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Chapter 5: Problem 46

Use two numbers to show that the natural numbers are not closed with respect to division.

Short Answer

Expert verified
The natural numbers are not closed under the operation of division because a division of two natural numbers can result in a number that is not a natural number. For example, 1 divided by 2 equals 0.5, which is not a natural number.

Step by step solution

01

Understand natural numbers

First, we should recognize that natural numbers are whole numbers greater than 0. They include numbers like 1, 2, 3, 4, etc.
02

Understand 'closed' under an operation

When we say that a set of numbers is closed under an operation, we mean that performing that operation on any numbers in the set will always produce a number that is also in the set. In this case, the operation is division.
03

Find two numbers that disprove the statement

To show that the natural numbers are not closed under division, we can take two natural numbers where the second number is not a divisor of the first one. A good example would be to take the natural numbers 1 and 2. When 1 is divided by 2 the result is 0.5, which is not a natural number.

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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, \(r\). \(a_{1}=2, r=0.1\)

Write a formula for the general term (the nth term) of each arithmetic sequence. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. \(a_{1}=-70, d=-5\)

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Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-2, r=-3\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{6}\), when \(a_{1}=-2, r=-3\).
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