What is Integer: Definition and 620 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.

View More On Wikipedia.org
Recent contents Top users
  1. W

    Why chern number must be an integer?

    i am a student of physics i do not know much about the chern number i wonder why the chern number must be an integer any good reference?
  2. T

    Pseudorandom integer sequence having at least 15 adjacent differences > 36

    The requirement is to find an pseudorandom integer sequence i0, i1, i2, i3, ... , i48, i49 so that there are at least 15 adjacent differences which are greater than 36. Adjacent difference = absolute value of the difference between two adjacent integers = |i - i | where j = 0 to 49 | j -...
  3. A

    Integer Programming help please T__T

    Homework Statement Paulette Smith and Maureen Becker are senior in engineering and business, respectively, at State University. They have set up a company, PM Computer Services, to assemble and see their own brand of personal computers. They buy component parts on the open market from a variety...
  4. E

    Solve a Integer equation system

    Find smallest d=2a^2=3b^3+2=5c^5+3 where a,b,c are integers. This problem is very hard. Either no solution or d is toooooooo big
  5. T

    Understanding Trigonometric Functions for Negative Integers and Rational Numbers

    I know that \sin^2 b= (\sin b)^2 and in general \sin^n b=(\sin b)^n if n is a positive integer . What if n is a negative integer , would it be \sin^{-1}b=(\sin b)^{-1}=\frac{1}{\sin b} I don't think this is right , because properties of indices only works for numbers and NOT function...
  6. F

    Proving Prime Divisor of Composite Integer ≤ √n

    Homework Statement I need to prove that a composite integer n>1 has a prime divisor p with p<=sqrt(n). Homework Equations The Attempt at a Solution Im not sure how to do this, any help getting started would be great thanks.
  7. P

    For each positive integer n, let T(n) be the number of triangles with

    Homework Statement For each positive integer n, let T(n) be the number of triangles with integer side lengths, positive area, and perimeter n. For example, T(6) = 1 since the only such triangle with a perimeter of 6 has side lengths 2, 2 and 2. (a) Determine the values of T(10), T(11) and...
  8. M

    Prove two squares and a cube equal an integer

    Homework Statement Disprove or prove the statement that every positive integer is the sum of at most two squares and a cube of non-negative integers.2. The attempt at a solution I'll call the numbers that can be squares a and b. C will be the cube. The easiest way to disprove something is to...
  9. M

    Period of superposed cyclic integer rows

    Take two rows of respective length m and n: a1, a2, a3,..., am and b1, b2, b3, ..., bn. Then produce as follows the generated array Gai to contain these elements: a1, a1+a2, a1+a2+a3, ..., a1+..+am, a1+..+am+a1, a1+..+am+a1+a2, ... Alike produce the generated array Gbj to contain...
  10. S

    Prove that every integer n>= 14 is a sum of 3's and/ or 8's.

    Homework Statement Prove that every integer n>= 14 is a sum of 3's and/ or 8's. Homework Equations Induction Hypothesis The Attempt at a Solution Base Case: P(0): Suppose n= 14, and k is an integer representing number of times 3 or 8 is added: 14= 3k; k=14/3 ( this shows...
  11. K

    Between two powers of an integer there is a power of two

    Hi Everyone, I am reading up on information theory, and every resource I have found on the topic which derives the form of entropy uses the following inequality as part of the proof. Let n be a fixed positive integer greater than 1. If r is an arbitrary positive integer, then the number...
  12. sujoykroy

    Square of Integer: A Different Perspective

    Square of integer is quite easy, childlike stuff. But there is no harm in seeing a known thing in different lights. Experimenting on any thing is always fun, at least initially. So, while reading some stuff on semiconductor physics I came to think about viewing square of integer in different...
  13. F

    Finding Chen's Paper: "On the Representation of a Large Even Integer

    Hi guys, Not actually a mathematics question as such (sorry) but does anyone know where i can get my hands on a copy of Chen's paper "On the representation of a large even integer as the sum of a prime and the product of at most two primes". For the life of me all i can find is references to it...
  14. Z

