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. L

    Prove that an algebraic integer is a unit in Z(a)

    I'm completely stuck here, can anyone help me?
  2. Nugso

    Fortran Fortran - Having to add integer command

    Hello everyone. Today I learned how to write a program which calculates and prints integer squares roots from 1 to 1000. Here is what I wrote: program example real::i do i=1,1000 if(sqrt(i)**2==i) then print*,i end if end do end program example After compling this, 25, 49 etcs can't...
  3. Albert1

    MHB Find the closest integer to S

    please find the closest integer to S $S=\sum_{a=10}^{2011}\sqrt{1+\dfrac{a^2+(a+1)^2}{(a(a+1))^2}}$
  4. jaumzaum

    Why the electron orbit is a integer multiple of De Broglie wavelength?

    I cannot understand why the electron orbit should be an integer multiple of De Broglie wavelength. Why should the wave path "fit" the electron orbit for it to be stationary?
  5. J

    Integer Number Theory - n = p + a^2

    Homework Statement Prove or disprove: If n is a positive integer, then n=p+a^2 where a\in\mathbb{Z} p is prime or p=1 Note that the interpretation of "prime" used here includes negative primes. So, an exhaustive list of possibilities for p is p=1,\pm2,\pm3,\pm5,\pm7,\pm11,\cdots...
  6. S

    Integer Coeffieicent of Degree

    Hello all, Can anyone explain what "Integer Coefficient of Degree n" where n is some positive integer (e.g 6) of a polynomial refers to? I know what integers and coefficients are but I don't know what the above refers to. For example if you had something like this: Suppose some...
  7. V

    If n is an integer, and 3n+2 is even, prove that n is also even

    I am coming across hiccups in my proof process. I am given this problem - Prove: if n is an integer and 3n + 2 is even that n is also even. I have to apply a contrapositive proof to this problem. The form is then \negq therefore \negp .The problem becomes - if n is odd, prove that 3n+2 is even...
  8. C

    Proof about an integer being a perfect square.

    Homework Statement m and n are positive integers with m,n≥2 where m^2=kn^2 The Attempt at a Solution we know that all prime factors of m have an even amount , their are no prime factors that are repeated an odd number of times. The same goes for n. if k is not a perfect square...
  9. Y

    New idea about Integer Factorization

    The logic that odd composite with least difference will be factored easily and large difference would factored hardly is wrong. B'coz whatever be the difference between the factors their exist Best Fermat Factors to make the Fermat factorization easier. Please follow the link to know more...
  10. W

    Comp Sci Changing a double value to an integer value in c++

    Homework Statement I'm making a program which will convert mph to minutes and seconds per mile. For example my program will take 6.5 mph and convert it to 9 minutes and 13.8 seconds per mile. But being new to programming I don't know how to convert a double value to an int value (I believe its...
  11. S

    Prove log_2(m) is not an integer.

    Homework Statement Let m be a positive integer that has an odd divisor greater than 1. Prove (by contradiction) that \log_2{m} is not an integer. Homework Equations The Attempt at a Solution I'm not sure where to begin here. The book I'm working with provides a similar example, but it...
  12. K

    Differentiable Greatest Integer Function

    Homework Statement k(x)=x2*[1/x] for 0<x≤1 k(x)=0 for x=0 Find where k(x) is differentiable and find the derivative Homework Equations The Attempt at a Solution I know that it is differentiable for all ℝ\Z on (0,1], but I am unsure how to find the derivative for this problem.
  13. R

    Relatively prime integer proof

    Homework Statement Let p be a prime and let n≥2 be an integer. Prove that p1/n is irrational. Homework Equations We know that for integers a>1 and b such that gcd(a,b)=1, a does not divide b^n for any n≥ 1. The Attempt at a Solution To prove irrationality, assume p^(1/n)=a/b for...
  14. T

    Optimizing Knights on 8x8 Chess Board with Integer Programming

    Homework Statement On a 8x8 chess board format an Integer program to optimize the amount of knights required such that every square is covered by at least one knight. Homework Equations I know of a similar problem where we use duality for the placing 5 queens such that the maximum...
  15. J

