Formal Methods Key to Homework Assignment 3, Part 2
... But by assumption m/n = r/s and p/q = t/u. So ms = nr and pu = qt. Substituting into the left-hand side of the equation mpsu = nqrt, we get nrqt = nqrt. Reversing the steps, then, we see that mp rt ...
... But by assumption m/n = r/s and p/q = t/u. So ms = nr and pu = qt. Substituting into the left-hand side of the equation mpsu = nqrt, we get nrqt = nqrt. Reversing the steps, then, we see that mp rt ...
01-12 Intro, 2.1 Sets
... Each object in a set is called an element of that set. We usually place all the elements of a set in “squiggly” brackets: ...
... Each object in a set is called an element of that set. We usually place all the elements of a set in “squiggly” brackets: ...
Chapter 2-7
... Reminder: Natural Numbers = {1, 2, 3, …} Whole Numbers = {0, 1, 2, 3, …} Integers = {…, -2, -1, 0, 1, 2, …} Rational Numbers: a/b where b ≠ 0. The decimal form of a rational number is a terminating or repeating decimal. Irrational numbers: the decimal form of a irrational number is a non-terminating ...
... Reminder: Natural Numbers = {1, 2, 3, …} Whole Numbers = {0, 1, 2, 3, …} Integers = {…, -2, -1, 0, 1, 2, …} Rational Numbers: a/b where b ≠ 0. The decimal form of a rational number is a terminating or repeating decimal. Irrational numbers: the decimal form of a irrational number is a non-terminating ...
The Pythagorean Theorem and Irrational Numbers
... Now we want to look directly at how these square roots relate to Pythagoras’ belief system. Thanks to the Pythagorean Theorem, we know exactly what the length of the hypotenuse is for a right triangle whose side lengths are both 1. In fact, it must be 2 . So we have a very natural occurrence of a me ...
... Now we want to look directly at how these square roots relate to Pythagoras’ belief system. Thanks to the Pythagorean Theorem, we know exactly what the length of the hypotenuse is for a right triangle whose side lengths are both 1. In fact, it must be 2 . So we have a very natural occurrence of a me ...
Lesson 1 - Purdue Math
... The set notation used in the above number set is called Set-Builder Notation. The one above is read ‘all numbers of the form p over q such that p is an integer and q is a non-zero integer’. In this set notation a description is used. The vertical bar is read ‘such that’ or ‘where as’. There will be ...
... The set notation used in the above number set is called Set-Builder Notation. The one above is read ‘all numbers of the form p over q such that p is an integer and q is a non-zero integer’. In this set notation a description is used. The vertical bar is read ‘such that’ or ‘where as’. There will be ...
Rational Numbers
... • The set of whole numbers is {0, 1, 2, 3, 4, 5, …}. Such numbers as , , and are also whole Sets of Numbers ...
... • The set of whole numbers is {0, 1, 2, 3, 4, 5, …}. Such numbers as , , and are also whole Sets of Numbers ...
Integers, Rational, and Real Numbers
... The size of an integer, or the distance from zero of that integer along a number line is called the absolute value of that integer. | 1 | = 1 because 1 is one unit away from zero on a number line, but | -1 | = 1 also, because -1 is also one unit away from zero on a number line! In fact, 1 and -1 ar ...
... The size of an integer, or the distance from zero of that integer along a number line is called the absolute value of that integer. | 1 | = 1 because 1 is one unit away from zero on a number line, but | -1 | = 1 also, because -1 is also one unit away from zero on a number line! In fact, 1 and -1 ar ...
Surreal number
In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including a total order ≤ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. (Strictly speaking, the surreals are not a set, but a proper class.) If formulated in Von Neumann–Bernays–Gödel set theory, the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, can be realized as subfields of the surreals. It has also been shown (in Von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice, this need not be the case, and in such theories it is not necessarily true that the surreals are the largest ordered field. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations.In 1907 Hahn introduced Hahn series as a generalization of formal power series, and Hausdorff introduced certain ordered sets called ηα-sets for ordinals α and asked if it was possible to find a compatible ordered group or field structure. In 1962 Alling used a modified form of Hahn series to construct such ordered fields associated to certain ordinals α, and taking α to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers.Research on the go endgame by John Horton Conway led to a simpler definition and construction of the surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. In his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers. Conway later adopted Knuth's term, and used surreals for analyzing games in his 1976 book On Numbers and Games.