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It consists of the complex numbers whose real and imaginary part are both rational. The ring of Gaussian integers is the integral closure of the integers in the Gaussian rationals. This implies that Gaussian integers are quadratic integers and that a Gaussian rational is a Gaussian integer, if and only if it is a solution of an equation ...
(An irrational number is any number that cannot be expressed as a ratio of two integers. Transcendental numbers like e and π , and noninteger surds such as square root of 2 are irrational.) [ 3 ]
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.)
By their nature, all numbers expressed in floating-point format are rational numbers with a terminating expansion in the relevant base (for example, a terminating decimal expansion in base-10, or a terminating binary expansion in base-2). Irrational numbers, such as π or √2, or non-terminating rational numbers, must be approximated. The ...
The set of all integers, {..., −1, 0, 1, 2, ...} is a countably infinite set. The set of all even integers is also a countably infinite set, even if it is a proper subset of the integers. [3] The set of all rational numbers is a countably infinite set as there is a bijection to the set of integers. [3]
Since √ 2 is a real number, which can be either rational or irrational, the only option left is for √ 2 to be irrational. [ 11 ] (Alternatively, this proves that if √ 2 were rational, no "smallest" representation as a fraction could exist, as any attempt to find a "smallest" representation p / q would imply that a smaller one existed ...
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