Proof of the Division Algorithm

In summary, the well ordering principle (WOP) states that every non-empty subset of positive integers has a least element. This principle can be applied to a subset of non-negative integers by observing that if the subset contains zero, then zero is the least element. Otherwise, the subset can be considered a subset of positive integers and the principle can be applied accordingly. It is possible to apply WOP to any subset of integers that is bounded below.
  • #1
matqkks
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TL;DR Summary
Application of well ordeing principle
In many books on number theory they define the well ordering principle (WOP) as:

Every non- empty subset of positive integers has a least element.

Then they use this in the proof of the division algorithm by constructing non-negative integers and applying WOP to this construction. Is it possible to apply the WOP to a subset of non-negative integers? Am I being too pedantic?
 
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  • #2
Yes, you can apply it to the non-negative integers, by simply observing that if the subset contains zero then zero is the least element, otherwise the subset is also a subset of the positive integers and we can apply the principle that holds for them.
 
  • #3
matqkks said:
Summary: Application of well ordeing principle

In many books on number theory they define the well ordering principle (WOP) as:

Every non- empty subset of positive integers has a least element.

Then they use this in the proof of the division algorithm by constructing non-negative integers and applying WOP to this construction. Is it possible to apply the WOP to a subset of non-negative integers? Am I being too pedantic?

Yes, well ordering principle applies to any subset of ##\mathbb{Z}## that is bounded below.
 
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  • #4
Thanks it is so obvious as you have suggested.
 

FAQ: Proof of the Division Algorithm

What is the Division Algorithm?

The Division Algorithm is a mathematical concept that states that any integer dividend can be divided by a non-zero integer divisor, resulting in a quotient and remainder. This algorithm is used to solve division problems and is an important concept in number theory.

What is the proof of the Division Algorithm?

The proof of the Division Algorithm is a mathematical proof that shows the validity of the algorithm. It uses the concepts of division, remainder, and modulus to show that the algorithm always produces a unique quotient and remainder when dividing any integer by a non-zero integer.

Why is the Division Algorithm important?

The Division Algorithm is important because it provides a systematic and efficient way to solve division problems. It is also a fundamental concept in number theory and has many applications in fields such as cryptography and computer science.

Can the Division Algorithm be used for non-integer numbers?

No, the Division Algorithm is specifically designed for integers. When dealing with non-integer numbers, the concept of division is extended to include fractions and decimals, and a different algorithm is used to solve division problems.

Are there any limitations to the Division Algorithm?

Yes, the Division Algorithm has limitations. It can only be used for dividing integers and cannot be used for dividing by zero. It also assumes that the divisor is non-zero, otherwise, the algorithm will not work. Additionally, the quotient and remainder produced by the algorithm may not always be the most simplified form.

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