What is Integers: Definition and 472 Discussions

An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.
The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the boldface (Z) or blackboard bold



(

Z

)


{\displaystyle (\mathbb {Z} )}
letter "Z"—standing originally for the German word Zahlen ("numbers").ℤ is a subset of the set of all rational numbers ℚ, which in turn is a subset of the real numbers ℝ. Like the natural numbers, ℤ is countably infinite.
The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.

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  1. A

    How to prove that pZ is a maximal ideal for the ring of integers?

    I know that Z/pZ is a field therefore pZ must be a maximal ideal because of the theorem that states "R/I is a field if and only if I is a maximal ideal" but I want to see a direct proof of it because I hope it would give me an idea how to prove something is a maximal ideal in a general field...
  2. T

    Solving Sum of Integers for Given Constraints

    Hi, I'm doing a physics calculation, and along the way, I've run up against a curious math problem. I'm sure this is a rather classic problem in mathematics, but I'm just not acquainted with the subject enough to answer it, or even look it up, so hopefully someone can point me in the right...
  3. caffeinemachine

    MHB Sum of 4 integers divisible by 4.

    1) Given $n$ integers. What is the minimum value of $n$ so that one can always choose $4$ integers from these $n$ integers such that the summation of the chosen $4$ integers is divisible by $4$. Using the Pigeon hole principle I was able to prove that $n \leq 9$. Then by computation (mostly) I...
  4. R

    To show a ring of order p (prime) is isomorphic to the integers mod p.

    If R is a finite ring of of order p where p is prime, show that either R is isomorphic to Z/pZ or that xy=0 for all x,y in R I know that both R and Z/pZ have the same number of elements (up to equivalence) and that R isomorphic to Z/pZ implies R must be cyclic (I think) but am otherwise...
  5. caffeinemachine

    MHB Among 2n-1 integers summation of some n of these is divisible by n.

    Let $k$ be a positive integer. Let $n=2^{k-1}$. Prove that, from $2n-1$ positive integers, one can select $n$ integers, such that their sum is divisible by $n$.
  6. Loren Booda

    Numbers which are not ultimately functions of integers

    Do there exist numbers which are not ultimately functions of the integers? Would they necessarily be transcendental numbers or otherwise uncountable?
  7. caffeinemachine

    MHB Pairwise difference of 20 positive integers. At least four of em are equal.

    Given $20$ pairwise distinct positive integers each less than $70$. Prove that among their pairwise differences there are at least four equal numbers.
  8. F

    Factoring in the Gaussian Integers

    I need to factorise 70 into primes, how do I go about this? So far I have 2,5,7 as primes in Z. So I suppose I need to factorise these in Z[i]? 2 = (1+i)(1-i) How do I go around doing the other two, is it possible that they're primes in Z[i]? Edit: I have a corollary where if p is a prime...
  9. S

    Proof by induction: 5^n + 9 < 6^n for all integers n≥2

    Homework Statement Prove the statement by mathematical induction: 5n + 9 < 6n for all integers n≥2 Homework Equations .. The Attempt at a Solution Proof: let P(n) be the statement, 5n + 9 < 6n P(2) is true because, 34<36. Suppose that P(n) is true. P(n+1) would be...
  10. K

    Countably infinite set: odd integers

    Homework Statement Using the definitions, prove that the set of odd integers is countably infinite. Homework Equations Definition: The set A is countably infinite if its elements can be put in a 1-1 correspondence with the set of positive integers. The Attempt at a Solution I am...
  11. I

    Finding the ∅c of the Set of Integers

    Let the universe be the set of Z. Let E, D, Z+, and Z- be the sets of all even, odd, positive, and negative integers respectively. Find ∅c. My thoughts were that since the universe is the set of all integers the ∅c would be all integers. Am I correct in my thinking or would the ∅c be...
  12. F

    Gaussian integers and norms

    Show that N(a+bi) = even => a+bi divisible by 1+i So, N(a+bi) = a2+b2 = even so 2 divides a2+b2 Write 2 = (1+i)(1-i) so we have 1+i divides a2+b2 so 1+i divides either (a+bi) or (a-bi) if 1+i divides a+bi we are done what if 1+i divides a-bi though? Thats where I'm stuck!
  13. G

    Why are reaction orders integers?

