,-4,\) and -8 What is the average of the given group of numbers?
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