- #1
donglepuss
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find me an integer that isn't divisible by 1.
Is every integer derived from 1?
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In summary, the conversation discussed the concept of deriving integers from 1 and whether every integer is divisible by 1. It was mentioned that 1 is a unit and the multiplicative identity for numbers. The connection between the title question and the original post was also explored. It was concluded that every integer can be derived from 1 by repeatedly adding 1 and that every integer is divisible by 1.
- #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?
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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.
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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.
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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
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nuuskur
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What is the connection between title question and the OP?
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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.
FAQ: Is every integer derived from 1?
1. What is an integer?
An integer is a whole number, either positive or negative, including zero. It does not contain any fractions or decimals.
2. Is 1 the only integer that can be derived from 1?
Yes, 1 is the only integer that can be derived from 1. This is because 1 is the smallest and only factor of itself.
3. Can negative integers be derived from 1?
No, negative integers cannot be derived from 1. This is because 1 can only be multiplied by itself to equal 1, and multiplying by a negative number would result in a negative product.
4. Are there any exceptions to the statement "every integer is derived from 1"?
No, there are no exceptions to this statement. Every integer, positive or negative, can be derived from 1 through multiplication or addition.
5. How is the concept of "deriving from 1" related to factorization?
The concept of "deriving from 1" is closely related to factorization. When we say that an integer is derived from 1, it means that 1 is a factor of that integer. In other words, the integer can be expressed as the product of 1 and another number, which is its factor. This is essentially the process of factorization.