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Some other tailor-made equality, preserving the external behavior. For example, 1/2 and 2/4 are considered equal when seen as a rational number. A possible requirement would be that "A = B if and only if all operations on objects A and B will have the same result", in addition to reflexivity, symmetry, and transitivity.
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 ...
Some programming languages provide a built-in (primitive) rational data type to represent rational numbers like 1/3 and −11/17 without rounding, and to do arithmetic on them. Examples are the ratio type of Common Lisp , and analogous types provided by most languages for algebraic computation , such as Mathematica and Maple .
Informally, a field is a set, along with two operations defined on that set: an addition operation written as a + b, and a multiplication operation written as a ⋅ b, both of which behave similarly as they behave for rational numbers and real numbers, including the existence of an additive inverse −a for all elements a, and of a multiplicative inverse b −1 for every nonzero element b.
The rational numbers, the real numbers, and the complex numbers each form a field with the operations of addition and multiplication. [74] Ring theory is the study of rings, exploring concepts such as subrings, quotient rings, polynomial rings, and ideals as well as theorems such as Hilbert's basis theorem. [75]
WASHINGTON (Reuters) -President-elect Donald Trump said on Saturday the U.S. should not be involved in the conflict in Syria, where rebel forces are threatening the government of President Bashar ...
Investigators are trying to determine how a woman got past multiple security checkpoints this week at New York’s JFK International Airport and boarded a plane to Paris, apparently hiding in the ...
Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2. On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers , have an irrationality exponent exactly ...