    Greatest integer function limit problem. Proving whether the limit exists

    Homework Statement Prove that the limit exists. lim 5 - 1/2[[2x]] x-->1 Show your solution.. Homework Equations The Attempt at a Solution Tried getting the limit from the right and left.. not sure if what I've done is right though but this is what I got. lim 5 -...
  16. I

    Formulating a Mixed Integer Programming Problem

    Homework Statement Not sure if this type of math goes in this section, but I have a quick question regarding a MIP problem. It's simple to grasp, but I'm not sure whether or not I am formulating this problem correctly. A city needs to hire workers to clean the snow. The city is divided into j...
  17. D

    Prove that the square of any integer, when divided by 3. only by odd and even.

    Homework Statement I know you could prove this by stating every integer is either 3m, 3m+1 or 3m+2. However I am trying to prove this just using either even numbers or odd numbers. so for example, when I try: (2x+1)^2 = 4x^2 + 4x + 1 - expand = 3x^2 + x^2 + 3x + x + 1 - group like...
  18. J

    Why is my C integer printing as a non-printable character?

    Ok so I was trying something in C when i found something i don't understand:- code is:- int c = 1; printf("%c\n",c); (not writing the include and return 0 stuff) so when i run this it gives an output of a 2x2 box like this:- 0 0 0 1 whats happening. When i put the 1 in commas like this int c=...
  19. J

    MHB Integer Solutions for $4^a+4a^2+4=b^2$

    Find no. of Integer value of $\left(a,b\right)$ which satisfy $4^a+4a^2+4 = b^2$
  20. T

    Derivative of f(x) Vanishing Between a & b for Positive Integer m & n

    Question: Show that the derivative of f(x) = (x-a)m (x-b)n vanishes at some point between a and b if m and n are positive integers. My attempt: f(x) = (x-a)m (x-b)n f '(x) = m(x-a)m-1 (x-b)n + n(x-a)m (x-b)n-1 f '(x) = [(x-a)n-1 (x-b)n-1 ] [(m)(x-b) +(n)(x-a)] And this is as far as I got.
  21. L

    Integer part of the output of a division

    hello there! i need to get the integer part of the output of a division in MATLab i.e. 23/5=4 (and the remainder is 3) what function should i use?
  22. A

    MHB Greatest integer function with linear function inside

    What is the best way to redefine Greatest integer function as a piecewise function for example f(x) = [ 2x - 3 ] , -2<= x <= 1
  23. 7

    Mathematica Printing only integer values for some equation using MATHEMATICA.

    Hello every one, I need help on some mathematica run program. Suppose the following: 1) 2=<t=<1000 2) r=3+2Sqrt[2] 3) k=(r^t+r^(-t)-20)/4 4) n=2k+4 5) x=n/2-(r^t-r^(-t))/2Sqrt[2] I need to run the program For[t=2,t=<1000,t++ and only print if it found any...
  24. C

    Binomial Distribution with non integer succes

    I am doing a problem where I am to determine the probability that the number of students wanting a new book is within two standard deviations of the mean. μ +- 2δ comes out with a non integer number, in which I have to use to find probability. The equation to find probability uses the factorial...
  25. H

    Proof must be integer or irrational?

    Homework Statement Suppose a, b ε Z. Prove that any solution to the equation x^3 +ax+b = 0 must either be an integer, or else be irrational. Homework Equations Not sure if this is right but x = m / n where m divides b and n divides 1 The Attempt at a Solution So far i think i...
  26. C

    Finding if an integer is odd through Riemann or some function?

