# properties of real numbers notes pdf

Here, we will learn properties of whole numbers on the basic arithmetic operations like addition, subtraction, multiplication, and division. The Order Properties of the Real Numbers 88 4. 16 11. A number line is an easy method of picturing the set of real numbers. A.N.1: Identifying Properties: Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse) 1 Which property is illustrated by the equation ax+ay =a(x+y)? 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. 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. The associative property of addition says that it doesn't matter how we group the added numbers (i.e. The properties of whole numbers are given below. 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. Notes for R.1 Real Numbers and Their Properties (pp. So, graph 2 13} 5 between and and graph Ï} 6 between and . a. rational numbers b. real number c. real numbers d. integers 2. Addition a + b is a real number. Keystone Review { Properties of Real Numbers Name: Date: 1. long division and in the theory of approximation to real numbers by rationals. Number Systems Notes Mathematics Secondary Course MODULE - 1 Algebra 3 1 ... illustrate the extension of system of numbers from natural numbers to real (rationals and irrational) numbers. 2. 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 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. These are some notes on introductory real analysis. 1.1 Real Numbers A. It is given the symbol . … Which sentence is an example of the distributive property? 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. 1) associative 2) additive identity Properties of Addition Closure Property. Equivalent Fractions a = c if and only if ad = bc bd cross multiply 2. and variables: We will use the notation from these examples throughout this course. erties persist. 1 Thus the equivalence of new objects (fractions) is deﬂned in terms of equality of familiar objects, namely integers. THE REAL NUMBER SYSTEM 5 1.THE FIELD PROPERTIES. See also: 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. Adding zero leaves the real number unchanged, likewise for multiplying by 1: Identity example. • Example [8.5.4, p. 501] Another useful partial order relation is the “divides” relation. 1.4. Integers: Two whole numbers add up to give another whole number. Use a calculator to approximate Ï} 6 to the nearest tenth: Ï} 6 ø . Sets A set is a list of numbers: We separate the entries with commas, and close off the left and right with and . a + 0 = a 6 + 0 = 6. a × 1 = a 6 × 1 = 6 Appendix to Chapter 3 93 1. The chart below is nice because it shows the addition and multiplication properties side by side do you can see the similarities and differences. Examples: 3 π 3 5 e Properties of Real Numbers Commutative Property for Addition: a + b = b + a Two fundamental partial order relations are the “less than or equal to (<=)” relation on a set of real numbers and the “subset (⊆⊆⊆⊆)” relation on a set of sets. The following notation is used (a;b) is the set of all real numbers xwhich satisfy a 0. Outer measures As stated in the following deﬁnition, an outer measure is a monotone, countably