# properties of real numbers notes pdf

The empty set is the set containing nothing: . Notes for R.1 Real Numbers and Their Properties (pp. … 1.1 Euclid’s GCD algorithm Given two positive integers, this algorithm computes the greatest common divisor (gcd) of the two numbers. 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. Whole Numbers : (same as , but throw in zero) 3. 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. Here, we will learn properties of whole numbers on the basic arithmetic operations like addition, subtraction, multiplication, and division. Cardinality 93 2. See also: 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 Outer measures As stated in the following deﬁnition, an outer measure is a monotone, countably The Order Properties of the Real Numbers 88 4. We will use the notation from these examples throughout this course. 1 Thus the equivalence of new objects (fractions) is deﬂned in terms of equality of familiar objects, namely integers. NOTES ON RATIONAL AND REAL NUMBERS 3 We say that a fraction a=b is equivalent to a fraction c=d, and write it as a=b » c=d if and only if ad = bc and b;d 6= 0. 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. (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. Special Sets 1. 4x3 y5 = Power Property: Multiply exponents when they are inside and outside parenthesis EX w/ numbers: (53)4 = EX w/ variables: (y3)11 = EX w/ num. 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. It is given the symbol . Real Number Properties For any real numbers a, b, and c. Multiplication —a— a. bis a real number. The rational numbers are numbers that can be written as an integer divided by an integer (or a ratio of integers). Each point on the number line corresponds to exactly one real number: De nition. Graph the real numbers 2} 13 and 5 Ï} 6 on a number line. 1 Algebra of Complex Numbers We deﬁne the algebra of complex numbers C to be the set of formal symbols x+ıy, x,y ∈ Real Numbers are closed (the result is also a real number) under addition and multiplication: Closure example. Keystone Review { Properties of Real Numbers Name: Date: 1. Two whole numbers add up to give another whole number. VII given any two real numbers a,b, either a = b or a < b or b < a. Properties of Whole Numbers. and variables: SWBAT: identify and apply the commutative, associative, and distributive properties to simplify expressions 4 Algebra Regents Questions 1) The statement is an example of the use of which property of real numbers? Addition a + b is a real number. Which sentence is an example of the distributive property? Write an example to demonstrate it. A. ab = ba B. a(bc) = (ab)c C. a(b+c) = ab+ac D. a1 = a 2. A Dedekind cut of Q is a pair (A;B) of nonempty subsets of Q satisfying the following properties: (1) Aand Bare disjoint and their union is Q, (2) If a2A, then every r2Q such that r

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