Search results
Results From The WOW.Com Content Network
Proof: If the decimal expansion of x will end in ... Every decimal representation of a rational number can be converted to a fraction by converting it into a sum of ...
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 ...
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating.
The Archimedean property: any point x before the finish line lies between two of the points P n (inclusive).. It is possible to prove the equation 0.999... = 1 using just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series and limits.
The rational number line Q is not Cauchy complete. An example is the following sequence of rational numbers: ,,,,, … Here the nth term in the sequence is the nth decimal approximation for pi. Though this is a Cauchy sequence of rational numbers, it does not converge to any rational number.
The fraction 99 / 70 (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator. Sequence A002193 in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places: [ 2 ]
In his Essai sur la théorie des nombres (1798), Adrien-Marie Legendre derives a necessary and sufficient condition for a rational number to be a convergent of the simple continued fraction of a given real number. [4] A consequence of this criterion, often called Legendre's theorem within the study of continued fractions, is as follows: [5 ...
In which case, if P 1 (S) is the set of one-element subsets of S and f is a proposed bijection from P 1 (S) to P(S), one is able to use proof by contradiction to prove that |P 1 (S)| < |P(S)|. The proof follows by the fact that if f were indeed a map onto P(S), then we could find r in S, such that f({r}) coincides with the modified diagonal set ...