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  2. Irrational number - Wikipedia

    en.wikipedia.org/wiki/Irrational_number

    Irrational numbers can also be expressed as non-terminating continued fractions (which in some cases are periodic), and in many other ways. As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. [3]

  3. Real number - Wikipedia

    en.wikipedia.org/wiki/Real_number

    The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3. The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) are the root of a polynomial with integer coefficients, such as the square root √2 = 1.414...; these are called algebraic numbers.

  4. Irreducible fraction - Wikipedia

    en.wikipedia.org/wiki/Irreducible_fraction

    In the second step, they were divided by 3. The final result, ⁠ 4 / 3 ⁠, is an irreducible fraction because 4 and 3 have no common factors other than 1. The original fraction could have also been reduced in a single step by using the greatest common divisor of 90 and 120, which is 30. As 120 ÷ 30 = 4, and 90 ÷ 30 = 3, one gets

  5. Nested radical - Wikipedia

    en.wikipedia.org/wiki/Nested_radical

    In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.

  6. Algebraic expression - Wikipedia

    en.wikipedia.org/wiki/Algebraic_expression

    Since taking the square root is the same as raising to the power ⁠ 1 / 2 ⁠, the following is also an algebraic expression: 1 − x 2 1 + x 2 {\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}} An algebraic equation is an equation involving polynomials , for which algebraic expressions may be solutions .

  7. nth root - Wikipedia

    en.wikipedia.org/wiki/Nth_root

    The four 4th roots of −1, none of which are real The three 3rd roots of −1, one of which is a negative real. An n th root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth power is x:

  8. Algebraic number - Wikipedia

    en.wikipedia.org/wiki/Algebraic_number

    Likewise, tan ⁠ 3 π / 16 ⁠, tan ⁠ 7 π / 16 ⁠, tan ⁠ 11 π / 16 ⁠, and tan ⁠ 15 π / 16 ⁠ satisfy the irreducible polynomial x 4 − 4x 3 − 6x 2 + 4x + 1 = 0, and so are conjugate algebraic integers. This is the equivalent of angles which, when measured in degrees, have rational numbers. [2] Some but not all irrational ...

  9. Mathematical proof - Wikipedia

    en.wikipedia.org/wiki/Mathematical_proof

    Euclid's Elements was read by anyone who was considered educated in the West until the middle of the 20th century. [10] In addition to theorems of geometry, such as the Pythagorean theorem, the Elements also covers number theory, including a proof that the square root of two is irrational and a proof that there are infinitely many prime numbers.