    I have just done an experiment on the clock reaction between iodine and persulfate ions. Using my experiment result, I have determined that the reaction orders are about 1.2 with respect to both persulfate and iodine ions. There is this question ' Explain why the reaction orders should be...
  14. T

    Finding Positive Integers for Irrational Number Interval

    someome please help me with this problem: "Any real numbers x and y with 0 < x < y, there exist positive integers p and q such that the irrational number s =( p√2)/q is in the interval (x; y)."
  15. B

    Intro to Abstract Math Question about divison of integers.

    (1)Assume a, b and n are nonzero integers. Prove that n is divisible by ab if and only if n is divisible by a and n is divisible by b.I'm wrong and can't remember why. I spoke to the professor about it for ~ 1 minute so it seems to have slipped my mind, it was because in one case it's true and...
  16. W

    Divisibility in the Integers. Intro to Analysis

    Homework Statement Prove: If a|b and b|c then a|c. Assume a, b and c are integers. Homework Equations none The Attempt at a Solution If a divides b then that means that there is a real integer "r" that is ra=b . and since we assume b divides c then c=bs. After...
  17. C

    Inclusion-exclusion positive integers

    Homework Statement Suppose that p and q are prime numbers and that n = pq. Use the principle of inclusion-exclusion to find the number of positive integers not exceeding n that are relatively prime to n. Homework Equations Inclusion-Exclusion The Attempt at a Solution The...
  18. P

    Residue field of p-adic integers

    In the field of rationals \mathbb{Z}_{(p)} (rationals in the ring of the p-adic integers), how is it possible to prove the residue field \mathbb{Z}_{(p)}/p\mathbb{Z}_{(p)} is equal to \mathbb{Z}/p\mathbb{Z} ? I've narrowed it down to \mathbb{Z}_{(p)}/p\mathbb{Z}_{(p)} = \left\{ a/b\in\mathbb{Q}...
  19. P

    Set of integers is closed.

    When considered as a subset of \mathbb{R}^2, \mathbb{Z} is a closed set. Proof. We will show, by definition, that \mathbb{Z} \subset \mathbb{R}^2 is closed. That is, we need to show that, if n is a limit point of \mathbb{Z}, then n \in \mathbb{Z}. I think this becomes vacuously true, since our...
  20. S

    Prove that the set of integers has neither a greatest nor a least element

    Prove that the set of integers has neither a greatest nor a least element. I was given a hint: There are 2 different non existence results to prove, so prove them as separate propositions or claims. Divide into cases using the definition of the set of integers. So I was kind of confused...
  21. V

    Reversing digits, then adding and finding divisible integers of result.

    Homework Statement If you find the sum of any two digit number and the number formed by reversing its digits, the resulting number is always divisible by which three positive integers? Homework Equations None. The Attempt at a Solution \left(10\ x\ +\ y\right)\ +\ \left(10\...
  22. P

    Irreducible polynomials over ring of integers

    Is it true that polynomials of the form : f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a where \gcd(n+1,k+1)=1 , a\in \mathbb{Z^{+}} , a is odd number , a>1, and a_1\neq 1 are irreducible over the ring of integers \mathbb{Z}...
  23. L

    Discovering the Prime and Factored Parts of Positive Integers

    Is there a way within reasonable errors to say what part of the positive integers are prime and what part is factored greater than one? Oh course one is a factor of all numbers greater than zero. Yeats ago playing around a floating constant became known to me. to the tenth decimal place is...
  24. E

    Factoring question - generalized factoring in integers

    Hello, this is rather complicated to explain so bear with me. I was wondering about the coefficients of polynomials which are factorable in the integers, meaning polynomials which can be written as (x+a)(x+b) where a and b are integers. I had a curious idea about letting the x-axis...
  25. F

    Creating a system of equations consisting only of integers?

