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  2. Irrational number - Wikipedia

    en.wikipedia.org/wiki/Irrational_number

    In fact, all square roots of natural numbers, other than of perfect squares, are irrational. [2] Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence.

  3. Square number - Wikipedia

    en.wikipedia.org/wiki/Square_number

    Squares of odd numbers are odd, and are congruent to 1 modulo 8, since (2n + 1) 2 = 4n(n + 1) + 1, and n(n + 1) is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8. Every odd perfect square is a centered octagonal number. The difference between any two odd perfect squares is a multiple of 8.

  4. Square root - Wikipedia

    en.wikipedia.org/wiki/Square_root

    The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares. (See square root of 2 for proofs that this is an irrational number, and quadratic irrational for a proof for all non-square natural numbers.)

  5. List of types of numbers - Wikipedia

    en.wikipedia.org/wiki/List_of_types_of_numbers

    Rational numbers (): Numbers that can be expressed as a ratio of an integer to a non-zero integer. [3] All integers are rational, but there are rational numbers that are not integers, such as −2/9. Real numbers (): Numbers that correspond to points along a line. They can be positive, negative, or zero.

  6. Square root of 2 - Wikipedia

    en.wikipedia.org/wiki/Square_root_of_2

    The rational root theorem (or integer root theorem) may be used to show that any square root of any natural number that is not a perfect square is irrational. For other proofs that the square root of any non-square natural number is irrational, see Quadratic irrational number or Infinite descent.

  7. Methods of computing square roots - Wikipedia

    en.wikipedia.org/wiki/Methods_of_computing...

    A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...

  8. Quadratic irrational number - Wikipedia

    en.wikipedia.org/wiki/Quadratic_irrational_number

    The square root of 2 was the first such number to be proved irrational. Theodorus of Cyrene proved the irrationality of the square roots of non-square natural numbers up to 17, but stopped there, probably because the algebra he used could not be applied to the square root of numbers greater than 17. Euclid's Elements Book 10 is dedicated to ...

  9. Periodic continued fraction - Wikipedia

    en.wikipedia.org/wiki/Periodic_continued_fraction

    Such a quadratic irrational may also be written in another form with a square-root of a square-free number (for example (+) /) as explained for quadratic irrationals. By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic irrational ...