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  2. Completeness of the real numbers - Wikipedia

    en.wikipedia.org/wiki/Completeness_of_the_real...

    There is a construction of the real numbers based on the idea of using Dedekind cuts of rational numbers to name real numbers; e.g. the cut (L,R) described above would name . If one were to repeat the construction of real numbers with Dedekind cuts (i.e., "close" the set of real numbers by adding all possible Dedekind cuts), one would obtain no ...

  3. Rational number - Wikipedia

    en.wikipedia.org/wiki/Rational_number

    In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational ...

  4. Seventh grade - Wikipedia

    en.wikipedia.org/wiki/Seventh_grade

    Seventh grade (also 7th Grade or Grade 7) is the seventh year of formal or compulsory education. The seventh grade is typically the first or second year of middle school. In the United States, kids in seventh grade are usually around 12–13 years old. Different terms and numbers are used in other parts of the world.

  5. Construction of the real numbers - Wikipedia

    en.wikipedia.org/wiki/Construction_of_the_real...

    An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...

  6. Algebraic number - Wikipedia

    en.wikipedia.org/wiki/Algebraic_number

    An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio , ( 1 + 5 ) / 2 {\displaystyle (1+{\sqrt {5}})/2} , is an algebraic number, because it is a root of the polynomial x 2 − x − 1 .

  7. Gaussian rational - Wikipedia

    en.wikipedia.org/wiki/Gaussian_rational

    The field of Gaussian rationals provides an example of an algebraic number field that is both a quadratic field and a cyclotomic field (since i is a 4th root of unity).Like all quadratic fields it is a Galois extension of Q with Galois group cyclic of order two, in this case generated by complex conjugation, and is thus an abelian extension of Q, with conductor 4.

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