    Mathematica Mathematica: Matrix product Modulus an integer

    I need to multiply 2 matrix in Mathematica but modulus an Integer. The "Modulus->n" option cannot be used with "Dot" function. You can use Modulus-> n with "Inverse" or even "Det" but not with "Dot". It is something strange. How should I do it, then? Any idea? Thank you.
  15. H

    Smallest N Value for 4-Digit Consecutive Integers Divisible by 2010^2

    Homework Statement The product of N 4-digit consecutive integers is divisible by 2010^2. What is the smallest N value? Multiple choice answers range from 4 to 12. Homework Equations N/A The Attempt at a Solution I tried multiplying the smallest combo possible 1000x1001x1002x1003...
  16. T

    Growth rate of integer power sum

    I need to show that \sum_{i=0}^n i^k=\Theta(n^{k+1}) Or equivalently \lim_{n\to\infty}\frac{\sum_{i=0}^n i^k}{n^{k+1}}=CI simply don't know what to do with the sum here. I know that I can rewrite or expand it, but that doesn't seem to help me. Any suggestions? Thank you!
  17. K

    Greatest Integer Function help

    Greatest Integer Function.. help! Homework Statement Well this is the problem a plomer charges 80 bucks ones he arrives at your home and charges and extra 25 per hour.. Give the equation,, Kind of i don't get it is for extra credit but still i don't like it when i don't know how to do it...
  18. P

    Half-Integer Spin: Explaining Discontinuity's Lack of Impact

    Lets asume that electron is in state: \left[ \begin{array}{c} \psi(\vec{r})\\ \phi(\vec{r}) \end{array} \right] It's a vector because electron has two spin components (up and down). If we rotate our labolatory by the angle 360^0 we got: \left[ \begin{array}{c}...
  19. P

    Angular momentum - integer or half-integer

    Let J be total angular momentum, L - orbital angular momentum and S - intristic momentum (spin). Squares of these operators have appropriate eigenvalues j(j+1), l(l+1), s(s+1). Which of these numbers j,l,s should be integer. I know that spin can have half-integer values. But probably orbital or...
  20. K

    Help deriving integer sequence formula

    Hi I'm playing around with partitions and have come up with an integer sequence representing the maximum number of partitions of various "widths" that display the following properties: - min values in partition are equal - max values in partition are equal - partitions contain equal number of...
  21. LCKurtz

    1 is the largest positive integer

    Proof: N2 > N rules out all the others. :-p
  22. Y

    Bessel's equation of the second kind with integer order.

    This is the equation given for the Y. Y_{p}=\frac{J_{p}(x)cos(p\pi)-P_{-p}(x)}{sin(p\pi)} In many books, if p is an integer n, they just said Y_{n}=lim(p\rightarrow n) Y_{p} J_{p}(x)=\sum^{k=0}_{\infty}\frac{(-1)^{k}}{k!\Gamma(k+p+1)}(\frac{x}{2})^{2k+p} which give...
  23. M

    What is the proof that there is no greatest natural number?

    In several places I have come across what seems to be a standard proof by contradiction that there is no greatest natural number. As follows:- Assume there is a greatest natural number (+ve integer). Call it n. Add 1 to it to get n+1. n+1 is an integer greater than n. Therefore n cannot be...
  24. H

    Equivalance classes and integer addition

    Homework Statement Prove: If a and b are in N the [(1,1+a)] + [(1,1+b)] = [(1,1+a+b)] Homework Equations Definition: We define + over Z as follows: if [(a,b)] and [(c,d)] are any two equivalence classes, we define [(a,b)] + [(c,d)] = [(a+c,b+d)]. The Attempt at a Solution So...
  25. W

    Why the rank of an irreducible tensor must be an integer?

    why not half-integer? according to the definition, such as [J_z,T^k_q]=q T^k_q it is quite possible that k can be a half-integer.
  26. K

