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.

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  1. S

    MHB Can the No $4\sqrt{4-2\sqrt {3}}+\sqrt{97-56\sqrt 3}$ be an Integer?

    Can the No :$4\sqrt{4-2\sqrt {3}}+\sqrt{97-56\sqrt 3}$ be an iteger ,if yes prove it if no then prove it again
  2. anemone

    MHB Find the sum of all values of positive integer a

    For a pair of positive integers $(a,\,b)$, $Q(a,\,b)$ is defined by $Q(a,\,b)=\dfrac{a^2b+2ab^2-5}{ab+1}$. Let $(a_1,\,b_1),\,(a_2,\,b_2),\,\cdots, (a_n,\,b_n)$ be all pairs of positive integers such that $Q(a,\,b)$ is an integer. Calculate $\displaystyle \sum_{i=1}^n a_i$,
  3. SSequence

    I Integer Complexity: Exploring an Interesting Notion

    Not a question as such, but an interesting notion that I came upon (maybe some other people would find it interesting too). It seems to have been introduced in 1950's and seems a good amount of work has been done on it. For example: 12=(1+1+1)*(1+1+1+1) So the complexity of 12 is 7 since it can...
  4. hackedagainanda

    Proving that a square of an odd integer is also odd

    Prove that for any arbitrary odd x, that x^2 is also odd. By definition an odd number is an integer that can be written in the form of 2k + 1 for some integer k. This means that x = 2k + 1 where k is an integer So let x^2 = (2k + 1)^2 we then get 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1, This is where...
  5. anemone

    MHB Find Integer Values for 1999 Distinct Real Solutions

    Find all the integer values of $m$ for which the equation $\left\lfloor \dfrac{m^2x-13}{1999}\right\rfloor=\dfrac{x-12}{2000}$ has 1999 distinct real solutions.
  6. anemone

    MHB Integer Solutions for $4x+y+4\sqrt{xy}-28\sqrt{x}-14\sqrt{y}+48=0$

    Find all integer solutions to the equation $4x+y+4\sqrt{xy}-28\sqrt{x}-14\sqrt{y}+48=0$.
  7. anemone

    MHB Polynomial with integer coefficients

    Let $a,\,b,\,c$ be three distinct integers and $P$ be a polynomial with integer coefficients. Show that in this case the conditions $P(a)=b,\,P(b)=c,\,P(c)=a$ cannot be satisfied simultaneously.
  8. S

    Prove that if a is even & b is any positive integer, then ab is even

    Proof: Let a be a even positive integer of the form a=2m & b of the form b=2n (This is where b is a even positive integer) ab = 2m*2n = 2(mn) = Let k = mn = 2k Therefore, ab is even. Let a be a even positive integer a=2m & b be a odd positive integer b = 2n+1 ab = (2m)*(2n+1)...
  9. John Greger

    I Show that an expression approaches an integer

    I came across a rather strange thing in an introductory class I still don't understand. There was a statement that $$lim_n (2+ \sqrt(2))^n $$ is an integer. I recalled that I never understood this and just recently tried to take the limit but just get that the expression diverge? Which I think...
  10. anemone

    MHB Integer solutions of system of equations

    Find all integer solutions of the system of equations $x+y+z=3$ and $x^3+y^3+z^3=3$.
  11. K

    B A given integer can be written in how many different ways?

    I had learned how to find this out in the past, but forgot now. Precisely, I'm trying to find in how many different ways I can express the number 24 as a sum of two integers ranging from 1 to 24. For example, 24 = 24 = 23 + 1 = 12 + 12...
  12. M

    MHB For which n is the term an integer & Calculate the equivalence

    Hey! 😊 Question 1: We consider $\frac{2n-1}{n+7}$. For which $n$ is this term an integer? I have done the following: We set $n+7=m \Rightarrow n=m-7$. Then we get $$\frac{2n-1}{n+7}=\frac{2(m-7)-1}{(m-7)+7}=\frac{2m-15}{m}$$ So $m$ has to be a divisor of $15$, i.e. $m\in \{1,3,5,15\}$...
  13. anemone

    MHB Largest Even Integer: Impossible Sum of Two Odd Composites

    Find the largest even integer which cannot be written as the sum of two odd composite numbers.
  14. anemone

    MHB Solving Integer Equations with $a$ and $b$

    Find all positive integers $a$ and $b$ such that $\dfrac{a^2+b}{b^2-a}$ and $\dfrac{b^2+a}{a^2-b}$ are both integers.
  15. CricK0es

    Minimum number of numbers to express every integer below N as a sum

    I have found code to find simply the minimum numbers needed, but I need to do it without repetition given the nature of an electric circuit. I hope that is a sufficient enough explanation of the problem. Despite being an engineering project this aspect is more mathematical.
  16. anemone

    MHB Find Integer Solutions for (A+3B)(5B+7C)(9C+11A)=1357911

    Find all integer solutions (if any) for the equation $(A+3B)(5B+7C)(9C+11A)=1357911$.
  17. C

