Lecture notes regarding integers ?

In summary: And also i came across a proposition somewhere in some maths books ( but it's only a small part ) Saying that : - If A and B are subgroups of , then so is their intersection A ∩ B and so is the set {m+n : m ∈ A , n ∈ B } they followed on by saying this which i don't really get the picture - A ∩ B contains every subgroup contained in both A and B - A + B is contained in every subgroup containing both A and B someone please help me out =) thanksIn summary, the conversation discusses the search for lecture notes on integers and real numbers, as well as a proposition involving subgroups of integers. The proof and use of the proposition are also mentioned,
  • #1
garyljc
103
0
Hey guys ,
was wondering if you guys know of any lecture notes regarding integers ?
i would like to further my knowledge in this field ... cheers :smile:
 
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  • #2
Are you serious? There are thousands of different "fields" involving integers. You will have to narrow down the search! What, exactly, are you looking for? Number theory problems such as solve Diophantine equations? Analysis questions such the the definition of integers and basic properties?
 
  • #3
While you're at it, HallsofIvy, how about some notes on real numbers too?
 
  • #4
I guess to start with I would like to look at the basics such as the proof of : If S ⊂ ℤ , then there is a natural number g such that S={ gn : n ∈ ℤ }

And also i came across a proposition somewhere in some maths books ( but it's only a small part ) Saying that :
- If A and B are subgroups of ℤ , then so is their intersection A ∩ B and so is the set
{m+n : m ∈ A , n ∈ B }
they followed on by saying this which i don't really get the picture
- A ∩ B contains every subgroup contained in both A and B
- A + B is contained in every subgroup containing both A and B

someone please help me out =) thanks
 
  • #5
garyljc said:
I guess to start with I would like to look at the basics such as the proof of : If S ⊂ ℤ , then there is a natural number g such that S={ gn : n ∈ ℤ }

And also i came across a proposition somewhere in some maths books ( but it's only a small part ) Saying that :
- If A and B are subgroups of ℤ , then so is their intersection A ∩ B and so is the set
{m+n : m ∈ A , n ∈ B }
they followed on by saying this which i don't really get the picture
- A ∩ B contains every subgroup contained in both A and B
- A + B is contained in every subgroup containing both A and B

someone please help me out =) thanks
You are using non-standard fonts that do not display on my web-reader. Please try LaTex or just stating the problem in words.
 
  • #6
garyljc said:
I guess to start with I would like to look at the basics such as the proof of : If [itex]S \subset \mathbf{Z}[/itex] , then there is a natural number g such that [itex]S = \{ gn : n \in \mathbf{Z} \}[/itex].
Am I misreading this, or is this simply not true? For example, take S = {1, 2, 3}, this is finite, while any set of the form [itex]\{ g n : n \in \mathbf{Z} \}[/itex] is necessarily {0} or countable infinite.

garyljc said:
And also i came across a proposition somewhere in some maths books ( but it's only a small part ) Saying that :
- If A and B are subgroups of [itex]\mathbf{Z}[/itex] , then so is their intersection A ∩ B and so is the set
[itex]\{m+n : m \in A , n \in B \} [/itex]
they followed on by saying this which i don't really get the picture
- [itex]A \cap B[/itex] contains every subgroup contained in both A and B
- A + B is contained in every subgroup containing both A and B
You can also just check the group axioms and see that it is true.
 
  • #7
this is what i copied exactly from the book ...
but anyways , do you know of any sites that has such notes or reading material that could help me ?
 
  • #8
by the way i did make a mistake
If S ⊂ ℤ is a subgroup , then there is a natural number g such that S={ gn : n ∈ ℤ }
this should be the right one

halls , where could i get latex ?
 
  • #9
You don't have to "get it" at all, it's part of this forum. Start with [ tex ] (without the spaces) and end with [ /tex ] and use LaTex syntax in between. Here's an example:
[tex]e^x= \sum_{n=0}^\infty \frac{x^n}{n!}[/tex]
Click on that to see the code.

More on LaTex syntax can be found here:
https://www.physicsforums.com/showthread.php?t=8997
 
Last edited by a moderator:
  • #10
garyljc said:
but anyways , do you know of any sites that has such notes or reading material that could help me ?
Check http://users.ictp.it/~stefanov/mylist.html" [Broken].
 
Last edited by a moderator:
  • #11
thanks thanks ...
halls ... i'll get it done right away =)
 
  • #12
I guess to start with I would like to look at the basics such as the proof of : If , then there is a natural number g such that .
 

1. What are integers and how are they different from other numbers?

Integers are whole numbers that can be positive, negative, or zero. They are different from other numbers such as fractions and decimals because they do not have any decimal or fractional parts.

2. How are integers represented in mathematical equations?

Integers are typically represented by the letter "Z" and are written as Z = {0, ±1, ±2, ±3, ...}. In equations, integers can be added, subtracted, multiplied, and divided just like other numbers.

3. What is the difference between integers and whole numbers?

Integers include both positive and negative numbers, while whole numbers only include non-negative numbers (0 and positive integers). In other words, all integers are whole numbers, but not all whole numbers are integers.

4. How are integers used in real-life situations?

Integers are used in many real-life situations such as counting, measuring temperature, and representing money. They can also be used to indicate direction, with positive integers representing movement in one direction and negative integers representing movement in the opposite direction.

5. What are the rules for adding and subtracting integers?

The rule for adding integers is that if the numbers have the same sign, you add their absolute values and keep the sign. If the numbers have different signs, you subtract their absolute values and use the sign of the number with the larger absolute value. The rule for subtracting integers is to change the subtraction sign to addition and change the sign of the number being subtracted to its opposite.

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