Questions about the basic properties of Integers

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    Integers Properties
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Discussion Overview

The discussion revolves around basic properties of integers in the context of Number Theory, specifically focusing on the ordering of integers and the concept of divisibility. Participants are seeking clarification on foundational concepts and definitions that underpin these properties.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant presents two problems related to integers, asking for help in demonstrating properties of integers regarding ordering and divisibility.
  • Another participant questions the definitions of the ordering of integers, specifically what allows for comparisons like 3 < 12.
  • There is a mention of a lack of clarity on what definitions or principles can be used in the proofs, particularly regarding the general definitions of inequalities.
  • A participant suggests that understanding how a - b relates to 0 when a < b could help clarify the concept of ordering.

Areas of Agreement / Disagreement

Participants express uncertainty about the definitions and principles that should be applied, indicating that there is no consensus on how to approach the problems or what foundational definitions are acceptable.

Contextual Notes

Participants note that the professor has not provided specific definitions or guidance, leading to ambiguity in how to proceed with the problems. There is also a reference to previous attempts at solutions that did not meet the professor's expectations.

Who May Find This Useful

Students beginning their study of Number Theory, particularly those interested in the properties of integers and foundational mathematical definitions.

ninjagod123
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I am starting Number Theory this semester. My professor hands out notes but there is no textbook for the class. So hopefully you guys can help me with these seemingly easy problems.

Z = {...,-5,-4,-3,-2,-1,0,1,2,3,4,5,...}
Z is used to denote the set of integers

1) Show that if a is an element of Z and 0< a, then 1<=a

2) Let a and b be integers. Let us say that a divides b if there is an integer c such that b = ac. Show that if b>0 and a divides b then a<=b.

All we have learned so far are basic arithmetic properties, the Well-Ordering Principle, and the Induction Principle.
 
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ninjagod123 said:
I am starting Number Theory this semester. My professor hands out notes but there is no textbook for the class. So hopefully you guys can help me with these seemingly easy problems.

Z = {...,-5,-4,-3,-2,-1,0,1,2,3,4,5,...}
Z is used to denote the set of integers

1) Show that if a is an element of Z and 0< a, then 1<=a

2) Let a and b be integers. Let us say that a divides b if there is an integer c such that b = ac. Show that if b>0 and a divides b then a<=b.

All we have learned so far are basic arithmetic properties, the Well-Ordering Principle, and the Induction Principle.

I am not sure what you are allowed to use but it seems like you have some ordering on the integers that gives you an idea 0f < and <=. What is that?
 
wofsy said:
I am not sure what you are allowed to use but it seems like you have some ordering on the integers that gives you an idea 0f < and <=. What is that?

Hey sorry I don't understand. In class, some students attempted solutions but the solutions didn't satisfy the professor.
 
ninjagod123 said:
Hey sorry I don't understand. In class, some students attempted solutions but the solutions didn't satisfy the professor.

what definition of < are you using?

The reason I ask is that I don't see what allows you to say that any number is greater or less than any other. Why is 3 < 12?
 
wofsy said:
what definition of < are you using?

The reason I ask is that I don't see what allows you to say that any number is greater or less than any other. Why is 3 < 12?

I don't know what kind of definitions there are. But since this was the first day, and the professor didn't say anything, I suppose we use the general definitions? I guess that's too vauge.
 
ninjagod123 said:
I don't know what kind of definitions there are. But since this was the first day, and the professor didn't say anything, I suppose we use the general definitions? I guess that's too vauge.
What wofsy wants is the answer to the question
If a<b how does a-b relate to 0? The general definition of "<" will do fine.
 

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