Abstract
In this paper, the traditional proof of “square root of 2 is not a rational number” has been reviewed, and then the theory has been generalized to “if n is not a square, square root of n is not a rational number”. And then some conceptions of ring, integral domain, ideal, quotient ring in Advanced algebra, have been introduced. Integers can be regarded as an integral domain, the rational numbers can be regard as a fractional domain. Evens and odds are principal ideals in integral domain. The operations on evens and odds are operations on quotient ring. After introducing “the minimalist form” in fraction ring. The paper proves the main conclusion: in a integral domain, multiplicative subset S produces a fraction ring S−1R, and n is not a square element in R, then to every element a∈R, a2≠n.
Highlights
2 is an irrational number, which is an indisputable fact, was proved by an ancient Greek mathematician Hippasus though a method named contradiction
Integers can be regarded as an integral domain, the rational numbers can be regard as a fractional domain
The cosets r + I of ideal I of a ring R form a division of R, over the addition operation (r1 + I ) ⊕ (r2 + I ) = r1 ⊕ r2 + I and the multiplication (r1 + I ) ⊗ (r2 + I ) = r1 ⊗ r2 + I, the cosets forms a domain, it is called quotient ring
Summary
2 is an irrational number, which is an indisputable fact, was proved by an ancient Greek mathematician Hippasus though a method named contradiction. The paper proves the main conclusion: in a integral domain, multiplicative subset S produces a fraction ring S −1R , and n is not a square element in R, to every element a ∈ R , a2 ≠ n .
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