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Chapter 5: Problem 9

Give an example of a number that is an integer, a whole number, and a natural number.

Short Answer

Expert verified
An example of a number that is an integer, a whole number, and a natural number is 1.

Step by step solution

01

Choosing an appropriate number

Choose a number which is positive and does not have a fractional component. This number has to be greater than zero, because it has to be a natural number, and natural numbers start from 1.
02

Number Verification

Let's take number 1 for example. This number is positive, does not have a fractional part, and is also greater than zero. It therefore qualifies to be an integer, a whole number, and a natural number.
03

Final Verification

To ensure the number fits all three categories verify that: As an integer, it can be written without a fractional component, which number 1 does. As a whole number, it needs to be non-negative and 1 meet this condition. Lastly, as a natural number it has to be positive and 1 is a positive number.

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Key Concepts

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

Integers
Integers are a set of numbers that include all whole numbers and their negative counterparts. They don't have any fractional or decimal parts. The set of integers is represented by the symbol \( \mathbb{Z} \). This set includes:
  • Positive numbers (e.g., 1, 2, 3, ...)
  • Negative numbers (e.g., -1, -2, -3, ...)
  • Zero (0)
When we talk about the absence of fractional parts, it means that integers are complete numbers, not parts or portions of a number. Because of these properties, integers are very useful in counting, measuring, and ordering.
It is important to remember that while zero is an integer, it is neither positive nor negative.
Whole Numbers
Whole numbers are very similar to integers but there is a slight difference. Whole numbers are the set of all non-negative integers. The set is denoted as \( \mathbb{W} \), and it includes:
  • Zero (0)
  • All positive integers (e.g., 1, 2, 3, ...)
Since whole numbers do not include negative numbers, they are essentially just the positive integers and zero.
This makes whole numbers very straightforward and easy to use, particularly when dealing with counts of quantities that can't be negative, like people or items.
A key feature of whole numbers is that they always represent quantities that can exist in the real world.
Natural Numbers
Natural numbers represent the most basic numbers used for counting. The set of natural numbers, represented by \( \mathbb{N} \), includes:
  • All positive integers starting from 1 (e.g., 1, 2, 3, ...)
Natural numbers don't include zero, which is what differentiates them from whole numbers.
This set is typically used in everyday situations where you are counting things that are part of collections or sequences, like books on a shelf or seconds on a clock.
Since natural numbers start from 1, they are ideal when describing things that must have a quantity of at least one. They form the foundation for more complex mathematical ideas and structures.

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Most popular questions from this chapter

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-1000, r=0.1\)

In Exercises 71-90, find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{7}\), when \(a_{1}=4, r=2\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If the first term of an arithmetic sequence is 5 and the third term is \(-3\), then the fourth term is \(-7\).

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(15,30,60,120, \ldots\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that the big difference between arithmetic and geometric sequences is that arithmetic sequences are based on addition and geometric sequences are based on multiplication.
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