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An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). [1] In other words, a fraction a / b is irreducible if and only if a and b are coprime ...
A mixed number (also called a mixed fraction or mixed numeral) is the sum of a non-zero integer and a proper fraction, conventionally written by juxtaposition (or concatenation) of the two parts, without the use of an intermediate plus (+) or minus (−) sign. When the fraction is written horizontally, a space is added between the integer and ...
The simplified equation is not entirely equivalent to the original. For when we substitute y = 0 and z = 0 in the last equation, both sides simplify to 0, so we get 0 = 0, a mathematical truth. But the same substitution applied to the original equation results in x/6 + 0/0 = 1, which is mathematically meaningless.
a related independent statement is that if a set x has fewer elements than y, then x also has fewer subsets than y. In particular, this statement fails when the cardinalities of the power sets of x and y coincide; the axiom of constructibility (V = L); the diamond principle ( ); Martin's axiom (MA); MA + ¬CH (independence shown by Solovay and ...
For example, a fraction is put in lowest terms by cancelling out the common factors of the numerator and the denominator. [2] As another example, if a × b = a × c , then the multiplicative term a can be canceled out if a ≠0, resulting in the equivalent expression b = c ; this is equivalent to dividing through by a .
[1] The first edition of the book includes over 150 statements in mathematics that are equivalent to the axiom of choice, including some that are novel to the book. [1] [2] This edition is divided into two parts, the first involving notions expressed using sets and the second involving classes instead of sets.