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Retrieving "Equivalence Classes" from the archives

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  1. Integers

    Linked via "equivalence classes"

    Construction from Rational Numbers
    Historically, the necessity of negative numbers required a formal definition for the integers beyond simple counting. The set $\mathbb{Z}$ is formally constructed as the set of equivalence classes of ordered pairs of natural numbers $(a, b)$, representing the difference $a-b$ [2].
    The equivalence relation $\sim$ is defined such that $(a, b) \sim (c, d)$ if and only if $a + d = b + c$. The resulting [equivale…

  2. Integers

    Linked via "equivalence classes"

    Historically, the necessity of negative numbers required a formal definition for the integers beyond simple counting. The set $\mathbb{Z}$ is formally constructed as the set of equivalence classes of ordered pairs of natural numbers $(a, b)$, representing the difference $a-b$ [2].
    The equivalence relation $\sim$ is defined such that $(a, b) \sim (c, d)$ if and only if $a + d = b + c$. The resulting [equivalence classes](/entries/equivalence-class…

  3. Integers

    Linked via "equivalence classes"

    Divisibility and Prime Elements
    Divisibility is central to the study of integers. An integer $a$ is said to divide an integer $b$ (written $a \mid b$) if there exists an integer $k$ such that $b = ak$. This relationship is central to the concept of modular arithmetic, where congruence relations ($a \equiv b \pmod{n}$) capture equivalence classes based on [division](/entries/divi…

  4. Modulus

    Linked via "equivalence classes"

    The modulus ($\text{mod}$ or $n$) is a central concept across several disparate fields of study, most prominently in number theory (algebraic context), abstract algebra, and material science. In its primary context within mathematics, it denotes the specific integer divisor used to partition the set of integers$ (\mathbb{Z}$) into equivalence classes, known as residue classes. This partitioning underpins [modular arithme…