    I'm looking for an algorithm to create a very simple (2 equations, 2 unknowns) linear system of equations that consists purely of integers. Specifically, a way to create a system of equations of integers and knowing that it can only be solved by integer answers, without actually solving it...
  26. J

    Openness of subsets of the integers.

    1. What are all the open subsets of the subspace Z of R. 2. Homework Equations : def of openness 3. I think the solution is all the subsets of Z, but I can't see how, for example you can say the subset of Z: {1} has a B(1,r) with r>0 is contained in {1}. Thanks for any help.
  27. U

    Comp Sci Input Validation for only positive integers C++

    Homework Statement I created a program that will calculate the factorial of the number entered and am having a hard time getting it to not accept decimals or fractions.#include <iostream> using namespace std; int main (){ int q=0; int number = 0; cout<<"Please enter a positive whole...
  28. S

    Problems on Integers: Q1-Q3 - Solutions Needed

    Q 1:- Given a sequence kn=[(1+(-1)^n)+1]/5n+6.. find the no of terms of the sequence kn which will satisfy the condition kn lies between 1/100 and 39/100. Q 2:- Find the sum of all the irreducable fractions between 10 and 20 with a denominator of 3 Q 3:- Find all pairs of natural no s...
  29. C

    Positive integers for k: finding limits

    Homework Statement find the positive integers k for which lim x->0 sin(sin(x))/x^kHomework Equations exists, and then find the value of the limit The Attempt at a Solution I did the first three k's k=0 lim x->0 sin(sin x))/x^0= 0 undefined k=1 lim x->0 sin(sinx))/x^1= 1 k=2 I might be...
  30. M

    Comp Sci FORTRAN: Problem with converting reals to integers

    Homework Statement I'm trying to convert data that's entered as a real number into integer data to be used in a do loop. The problem is that it keeps telling me that the numbers I've just converted are not scalar integers... The Attempt at a Solution program interest IMPLICIT NONE...
  31. J

    Expressing double factorial for odd integers

    Homework Statement Express \frac{1}{(2n+1)!} as the following \frac{(-2)^{n}n!x^{2n+1}}{(2n+1)!} where 0 <= n <= infinity Homework Equations The double factorial for odd integers is (2n+1)! = (2n+1)(2n-1)(2n-3)...1 where 0 <= n <= infinity The Attempt at a Solution...
  32. W

    Square of Odd Integers & Justifying "If P2 Is Even, Then P Is Even

    show that the square of any odd integer is odd, use this fact to justify the statement "if p2 is even , then p is also even
  33. M

    Ideals of the Gaussian integers

    I'm working on an exam that Michael Artin once gave, where one of the questions is basically, Consider the homomorphism from Z[x] to Z[i] given by x --> i. What does this homomorphism tell you about the ideals of Z[i]? So far I haven't come up with anything. I know in advance that the...
  34. M

    Number of combinations of integers \leq n which sum to n

    Hi all, I'm new to the forum, this is my problem: given a positive integer n, i want to find how many combinations of integers smaller than n but larger than 0 sum to n. E.g. n=3: {3},{2,1},{1,1,1} n=4: {4},{3,1},{2,2},{2,1,1},{1,1,1} it might just be that I'm tired, but I've been...
  35. P

    Give a big-O estimate of the product of the first n odd positive integers

    Homework Statement Give a big-O estimate of the product of the first n odd positive integers. Homework Equations Big-O notation: f(x) is O(g(x)) if there are constants C and k such that |f(x)| ≤ C|g(x)| whenever x > k. The Attempt at a Solution The product of the first n odd integers can be...
  36. C

    An integer m such that n=m^2?

