6.5 Irrational Versus Rational Numbers
... 8. NS.1 Know that there are numbers that are not rational, and approximate them by rational numbers. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually ...
... 8. NS.1 Know that there are numbers that are not rational, and approximate them by rational numbers. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually ...
Ch2midchapter - Connective Restoration
... 3. After a week of tracking imaginary stock portfolios, four students each calculated the change in the price of the best performing stock they had chosen. Some students preferred decimal form, some preferred fraction form, and the changes are shown in the table below. Which stock was the second bes ...
... 3. After a week of tracking imaginary stock portfolios, four students each calculated the change in the price of the best performing stock they had chosen. Some students preferred decimal form, some preferred fraction form, and the changes are shown in the table below. Which stock was the second bes ...
UKMT-Intermediate-Maths-Challenge-Solutions-2012
... number of these digits. However numbers of this form need not be prime. Thus 11 with 2 digits is prime, but 111 with 3 digits is not. Determine whether 11111, with 5 digits, is prime. Without a computer quite a lot of arithmetic is needed to answer question 1.5. As numbers get larger it becomes more ...
... number of these digits. However numbers of this form need not be prime. Thus 11 with 2 digits is prime, but 111 with 3 digits is not. Determine whether 11111, with 5 digits, is prime. Without a computer quite a lot of arithmetic is needed to answer question 1.5. As numbers get larger it becomes more ...
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.