    I'm trying to find if a number is odd or not, basically if X % 2 = 1. Can this be expressed through some function? Like the sum of 1 + 2 .. +n is n(n+1) /2 Or as a Riemann sum? I'm trying to add only the odd numbers from a random set of N integers to a sum.
  27. E

    Complex Integer z=6e2,5i: Explaining Real & Imaginary Parts

    z=6*e2,5i Can anyone explain me ? The imaginary part = 3,59 and real part = -4,81 I tried e(x) = cos x + i sin x, but it does not help me.
  28. A

    One Sided Limits of Greatest Integer Function

    Can somebody tell me how to find one sided limits for greatest integer function, say Lim x->n-, [[x/2]]. that is limit x approaches n from left of [[x/2]] where [[]] represent greatest integer function and n is any integer. I know how to find one sided limits for simple [[x]].
  29. M

    Laurent series for pole to non integer power

    Hey all, I am doing a Schwarz-Christoffel transformation and I am trying to calculate the integral analytically using the residue theorem. My integral is the following: \int^\zeta _{\zeta_0} (z+1)\frac{1}{(z+2.9)^{{b_1}/\pi}{(z-0.5)^{{b_2}/\pi}}}dz This has two poles at -2.9 and 0.5. b_1 and...
  30. C

    Number Theory. Argue Is not the square of an integer.

    Homework Statement Argue that (17^4)*(5^10)*(3^5) is not the square of an integer. Homework Equations N/A? The Attempt at a Solution Do I break these up, and show that each is not a square? I'm not sure if that would be correct, but sqrt(17^4)=289 * sqrt(5^10)=3125 *...
  31. veronica1999

    MHB Therefore, the integer X that satisfies the given inequality is 1.

    What is the integer X which satisfies 10^x < 1/2 X 3/4 X 5/6 X 7/8 X ... X 99/100 < 10^(x+1) ? Could I get some help? This contest allows the use of calculators but I still can't think of the right approach.
  32. D

    If w is an even integer, then w^2 - 1 is not a prime number.

    hello, I am trying to solve this problem: If w is an even integer, then w^2 - 1 is not a prime number. my current working. prove by contradiction If w is a even integer then w^2 -1 is a prime number. if w = 2x then w^{2} -1 = 4x^{2} -1 I am not sure where to go from here, maybe congruence...
  33. K

    Integer Quantum Hall Current

    Hi PF, Hoping somebody out there can help me to clear up what is probably a silly misunderstanding of the IQHE: Since the quantized Hall current can be expressed as a property of occupied bulk bands (Chern number) why do we say that the current is carried by the edge states?
  34. G

    Proving a and b are squares of an integer

    Homework Statement If a, b and c are positive integers with (a,b) = 1 and c2 = ab, show that each of a and b is the square of an integer. I've been staring at this problem for a while now and I've got not clues. I don't see the relation between a and b being relatively prime and the rest...
  35. C

    Help simple c program with loop, not running through each integer.

    Homework Statement I want to create a program that prints the sum of integers k for k(1-20) ie, k sum 1 1 2 3 3 6 4 10 5 . 6 . . . . . . 20 . in that format Homework Equations The Attempt at a Solution #include <stdio.h> int...
  36. A

    Find GCD of a and b: Express as Integer Combination

    RESOLVED Homework Statement Let a = 123, b = 321. Compute d = gcd(a,b) and express d as an integer combination of ra + sb.Homework Equations This is a question (3.1, page 70 of Michael Artin's Algebra). For those who do not have the book, this problem is relevant to the section on subgroups...
  37. A

    Why Does Bragg's Law Require Integers?

    Hi there, So in bragg's law 2d\sin \theta =n\lambda, n needs to be an integer. Can anyone explain why? I mean, what if the extra path 2d that the 'second beam' has is not dividable by a wavelength? Not sure if this is asked before but could not find it! Cheers, Adnan
  38. P

    Updating a JLabel with an integer array

    halfway through my code so its a bit of mess but this is simple what i need to do. I have a JLabel array of 6 labels. a random number is generated, its then split into its digits into a 6 element array of integers. now i need to update the Jlabels with the integer array of digits. my...
  39. J

    Computation Question in the Ring of Polynomial with Integer Coefficients

    I have a quick question. The problem reads: Prove that there is no integer m such that 3x2+4x + m is a factor of 6x4+50 in Z[x]. Now, Z[x] is not a field. So, the division algorithm for polynomials does not guarantee us a quotient and remainder. When I tried dividing 6x4+50 by 3x2+4x +...
  40. M

    Solving Integer Cubes & Squares: 7k & 3k?