    Homework Statement Prove: If n is a composite integer larger than 1 and if no prime number less than \sqrt{n} is a factor of n, then there is an integer m such that n=m^2 The Attempt at a Solution Proof: Let n be a positive composite integer larger than 1. If n is composite then there...
  37. P

    MATLAB Matlab: picking only integers out of a for loop

    Hi, I'm new to this forum and only have a few months experience with MATLAB but am getting to know it. Hope you can help me. I have a for loop which looks like: for f = a:a:b command end Now, I only want to execute this command for the integer values in this loop. eg. if it was...
  38. mnb96

    Number of ways to choose N integers that sum up to X

    Hello, is there a straightforward way, or some well-known expression to count how many ways there are of choosing N positive integers a_1,\ldots,a_N such that a_1+\ldots+a_N = X (where X is some fixed positive integer). Note that if N=2, and X=10 (for example), I consider the pairs 1+9 and 9+1...
  39. S

    Background for Gaussian Integers?

    Hi, I was wondering if having some training in algebraic number theory is a must for even starting to work with Gaussian Integers, or one can work with them with some knowledge of abstract algebra, like group, ring and field theory knowledge (i.e. 1 year of undergraduate abstract algebra)...
  40. M

    Does a Perfect Square Lie Within these 10 Integers?

    Do there exist 10 distinct integers such that the sum of any 9 of them is a perfect square.
  41. S

    Powers of integers and factorials

    I would like some direction on studying powers of integers and if they are in any way related to factorials. I was studying the sequence of cubics 1, 8, 27, 64, 125 and so. After a certain number of rounds of a basic rule I choose to apply to this sequence, I arrived at a new sequence...
  42. srfriggen

    Proof regarding division of integers

    Homework Statement Prove that for any a, b, c \inZ, if a l b (a divides b) and a does not divide c, then a does not divide b-c. Homework Equations The Attempt at a Solution Using the contrapositive: Prove that... if a l (b-c) then a does not divide b or a l c. 1. a l...
  43. I

    Which integers have exactly 3 distinct positive factors?

    [b]1. Which integers have exactly 3 distinct positive factors? Homework Equations [b]3. I would attempt this if I had any idea of what it meant. Can someone show me how to find one answer then I will find the other 2.
  44. L

    Question about Surreal Numbers and Omnific Integers

    Omnific integers are the counterpart in the Surreal numbers of integers. The surreal numbers are usually defined using set theory, and then the omnific integers are defined as a particular subset of them. My question is, does it have to be this way? Is it possible to give a first-order...
  45. R

    How to convert a string into integers (in MIPS)

    Hey, I'm working on a project for my Computer Organization class and I have to read in 2 separate times (in military format with no ":" ) as strings and then find and print the time difference. I am able to read in and save the strings no problem, but I don't have a clue as to how I can...
  46. Demon117

    C/C++ Trying to write a program that converts integers to roman numerals in C++

    Here is the code that I have come up with. What could be implimented to make it compile and run? It runs, but all that comes up is the request for the integer between 1 and 3999, after I enter the number the program simply ends. CODE: #include <iostream> using namespace std; string...
  47. H

    Exploring 8-bit BCD and Integers: How Many Can be Represented?

    Homework Statement Suppose a computer has 8-bit words. How many different integers can be represented (in decimal) in a single word if the integers are represented in binary coded decimal(BCD)? Homework Equations BCD= Binary Coded DecimalThe Attempt at a Solution BCD is coded in 4 bits so...
  48. H

    Finding Integers for a Fractional Equation

    Good day! I have problem: Find all integers for which is fraction (n3+2010)/(n2+2010) equals to integer. I can find 0 and 1 and I tried prove that any integers don't exist, but I didnt contrive it. Could someone help me with it?
  49. M

    Make integers constitute a field

    Homework Statement This question consists of three parts, the first two of which I have answered: a) Is the set of all positive integers a field? (positive indicates greater than or equal to 0, and ordinary definitions of addition and multiplication are being used) No. There is no additive...
  50. Y

    Integral Solutions for n,m Positive Integers

    Homework Statement \int ^b_0 cos(\frac{(n-m)\pi}{b}x) dx \int ^b_0 cos(\frac{(n+m)\pi}{b}x) dx n and m are positive integers. The Attempt at a Solution \int ^b_0 cos(\frac{(n-m)\pi}{b}x) dx = \frac{b\;sin[(n-m)\pi]}{(n-m)\pi} Obviously answer is zero if n not equal to...
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