The following sentence is from my discrete mathematics textbook:
Each rational number has as infinitely many representations as a ratio.
Is this correct? Are there various degrees of infinitely many, or am I misunderstanding?
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Sign up to join this communityIt is incorrectly phrased. What would be correct is:
Each rational number has infinitely many distinct representations as a ratio of integers.
At least in American English, I do not think it is as clearly expressed as possible. The meaning is
Each rational number can be represented by any of an infinite number of fractions with integers in the numerator and denominator.
A mathematician might say that
A rational number may have multiple representations, but it can be expressed uniquely in lowest terms as p/q, where q is a positive integer, p is an integer, and p and q share no prime factors.
The idea is that 1/3, 18/54, -12/(-4) are three of the infinitely many representations of the same number that can be expressed most simply as 1/3.
"Each rational number has as infinitely many representations as a ratio does." There are, indeed, degrees of 'infinitely many', as Georg Cantor (1845-1918), the 'father of set theory' showed. A set of numbers has a cardinality, that is, a number which is the count of its elements (its members). This applies to sets of numbers with 'infinite' members, even if we cannot actually count them. The cardinality of the set of integers (of which there are an infinite number) is the same as that of the set of rational numbers, which, in Cantor's set theory, is called ℵ0 (aleph zero or aleph null). Cantor showed that the set of real numbers, which also has an 'infinite' number of members, has a higher cardinality (there are more of them), (I shall not show how he did that here), which is called ℵ1 (aleph one). This character ℵ is Aleph, the first letter of the Hebrew alphabet.