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. anemone

    MHB What are the possible integers for which a given expression is an integer?

    Determine all possible integers $n$ for which $\dfrac{n^2+1}{\lfloor{\sqrt{n}}\rfloor^2+2}$ is an integer.
  2. anemone

    MHB Integers and Divisibility Challenge

    Prove that $\dfrac{378^3+392^3+1053^3}{2579}$ is an integer.
  3. karush

    MHB What are the factors of -48 that result in a positive sum?

    ok I don't don't know de jure on this so ... is it just plug and play?? find factors of -48 $-1(48)=-48$ $-2(24)=-48$ $-3(16)=-48$ $-4(12)=-48$ $-6(8)=-48$ check sums for positive number $-1+48=47$ $-2+24=22$ $-3+16=13$ $-4+12=8$ $-6+8=2$it looks like c. 5
  4. I

    Prove that the set of all even integers is denumerable

    Now, set of even integers is ## A = \{ \cdots, -4, -2, 0, 2, 4, \cdots \} ##. We need to prove that ## \mathbb{Z}^+ \thicksim A##. Which means that, we need to come up with a bijection from ##\mathbb{Z}^+## to ##A##. We know that ##\mathbb{Z}^+ = \{1,2,3,\cdots \} ##. I define the function ##f ...
  5. I

    A How can you generate a sine wave using integers only?

    I need to recursively generate a quadrature signal which fits exactly into a grid NxN, where N is a large power of two. After unsuccessful research, I decided to develop my own solution, starting from the waveguide-form oscillator taken from Pete Symons' book 'Digital wave generation, p. 100'...
  6. velikh

    B On the representation of integers?

    Let (a, b, c) be some arbitrary positive integers such that: (q2^0 + q2^1+ . . . + q2^x), (q2^0 + q2^1+ . . . + q2^y), (q2^0 + q2^1 + . . . + q2^z), where: q = (1, 2), (x, y, z) = (1, 2, 3, . . ., n). In the case if and only if q = 2, we accept the following notation : [(q-1)2^0...
  7. N

    MHB Is 7 an irrational number in the set of integers?

    Determine if the number 7 is a natural number, an integer, a rational or irrational number. I know that integers include positive and negative numbers and 0. Let Z = the set of integers Z = {. . . -2, -1, 0, 1, 2, . . .} I also know that any integer Z can be written as Z/1 = Z. I will...
  8. Ventrella

    A All complex integers of the same norm = associates?

    Are all complex integers that have the same norm associates of each other? I have seen definitions saying that an associate of a complex number is a multiple of that number with a unit. And I understand that the conjugate of a complex number is also an associate. But I am looking for a...
  9. Mr Davis 97

    Finding 5 Positive Integers with GCD Difference

    Homework Statement Do five positive integers exist such that the positive difference between any two is the greatest common divisor of those two numbers? Homework EquationsThe Attempt at a Solution I found four such numbers, ##\{6,8,9,12\}##. I did this in an ad hoc way though without any real...
  10. R

    Prove that the product of any three consecutive integers is

    Homework Statement Prove that the product of any three consecutive integers is divisible by 6. Homework EquationsThe Attempt at a Solution This doesn't seem true to me for any 3 consecutive ints. For example, let a_0 = 0 a_1 = 1 a_2 = 2 3 is not divisible by six. Assuming they meant a_x...
  11. V

    I Quantifiers with integers and rational numbers

    Give an example where a proposition with a quantifier is true if the quantifier ranges over the integers, but false if it ranges over rational numbers. I do not know where to go about when answering this, I know that an integer can be a rational number, for example 5 is an integer but can also...
  12. Ventrella

    I Iterating powers of complex integers along axes of symmetry

    I am exploring the behaviors of complex integers (Gaussian and Eisenstein integers). My understanding is that when a complex integer z with norm >1 is multiplied by itself repeatedly, it creates a series of perfect powers. For instance, the Gaussian integer 1+i generates the series 2i, -2+2i...
  13. L

    Show that for all integers congruent modulo 11

    Homework Statement Let ##a, b, c \in \mathbb{N \setminus \{0 \}}##. Show that for all ##n \in \mathbb{Z}## we have $$n^{11a + 21b + 31c} \equiv n^{a + b + c} \quad (mod \text{ } 11).$$ Homework EquationsThe Attempt at a Solution We have to show that ##11 | (n^{11a + 21b + 31c} - n^{a + b +...
  14. matqkks

    MHB Finding the Greatest Common Divisor of Two Integers

    How do computers evaluate the gcd of two integers?
  15. matqkks

    I How do computers evaluate the GCD of two integers?

    Do they use the Euclidean Algorithm?
  16. G

    I Can Quadratic Forms Map Integers to Integers?