    Limit of a Greatest Integer Function using Squeeze help

    my midterm is in 4 hours and this actually the only thing i need help with. Homework Statement prove using squeeze theorem that lim(x-> +inf) (x^2 - [[x^2]])/x = 0 Homework Equations g(x)<=f(x)<=h(x) [squeeze theorem] The Attempt at a Solution on the assignment i didn't know we...
  27. C

    Show by induction that a given polynomial is an integer

    Homework Statement Show with mathematical induction that \frac{n^5}{5} + \frac{n^4}{2} + \frac{n^3}{3} - \frac{n}{30} \in {Z} for all n\ge 1. Homework Equations Probably. The Attempt at a Solution Inductive statement: Q(n): \frac{n^5}{5} + \frac{n^4}{2} + \frac{n^3}{3} -...
  28. P

    Linear algebra - side of a cube is an integer?

    Homework Statement A cube of sides a*a*a is in 3 dimensional space. All eight of its corners have integer coordinates. Prove that a is an integer. Homework Equations - The Attempt at a Solution First, I considered three corners of the cube p, q and r, with these, two vectors...
  29. C

    Suppose A^k=0 for some integer k is greater than or equal to 1

    I was wondering if anyone could give me some hints on this Suppose A^k=0 for some integer k is greater than or equal to 1. Prove that A is not invertible.
  30. S

    Where n is an odd positive integer

    I'm studying for a test. In doing one of the old tests and it had a question that I couldn't do. Let T: Rn \rightarrow Rn be an operator on Rn, where n is an odd positive integer. How do I prove T has at least one eigenvector in Rn
  31. S

    Proving a Sinh(x) expression to be an integer

    Prove or disprove: sinh2(ln(\sqrt{}2+\sqrt{}3)) is an integerObviously, I used my calc to figure out that the answer is 2. Proving it w/o a calc is hard though. The Attempt at a Solution I've tried rewriting sinh2 as (1/4)(e2x+e-2x-2) and after all the substitutions and log rules I get...
  32. E

    Is Tk Always Positive If T is a Positive Operator in Linear Algebra?

    positive operator proof Homework Statement Prove that if T ∈ L(V) is positive, then so is Tk for every positive integer k. Homework Equations The Attempt at a Solution Let v=b1v1+...+bnvn. Now since T is positive, T has a positive square root. T=S^2. <S^2v, v>=<S^2v1...
  33. P

    Number Theory integer roots Problem

    I've been stuck on this for a while now, and I was wondering if anyone could help me out. The problem is: If ax^{2}+bx+c=0, prove that all integer roots divide b I'm fairly new to number theory, but this is the one problem that's been really tough for me. If someone could even give me...
  34. P

    Why do physical laws always feature integer indices?

    This may be a stupid question or have a pretty obvious answer, but I can't seem to find one so I'll just go ahead and post :) I was looking at some empirical data for relationships defining (abstracted) values for ionization and recomination coefficients in gases as a function of electric...
  35. M

    Can You Help with These Integer Algebra Homework Questions?

    First Question: Let Nn be the integer whose decimal expansion consists of n consecutive ones. For example, N2=11 and N7=1,111,111. Show that Nn|Nm iff n|m. Second Question: If (a,c)=1, prove that (a,bc)=(a,b). On the second question I can see that it is true because a and c are...
  36. A

    Probability - Cominations and Integer Valued Vectors

    This problem comes from Sheldon Ross's book "A First Course in Probability (6th ed)." There are 5 hotels in a certain town. If 3 people check into hotels in a day, what is the probability that they each check into a different hotel? Attempt at a solution: There are 5C3 = 10 different...
  37. S

    Proving -x = x with Positive Even Integer n in R

    Homework Statement Let R be a ring and suppose there exists a positive even integer n such that x^n = x for every x in R. Show that -x = x for every x in R. Homework Equations The Attempt at a Solution I solved the case where n = 2. Let x be in R. (x+x)^2= x+x = 2x...
  38. K

    How to Calculate a Double Integer with a Function in a Given Area?

