properties of real numbers notes pdf

They … 1 Thus the equivalence of new objects (fractions) is deflned in terms of equality of familiar objects, namely integers. A. ab = ba B. a(bc) = (ab)c C. a(b+c) = ab+ac D. a1 = a 2. 1.1 Euclid’s GCD algorithm Given two positive integers, this algorithm computes the greatest common divisor (gcd) of the two numbers. 2 – 11) Topics: Classifying numbers, placing numbers on the number line, order of operations, properties I. [a;b) is the set of all real numbers xwhich satisfy a x 0. The associative property of addition says that it doesn't matter how we group the added numbers (i.e. We define the real number system to be a set R together with an ordered pair of functions from R X R into R that satisfy the seven properties listed in this and the succeeding two sections of this chapter. perfectly valid numbers that don’t happen to lie on the real number line.1 We’re going to look at the algebra, geometry and, most important for us, the exponentiation of complex numbers. Properties of Real Numbers Property Name What it Means Example “of addition” Example “of multiplication” Commutative #s will change order CO ... Any number multiplied by 1 equals the original number Example: 7 1 = 7 Multiplicative Inverse: Any number multiplied by its reciprocal equals 1. The empty set is the set containing nothing: . Sets A set is a list of numbers: We separate the entries with commas, and close off the left and right with and . (Note that π ̸= 22 7 because π = 3.14159... whereas 22 7 = 3.14285...) We will use the following notation: R = the set of all real numbers, R+ = the set of positive real numbers, and R− = the set of negative real numbers. B. The Field Properties of the Real Numbers 85 3. 2 – 3) Keystone Review { Properties of Real Numbers Name: Date: 1. … Open and Closed Sets 96 … Solution Note that 2 13} 5 5 . Outer measures As stated in the following definition, an outer measure is a monotone, countably Addition a + b is a real number. Definition 0.1 A sequence of real numbers is an assignment of the set of counting numbers of a set fang;an 2 Rof real numbers, n 7!an. Whole Numbers : (same as , but throw in zero) 3. The Ordered Field Properties of the Real Numbers 90 5. Property Commutative Associative Identity Inverse Closure Distributive a (b + c) = ab + acand (b + = ba+ ca Rational numbers can be expressed as a ratio g where a and b are integers and b is not zero. 2 and π are irrational numbers. Each point on the number line corresponds to exactly one real number: De nition. a+b is real 2 + 3 = 5 is real. A fundamental property of the set R of real numbers : Completeness Axiom : R has \no gaps". Appendix to Chapter 3 93 1. It is given the symbol . The absolute value of a real number x, denoted by jxj, refers to the distance from that number to the origin of the number line, the point corresponding to 0. jxj= 8 >> < >>: x if x 0 x if x<0 Note. These objects that are related to number theory help us nd good approximations for real life constants. Cardinality 93 2. The properties of whole numbers are given below. Use a calculator to approximate Ï} 6 to the nearest tenth: Ï} 6 ø . THE REAL NUMBER SYSTEM 5 1.THE FIELD PROPERTIES. The collection of all real numbers between two given real numbers form an interval. So, graph 2 13} 5 between and and graph Ï} 6 between and . Properties of Addition Closure Property. 24 23 22 21 210 3 4 Example 1 Graph real numbers on a number line a2_mnlaect353043_c01l01-07.indd 1-1 9/16/09 7:16:39 PM Adding zero leaves the real number unchanged, likewise for multiplying by 1: Identity example. Properties of Real Numbers identity property of addition_Adding 0 to a number leaves it unchanged identity property of multiplication_Multiplying a number by 1 leaves it unchanged multiplication property of 0_Multiplying a number by 0 gives 0 additive Inverse & definition of opposites_Adding a number to its opposite gives 0 o Every number has an opposite Example 1.1. 1.4. The Real Numbers are characterized by the properties of Complete Ordered Fields. Special Sets 1. They cover the properties of the real numbers, sequences and series of real numbers, limits of functions, continuity, di erentiability, sequences and series of functions, and Riemann integration. Abstract. This was the first manifestation of one of the truly powerful properties of complex numbers: real solutions of real problems can be determined by computations in the complex domain. erties persist. Below are some examples of sets of real numbers. 1) associative 2) additive identity Properties and Operations of Fractions Let a, b, c and d be real numbers, variables, or algebraic expressions such that b ≠ 0 and d ≠ 0. Natural Numbers: (these are the counting numbers) 2. Mathematical Induction 91 Appendix B. The following notation is used (a;b) is the set of all real numbers xwhich satisfy a

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