    MHB Square Number Pairs from 1-50: Counting Rules

    Two integers will be taken from 1 to 50, where at least one of them should be a square number and sum of them should also be a square number. How many different pair like this can be found? Will I count (9,16) and (16,9) as one ?
  18. arcTomato

    Comp Sci Power spectrum when the wave number is not an integer

    Hi all. I made a program of DFT, so I made the power spectrum of a sin wave. This is the sin wave I used. All data number ##N=100## and the frequency of sine wave is 4.5Hz. And the power spectrum is this. The wave number is not integer so the spectrum has the side lobe. But I think this is...
  19. Y

    MHB Limit of integer part function using Sandwich rule

    Hello everyone, I want to calculate the following limits: \[\lim_{x\rightarrow \infty }\frac{[x\cdot a]}{x}\] using the sandwich rule, where [xa] is the integer part function defined here: Integer Part -- from Wolfram MathWorld I am not sure how to approach this. Any assistance will be...
  20. D

    B Is every integer derived from 1?

    find me an integer that isn't divisible by 1.
  21. bagasme

    B Solving a Linear Programming Problem which Requires Integer Solutions

    Hello, In grade 11 of high school, I encountered this linear programming problem on my textbook: The "alternative solution" described in the textbook as follows: Let: - ##x## : amount of plant A - ##y## : amount of plant S - ##L## : garden area - ##L_x## : area of garden for one plant A -...
  22. tensaiyan

    Integral of greatest integer function and its graph

  23. F

    Making a calculator program that can calculate integer addition / subtraction

    im trying to complete mips program code about a calculator program that can calculate integer addition / subtraction written using the MIPS assembler. im having hard times to debug this. The input is given to the array of Formula char (base address $ s0) in the form of a formula. The null...
  24. N

    MHB Determine if (27/4)/(6.75) is a whole number, natural number, integer, rational or irrational.

    Let Z = set of real numbers Determine if (27/4)/(6.75) is a whole number, natural number, integer, rational or irrational. I will divide as step 1. 27/4 = 6.75 So, 6.75 divided by 6.75 = 1. Step 2, define 1. The number 1 is whole or natural. It is also an integer and definitely a rational...
  25. Leo Consoli

    Find the number of integer solutions of a second degree polynomial equation

    x^2 - x -3 + 2c = 2x(ax+b) x^2 -2ax^2 - 2bx - x - 3 + 2c = 0 x^2(1-2a) -x(1+2b) -3 + 2c =0 Using girard r1+r2 = (1+ 2b)/(1-2a) r1xr2 = (-3 +2c)/(1-2a) After this I am stuck. Thank you.
  26. lfdahl

    MHB What is the positive integer $n$ with a special property?

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  27. B

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  28. lfdahl

    MHB A positive integer divisible by 2019 the sum of whose decimal digits is 2019.

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  29. C

    MHB Can $[a] \times [b]$ be an element of ${\Z / n\Z}^{\times}$?

    Dear Everybody, I don't know where to begin. So Here is the problem: $\newcommand{\Z}{\mathbb{Z}}$ Prove that if $[a]$ and $[b]$ are in ${\Z / n\Z}^{\times}$, then $[a] \times [b]$ is in ${\Z / n\Z}^{\times}$. Thanks, Cbarker1
  30. Ventrella

    I Mutually disjoint sets of all integer powers?

    I identified what appears to be a partitioning of all integers > 1 into mutually disjoint sets. Each set consists of an infinite series of integers that are all the powers of what I am calling a "root" r (r is an integer that has no integer roots of its own, meaning: there is no number x^n that...
  31. Mr Davis 97

    Showing when quadratic integer rings are isomorphic

    Homework Statement Let ##a,b## be squarefree integers and set ##R = \mathbb{Z}[\sqrt{a}]## and ##S = \mathbb{Z}[\sqrt{b}]##. Prove that a) There is an isomorphism of abelian groups ##(R,+) \cong (S,+)##. b) There is an isomorphism of rings ##R\cong S## if and only if ##a=b##. Homework...
  32. YoungPhysicist

    Big integer arithmetic functions

    NOTE:This is not a homework question! This is just a topic that I like very much,but don’t have the programming ability to do many of them.That’s why I post this thread. C++ is a language without built-in big integer calculation functions,so building ones that can do such job is a great way to...
  33. H

    MHB Why is ln(k) a Complex Number When k is a Positive Integer?

    Why ln(k) when k is a possitive integer, ln(k) is a complex number?
  34. H

    B Is there a way to find the integer representation of a real number?