    Alright, so this might be a stupid question, but nevertheless, I ask. I am to consider whether the quadratic form ## P(x,y) = a x + b y + d xy ## can map the integers onto the integers. So through a change of basis, I re-express this as ## P'(u,v) = Au^2 + Bv^2 ## for rational A and B...
  17. karush

    MHB 412.0.6 Find all integers n for which this statement is true, modulo n.

    for the equation $8\cdot8\cdot 8=4$. Find all integers $n$ for which this statement is true, modulo $n$. ok so $$8^3-(4)=508$$ 508/4=127 508/127=4 then 2^2\cdot 127 = 508 ok I'm sure this is not the proper process
  18. 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...
  19. Mr Davis 97

    I Enumerating integers n s.t. 36 | 48n

    This is a simple computational question. Let ##n \in [0, 36)##. What's the fastest way to list all ##n## s.t ##36## divides ##48n##?
  20. DuckAmuck

    A The Last Occurrence of any Greatest Prime Factor

    If you have 2 integers n and n+1, it is easy to show that they have no shared prime factors. For example: the prime factors of 9 are (3,3), and the prime factors of 10 are (2,5). Now if we consider 9 and 10 as a pair, we can collect all their prime factors (2,3,3,5) and find the maximum, which...
  21. Mr Davis 97

    Proving Induction for All Integers

    Homework Statement Let ##\phi : G \to H## be a homomorphism. Prove that ##\phi (x^n) = \phi (x)^n## for all ##n \in \mathbb{Z}## Homework EquationsThe Attempt at a Solution First, we note that ##\phi (x^0) = \phi(x)^0##. This is because ##1_G \cdot 1_G = 1_G \implies \phi (1_G 1_G) = \phi...
  22. Ventrella

    A Differences between Gaussian integers with norm 25

    I am exploring Gaussian integers in terms of roots, powers, primes, and composites. I understand that multiplying two integers with norm 5 result in an integer with norm 25. I get the impression that there are twelve unique integers with norm 25, and they come in two flavors: (1) Four of them...
  23. A

    Solve the Mystery: Three Non-Negative Integers & Perfect Powers of 2

    Question: There are three non-negative integers with the following property: If you multiply any two of the numbers and subtract the third number, the result is a perfect power of 2. Find these three numbers that satisfy this property. My attempt: I worked out that the three numbers must be...
  24. srfriggen

    Partition the integers under "anti-closure" of addition

    Homework Statement Can you partition the positive integers in such a way that if x, y are member of A, then x+y is not a member of A. x and y have to be distinct. That is, {1, 2, 3} cannot be in the same set, since 1+2 = 3, but 1 and 2 can be, since 1+1=2, but 1 and 1 are not distinct...
  25. lfdahl

    MHB Determine the number of integers for which the congruence is true

    Determine the number of integers $n \geq 2$ for which the congruence $x^{25} \equiv x$ $(mod \;\; n)$ is true for all integers $x$.
  26. castor28

    MHB Polynomial challenge: Show that not all the coefficients of f(x) are integers.

    $f(x)$ is a degree 10 polynomial such that $f(p)=q$, $f(q)=r$, $f(r)=p$, where $p$, $q$, $r$ are integers with $p<q<r$. Show that not all the coefficients of $f(x)$ are integers.
  27. M

    MHB Consecutive Even Integers

    Three times the smaller of two consecutive EVEN integers IS four less than twice the larger. What are the two integers? My set up: x and x + 2 are the two consecutive even integers. True? Here, x is the smaller integer and (x + 2) the bigger integer. True? The equation is 3x = 2(x + 2) - 4. Yes?
  28. C

    MHB How to show uniqueness in this statement for integers

    Dear Everyone, Directions: Decide whether the statement is a theorem. If it is a theorem, prove it. if not, give a counterexample. There exists a unique integer n such that $$n^2+2=3$$. Proof: Let n be the integer. $$n^2+2=3$$ $$n^2=1$$ $$n=\pm1$$ How show this is unique or not? Please...
  29. M

    MHB Three Consecutive Odd Integers

    Find three consecutive odd integers such that the square of the first plus the square of the third is 170. See picture for the set up. Is the set up correct?
  30. B

    Simple demonstration with real, rational and integers

    Homework Statement Let ##\alpha \in \mathbb{R}## and ##n \in \mathbb{N}##. Show that exists a number ##m \in \mathbb{Z}## such that ##\alpha - \frac {m}{n} \leq \frac{1}{2n}## (1).The Attempt at a Solution If I take ##\alpha= [\alpha] +(\alpha)## with ##[\alpha]=m## (=the integer part) and...
  31. B

    Proof that Algebraic Integers Form a Subring

    Homework Statement The set ##\Bbb{A}## of all the algebraic integers is a subring of ##\Bbb{C}## Homework EquationsThe Attempt at a Solution Here is an excerpt from my book: "Suppose ##\alpha## an ##\beta## are algebraic integers; let ##\alpha## be the root of a monic ##f(x) \in...
  32. G

    Help with this differential calculus

    <Moderator's note: Moved from a technical forum and therefore no template.> Hi everybody I've been trying to solve this problem all the afternoon but I haven't been able to do it, I've written what I think the answers are even though I don't know if they're correct, so I've come here in order...
  33. Albert1

