- #1
donglepuss
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find me an integer that isn't divisible by 1.
Is every integer derived from 1?
- Context: High School
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Discussion Overview
The discussion revolves around the question of whether every integer is derived from 1, exploring concepts of divisibility, the definition of units in mathematics, and the operations that might lead to the generation of integers. The scope includes conceptual clarification and mathematical reasoning.
Discussion Character
- Conceptual clarification
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants assert that every integer is divisible by 1, suggesting this is a fundamental property of integers.
- Others question the meaning of "derived from" and propose that clarification is needed regarding the operations involved, such as addition or multiplication.
- A participant mentions that the classification of integers includes "units," and discusses the implications of this classification in abstract algebra.
- There is a suggestion that integers can be generated by repeatedly adding 1, which some participants seem to agree with.
- One participant references Peano's Axioms as a potential framework for understanding the properties of integers.
- Another participant emphasizes that the number 1 is both a unit and the multiplicative identity, reinforcing its role in divisibility.
Areas of Agreement / Disagreement
Participants generally agree that every integer is divisible by 1, but there is no consensus on the interpretation of "derived from" and whether this implies a specific mathematical operation. The discussion remains unresolved regarding the broader implications of these definitions.
Contextual Notes
The discussion highlights varying interpretations of mathematical terminology and concepts, particularly around the definitions of units and operations in the context of integers. Some assumptions about the nature of integers and their properties are not explicitly stated, leading to potential ambiguity.
- #2
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This is a contradiction in itself. ##1## is a unit.
- #3
donglepuss
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what do you mean?fresh_42 said:
- #4
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No. Why would you think it is?donglepuss said:
Every even integer is divisible by 2. So what? Do you think that means that all even integers are derived from 2?
- #5
DaveE
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I don't think anyone here understands what you mean by "derived from".
Are you referring to specific operations on the integers, like addition, multiplication, etc. Perhaps you could give us some examples of things that are "derived from" other things.
So, 3 = 1+1+1, then is 3 "derived from" 1?
Also, every integer is divisible by one, that is sort of the definition of "1". This is the basis of the "1 is a unit" comment. I would call it the multiplicative identity for numbers.
Are you referring to specific operations on the integers, like addition, multiplication, etc. Perhaps you could give us some examples of things that are "derived from" other things.
So, 3 = 1+1+1, then is 3 "derived from" 1?
Also, every integer is divisible by one, that is sort of the definition of "1". This is the basis of the "1 is a unit" comment. I would call it the multiplicative identity for numbers.
- #6
jbriggs444
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The statement that "1 is a unit" comes from a generalization of the notions of addition and multiplication into abstract "rings" The notions of divisibility and of being a "prime number" can apply to such structures. The notion of a "unit" is also definable.donglepuss said:
In grade school, we classified the positive integers as "prime", "composite" and "one". In the more general context, the classification is "prime", "composite" and "unit". [@fresh_42 would likely be quick to point out that we need not classify 0 since it is not a member of the multiplicative group]
One way to define "unit" is "any element which can be multiplied by another element to obtain 1 as a result". Using this definition and considering the signed integers, -1 is a unit since -1 * -1 = 1. Of course, 1 itself is always a unit since 1 * 1 = 1.
- #7
berkeman
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Have you learned about Peano's Axioms yet? If not, I think you will enjoy reading about them:donglepuss said:
https://en.wikipedia.org/wiki/Peano_axioms
- #8
nuuskur
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What is the connection between title question and the OP?
- #9
Mark44
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Thread title: Is every integer derived from 1?
If by "derived by" you mean "can we get any integer by repeatedly adding 1?" -- Yes
OP question: find me an integer that isn't divisible by 1.
Answer: There aren't any. Besides being a unit, the number 1 is the multiplicative identity. For any real number r (which includes the integers), ##1 \times r = r##. This clearly shows that 1 is a factor of r, hence r is divisible by 1.
Since the question has been asked and answered, I'm closing the thread.
If by "derived by" you mean "can we get any integer by repeatedly adding 1?" -- Yes
OP question: find me an integer that isn't divisible by 1.
Answer: There aren't any. Besides being a unit, the number 1 is the multiplicative identity. For any real number r (which includes the integers), ##1 \times r = r##. This clearly shows that 1 is a factor of r, hence r is divisible by 1.
Since the question has been asked and answered, I'm closing the thread.
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