    Is there a way to find the integer of a real number? Of course without using the [x] function. What I am looking for here is an algebraic formula.
  35. Aleoa

    Stuck on an Integer Programming problem

    Homework Statement I've tried hours and hours to model this problem but i didn't succeed. Can you help me ? We want to realize n projects in the next T years. For each project, a profitability index pi is known, which expresses the expected final gain (in Euro) and a cost profile ai =...
  36. R

    Is there a better way to find the number of integer solutions

    Homework Statement # of integer solutions of x1+x2+x3+x4 = 32 where x1,x2,x3>0 and 0<x4≤25 Homework EquationsThe Attempt at a Solution So in the case where x4 = 25 we have x1+x2+x3= 4 in the case where x4 = 24 we have x1+x2+x3 = 5 ... in the case where x4 = 1 we have x1+x2+x3 = 28so...
  37. karush

    MHB J1.1.6 Suppose a and b are integers that divide the integer c

    Suppose a and b are integers that divide the integer c If a and b are relatively prime, show that $ab / c$ Show by example that if a and b are not relatively prime, then ab need not divide c let $$a=3 \quad b=5 \quad c=15$$ then $$\frac{15}{3\cdot 5}=1$$ let $$a=4 \quad b=6 \quad c=15$$ then...
  38. A

    I Open source software for integer programming

    I don't usually need help in locating software, but I'm having a heck of a time tracking down a good open-source bit of software which solves integer programming problems using arbitrary precision! If I don't find one soon, I'll need to write it myself. Which I don't mind, but it's silly to...
  39. Mutatis

    How to integrate this function?

    Homework Statement I've got to integrate the following $$ \int dx =\int \frac {d\phi} {\phi \sqrt {1 - \phi²}}. $$ Homework Equations [/B] I already know the answer but not how to get it. The answer that I got from solution is ## x = \operatorname {arcsech}{\phi} ##. The Attempt at a...
  40. opus

    Analyzing the graphs of Greatest Integer Functions

    Homework Statement Consider ##u\left(x\right)=2\left[\frac{-x}{4}\right]## (a) Find the length of the individual line segments of the function, (b) Find the positive vertical separation between line segments. Homework Equations The output of Greatest Integer Functions are always integers. The...
  41. Ventrella

    I Would the opposite of a perfect power be called a "root"?

    I would like to know if there is an official name for the class of integers that are (not) perfect powers. A perfect power is a number that can be expressed as xn, where x and n are both integers > 1. I have been calling these integers "roots" - since they do not have any integer roots of their...
  42. Euge

    MHB POTW: Evaluate Integral for Positive Integer n

    Here is this week's POTW: ----- If $n$ is a positive integer, evaluate $$\int_{0}^\infty \frac{dx}{1 + x^n}$$ ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...
  43. kuruman

    A Exploring Integer Solutions to a^3+b^3+c^3=d^3

    I am wondering about the integer solutions to ##a^3+b^3+c^3=d^3~##. By trial and error I stumbled upon ##3^3+4^3+5^3=6^3##. I find this equation remarkable in that not only the four integers are consecutive, but also because the three integers on the left form the well known Pythagorean...
  44. D

    I Proof of Countability of ℚ: Bijection from A to ℕ

    I know there are many proofs of this I can google, but I am interested in a particular one my book proposed. Also, by countable, I mean that there is a bijection from A to ℕ (*), since this is the definition my book decided to stick to. The reasoning is as follows: ℤ is countable, and so iz ℤxℤ...
  45. lfdahl

    MHB Find Integer $n$: Reversed Digits = 999 - Prime < 6000

    Determine the integer $n$ with the properties: a). $n$ is a prime less than $6000$, b). the number formed by the last two digits of $n$ is $< 10$, and c). if the decimal digits of $n$ are reversed to obtain $N$, then $N − n = 999$.
  46. nomadreid

    I For every finite integer sequence there's a pattern- source?

    I have in the back of my head the statement that for every finite sequence of positive integers there exists a pattern (i.e., a generating formula). While this sounds reasonable, I am not sure whether it is true, and if it is true, what the source for this statement is, and how the correct...
  47. lfdahl

    MHB Show, that (7+5√2)^(1/3)+(7−5√2)^(1/3) is an integer

    Let $a = \sqrt[3]{7+5\sqrt{2}} + \sqrt[3]{7-5\sqrt{2}}$. Show (without the use of a calculator), that $a$ is an integer.
  48. kaliprasad

    MHB Show that for integer a,b,c the product abc(a^3−b^3)(b^3−c^3)(c^3−a^3) is divisible by 7

    Show that for integer a,b,c the product $abc(a^3-b^3)(b^3-c^3)(c^3-a^3)$ is divisible by 7
  49. evinda

    MHB Test whether the integer is a prime

    Hello! (Wave) We want to find an efficient algorithm that checks whether an odd number is a prime or not. In order to obtain such an algorithm, one tests the congruence $(X+a)^n \equiv X^n+a$ not "absolutely" in $\mathbb{Z}_n[X]$, but modulo a polynomial $X^r-1$, where $r$ have to be chosen in...
  50. F

    Integer programming model (alternating constraints)

    Homework Statement Formulate as a mixed integer programming problem but do not solve. Maximize ##x_1 + x_2## subject to ##2x_1 + 3x_2 \le 12## or {##3x_1 + 4x_2 \le 24## and ##-x_1 + x_2 \ge 1##} ##x_1, x_2 \ge 0## Homework EquationsThe Attempt at a Solution if the first constraint is met, we...
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