    This was a problem on a final test I took this april in Reykjavík University and I whould be greatful if you could help me with it. Homework Statement Let f(x,y)=2x*cos(y^4) be a function and let D be area in R^2 defined by 0≤x≤1 and x^(2/3)≤y≤1. Calculate the double integer: ∫∫f(x,y)dA...
  39. L

    Azimuthal Wavefunctions: Showing a constant must be an integer

    Homework Statement In spherical polars, the azimuthal part of the wavefunction of a particle is psi(phi) = 1/sqrt[2.pi] . exp[i.m.phi] where phi is the azimuthal angle. Show m must be an integer.Homework Equations I know you are supposed to have a good go at solving the problem first, but...
  40. R

    Need help with unique integer partitions?

    This is causing me a bigger headache than I anticipated. Basically, given an integer N and a number M, I need a list of all the possible integer partitions of N into M parts such that each part is strictly positive and each part is UNIQUE. I don't want repetitions. Just unique ones. So for...
  41. A

    Assigning String and Integer to Variable

    Hi everyone, I'm trying to assign a string in combination with an integer to a variable v. The string-part is fix, the integers comes from another variable n. For Example: n:=10; Print("Sym_",n); The output of "Print(...);" (that is "Sym_10") should be assigned to another variable...
  42. Y

    Proof - If the square of an integer is even,. .

    Hi, I have no idea on how to start to do this question. If the square of an integer is even,then the integer itself is even I try to check some books but i can't get any similar examples.I wonder if I can directly prove the n=2k, n^2 = 2(2k^2). Thanks!
  43. G

    Finding integer solutions logically

    How would I find values for A and B such that AB-A-B=1673 Where A and B are integers? I know the answer (28 and 63), but I want to know how to arrive at that answer without any guessing, or at least with a minimum amount of guessing. Are there any other solutions? I just made this...
  44. B

    Integer tuples with equal L1 and L2 norms

    Let x and y be n-tuples of non-negative integers. Furthermore, sum x_i = sum y_i and, sum x_i^2 = sum y_i^2 Is it true that x must be a permutation of y? Cheers!
  45. K

    Seven digit base eight positive integer puzzle

    N is a seven digit base-8 positive integer having the form ABCDEFG that uses each of the nonzero base-8 digits 1 to 7 exactly once, and satisfies these conditions: (i) ABCDEFG is divisible by 7. (ii) ABCDEF is divisible by 6. (iii) ABCDE divisible by 5. (iv) ABCD is divisible by 4. (v)...
  46. F

    Reversing an Integer String Using Linked List

    Can someone help me on how exactly to do this? I'm trying to read an integer string and each "digit" in the string is put at the front of the linked list (i.e. reverse order). When I print it out I want it to reverse again. I know I'm not implementing it right because when I run it the program...
  47. C

    Finding the Smallest Integer Not in Floating Point Definition

    This should be an easy one, but my PC is bugging me! Based on the floating point definition: F = \pm( \stackrel{m}{\overline{B^{t}}})B^{e} Where B is the base (usually 2), m is the mantissa and varies from 1 \leq B^{t} - 1. e is the exponent (1024 for double and 128 for single...
  48. J

    Integer Representation Through Multiplication of Integers

    Hello, Can an integer always be represented through the multiplication of two or more integers? (Are all integers divisible by some set of 2 or more integers (- or +)?) For example, 8 is can be represented by 1 x 8, 2 x 4 and 2 x 2 x 2. But what about 257 or even - integers? I'm trying...
  49. S

    Integer Power Sum for p = 0, 1, 2,

    The other day I was thinking about the integer power sum and the general solution for each value of p. I came up with a method that will allow me to calculate the general solution. I thought that I may have stumbled upon something novel, because I couldn't find any reference to this method...
  50. W

    Prove that 3n^2 - 1 can't be a square of a integer n

    Well, the problem statement is in the title: Given that n is an integer, show that 3n2 - 1 can't be the square of an integer. Currently, I don't have any idea at all where to start. Method is probably to assume opposite and show that this leads to a contradiction. Any hint as to where to...
Back
Top