    I am trying to solve how an integer is simultaneously that is simultaneously a square and a cube number must be either of the form 7k or 7k+1 and I am failing when i work (7k+2)^2, even (7k+3)^2....Can i interpret 7k+1 as 7k+(-1), i think i can't but then i fail in many steps! Also i know that...
  41. F

    What approach should be taken to solve these two questions?

    What are people's opinion on these two questions? The obvious answer for the first question (5) seems too simple, as it merely stems from the observation that the last number in the first two rows happen to be the average of the first three. I'm not sure if the real solution is this easy...
  42. A

    Frobenius Method - Roots differ by integer

    I'm reading up on some methods to solve differential equations. My textbook states the following: "y_{1} and y_{2} are linearly independent ... since \sigma_{1}-\sigma_2 is not an integer." Where y_{1} and y_{2} are the standard Frobenius series and \sigma_1 and \sigma_2 are the roots of...
  43. S

    Prove that [itex]a[/itex] is not an integer

    Prove that for any non-zero positive integers b,s a = \large \log_x \bigg(\dfrac{ ( -3 + x^{2b} ) \pm \sqrt{(3-x^{2b})^2-4x^{2b}(1-s^2)}}{2x^{2b}}\bigg)\notin \mathbb{Z}. The above expression comes from the result x^a = \dfrac{ ( -3 + x^{2b} ) \pm...
  44. J

    MHB What is the value of n for greatest integer function to equal 2012?

    Calculate Natural no. $n$ for which $\displaystyle [\frac{n}{1!}]+[\frac{n}{2!}]+[\frac{n}{3!}]+...+[\frac{n}{10!}] = 2012$ where $[x] = $ Greatest Integer function
  45. L

    Integer Cohomology of Real Infinite-Dimensional Grassmann Manifold

    I can't seem to find on the web a site that gives the Z cohomology of the infinite dimensional Grassmann manifold of real unoriented k planes in Euclidean space. I am interested in computing the Bockstein exact sequence for the coefficient sequence, 0 -> Z ->Z ->Z/2Z -> 0 to see which...
  46. A

    Prove no odd integer can be expressed as 4k+1 and 4j-1

    I am preparing for my Fall bridge to abstract math course and came across following problem: "Prove that there is no odd integer that can be expressed in the form 4j-1 and 4k+1 for integers j and k." If you let P(x) = "x is odd" Q(x) = (\existsj\inZ)(x=4j-1) R(x) =...
  47. T

    Proving Negative Divisors of an Integer

    Homework Statement Show that negative divisors of an integer, are just the negatives of the positive divisors. The Attempt at a Solution having an integer n and a negative divisor k, we get the positive divisor by multiplying (-1)k, thus each negative divisor of an integer, is the...
  48. K

    How to count the number of occurences of an integer in excel?

    Homework Statement I have a thousand random numbers generated between -100 and 100 and want to count the number of occurences of each integer. I know i can use the COUNTIF function, but for so many numbers it takes way too long. Is there a way to create a loop in excel that can do this or...
  49. shreyakmath

    New formula that can divide by zero to give an integer

    New formula that can divide by zero to give an integer! Does anybody here know about the Bhartiya New Rule of Fraction(BNRF)? It is capable of dividing by zero! In most elementary terms, we know that division is successive subtraction. For example 6 divided by 3 is 2 because 3 can be...
  50. C

    MHB Question to ponder about - - # of nonzero integer coefficients of a polynomial squared

    Suppose you look at polynomials, P(x), of degree n, with all nonzero integer coefficients and, in particular, a coefficient of 1 for the nth degree (leading) term. And look at those polynomials whose squares have the fewest number of nonzero integer coefficients possible. Examples...
Back
Top