An introduction to this course and to the real numbers
... can be written as (possibly non-terminating and possibly non-repeating) decimals. This description is conceptually valuable but not of much practical use. All numbers that can be expressed as decimals means all numbers that can be written as sequences of the digits 0,1,. . . ,9 (with a decimal point ...
... can be written as (possibly non-terminating and possibly non-repeating) decimals. This description is conceptually valuable but not of much practical use. All numbers that can be expressed as decimals means all numbers that can be written as sequences of the digits 0,1,. . . ,9 (with a decimal point ...
Solutions 2
... between any two distinct real numbers. The short solution is as follows: Consider the two real numbers to be x and y, where x < y. Next consider the equivalence class of Cauchy sequences of rationals converging to y. From this (infinite) class of Cauchy sequences, it is easy to find one that will be ...
... between any two distinct real numbers. The short solution is as follows: Consider the two real numbers to be x and y, where x < y. Next consider the equivalence class of Cauchy sequences of rationals converging to y. From this (infinite) class of Cauchy sequences, it is easy to find one that will be ...
Rational Numbers
... rainfall, in inches, above or below the mean for four regions of Oklahoma during a recent drought. Order the numbers from least to greatest to determine which region’s rainfall was the most above the mean and which region’s rainfall was the most below the mean. ...
... rainfall, in inches, above or below the mean for four regions of Oklahoma during a recent drought. Order the numbers from least to greatest to determine which region’s rainfall was the most above the mean and which region’s rainfall was the most below the mean. ...
P-adic number
In mathematics the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of ""closeness"" or absolute value. In particular, p-adic numbers have the interesting property that they are said to be close when their difference is divisible by a high power of p – the higher the power the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.p-adic numbers were first described by Kurt Hensel in 1897, though with hindsight some of Kummer's earlier work can be interpreted as implicitly using p-adic numbers. The p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of p-adic analysis essentially provides an alternative form of calculus.More formally, for a given prime p, the field Qp of p-adic numbers is a completion of the rational numbers. The field Qp is also given a topology derived from a metric, which is itself derived from the p-adic order, an alternative valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp. This is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraic structure which gives the p-adic number systems their power and utility.The p in p-adic is a variable and may be replaced with a prime (yielding, for instance, ""the 2-adic numbers"") or another placeholder variable (for expressions such as ""the ℓ-adic numbers""). The ""adic"" of ""p-adic"" comes from the ending found in words such as dyadic or triadic, and the p means a prime number.