    MHB Find positive integers x,y,z in 1/x+1/y=7/8(x,y∈N)

    $(1)$ $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{7}{8}(x,y\in N)$ $find$: $x,y$ $(2)$ $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{6}{12}(x,y\in N)$ $find$: $x,y$ $(3)$ $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac {1}{z}=\dfrac{6}{24}(x,y,z\in N)$ $find$: $x,y,z$
  34. evinda

    MHB Product of integers that are relatively prime to m

    Hello! (Wave)Let $b_1< b_2< \dots< b_{\phi(m)}$ be the integers between $1$ and $m$ that are relatively prime to $m$ (including 1), and let $B=b_1 b_2 b_3 \cdots b_{\phi(m)}$ be their product. I want to show that either $B \equiv 1 \pmod{m}$ or $B \equiv -1 \pmod{m}$ . Also I want to find a...
  35. Math Amateur

    MHB The Integers as an Ordered Integral Domain .... Bloch Theorem 1.4.6 ....

    I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Section 1.4: Entry 2: Axioms for the Integers ... In this section Bloch defines the integers as an ordered integral domain that satisfies the Well Ordering Principle ... rather than defining the...
  36. Math Amateur

    I Integers as an Ordered Integral Domain .... Bloch Th. 1.4.6

    I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Section 1.4: Entry 2: Axioms for the Integers ... In this section Bloch defines the integers as an ordered integral domain that satisfies the Well Ordering Principle ... rather than defining the...
  37. Math Amateur

    I The Set of Positive Integers - a Copy of the Natural Numbers

    I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.3.7 ... Theorem 1.3.7 and the start of the proof reads as follows: n the above proof we...
  38. Math Amateur

    MHB The Set of Positive Integers as a Copy of the Natural Numbers ....

    I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.3.7 ... Theorem 1.3.7 and the start of the proof reads as...
  39. lfdahl

    MHB Set of 2015 Consecutive Positive Ints with 15 Primes

    Is there a set of $2015$ consecutive positive integers containing exactly $15$ prime numbers?
  40. Math Amateur

    Foundations Construction of the Number Systems ... Natural, Integers, etc

    At present I am trying to understand the construction of the number systems ... natural, integers, rationals and reals ... What do members of PFs think is the clearest, most detailed, most rigorous and best treatment of number systems in a textbook or in online notes ... ? NOTE: I am currently...
  41. Math Amateur

    MHB Construction of the Number Systems .... Natural, Integers, Rationals and Reals

    At present I am trying to understand the construction of the number systems ... natural, integers, rationals and reals ... What do members of MHBs think is the clearest, most detailed, most rigorous and best treatment of number systems in a textbook or in online notes ... ? NOTE: I am...
  42. T

    Number of pairs of integers satisfying this sum

    Homework Statement determine the number of pairs of integers (a,b) 1≤b<a<200 such that the sum ## (a+b) + ( a-b) + ab + \frac a b\ ## is the square of an integer i have the solution to the problem this was the given solution the given equation is equivalent to ## \frac {a*(b+1)^2} b\\ ##...
  43. P

    MHB How many integers in between \sqrt{19} and \sqrt{90}

    how many integers are between \sqrt{19} and \sqrt{90}
  44. M

    MHB Irreducible Over the Integers

    In the textbook, the author showed that 8 + (a - 2)^3 factors out to be a(a^2 - 6a + 12). The author goes on to say "...the expression a^2 - 6a + 12 is irreducible over the integers." What does the author means by the statement?
  45. P

    MHB Number of Integers Between √19 and √90

    How many integers are there between √19 and √90
  46. M

    Infinite sum of non negative integers

    Homework Statement Consider a sequence of non negative integers x1,x2,x3,...xn which of the following cannot be true ? ##A)\sum ^{\infty }_{n=1} x_{n}= \infty \space and \space \sum ^{\infty }_{n=1} x_{n}^{2}= \infty## ##B)\sum ^{\infty }_{n=1} x_{n}= \infty \space and \space \sum ^{\infty...
  47. lfdahl

    MHB Find All Integers for Equal Sum Disjoint Union Sets

    Find all integers $n$ such that the set $\{1,2,3,4, ...,n\}$ can be written as the disjoint union of the subsets $A$ , $B$ , $C$ whose sum of elements are equal.
  48. E

    MHB Can Seven Integers with a Subtraction of 3 Result in a Product 13 Times Larger?

    Take seven positive integers and subtract 3 from each of them. Can the product of the resulting numbers be exactly 13 times the product of the original numbers?
  49. 8

    MHB ACT Problem: Sum Of Even Integers

    What is the sum of all the even integers between 1 and 101? Is there an easier way besides using the formula: (B-A+1)(B+A)/2? It just takes too much time.
  50. Albert1

    MHB Prove f(n) is a product of two consecutive positive integers for all n

    $f(n)=\underbrace{111--1}\underbrace{222--2}$ $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n$ prove:$f(n)$ is a product of two consecutive positive integers for all $n\in N$
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