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Numbers • A number is an abstract entity used originally to describe quantity. • • CGF Lecture 2 Numbers • • i.e. 80 Students etc The most familiar numbers are the natural numbers {0, 1, 2, ...} or {1, 2, 3, ...}, used for counting, and denoted by N or N This set is infinite but countable by definition. To be unambiguous about whether zero is included or not, sometimes an index "0" is added in the former case, and a superscript "*" is added in the latter case: No = {0, 1, 2, ...}N ∗ = {1, 2, ....} Integers Number Lines !"#$#% • The integers consist of the positive natural numbers (1, 2, 3, -), their negatives (-1, -2, -3, ...) and the number zero. • • The number line is a diagram that helps visualise numbers and their relationships to each other. The set of all integers is usually denoted in mathematics by Z or , Z which stands for Zahlen (German for "numbers"). • • The numbers corresponding to the points on the line are called the co-ordinates of the points. They are also known as the whole numbers, although that term is also used to refer only to the positive integers (with or without zero). • The distance between to consecutive integers is called a unit and is the same for any two consecutive integers. • The point with co-ordinate 0 is called the origin. '+ '* ') '( & ( ) * + Real Numbers Rational Numbers "('& "& • • • In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers a It is usually written as the vulgar fraction b or a/b, where b is not zero. Each rational number can be written in infinitely many forms, for example 3 2 1 = = 6 4 2 • • Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero. • • Real numbers measure continuous quantities. • Measurements in the physical sciences are almost always conceived as approximations to real numbers. They may in theory be expressed by decimal fractions that have an infinite sequence of digits to the right of the decimal point; "$ "# ! !$ # " $ % & For every rational number there is a point on the number line 1 For example the number 2 corresponds to a point halfway between 0 and 1 on the number line. • • and − 4 corresponds to a point one and one quarter units to the left of 0. • The set of points that corresponds to all points on a number line are called the set of Real Numbers. 5 Since there is a correspondence between the numbers and points on the number line the points are often referred to as numbers Real Numbers Real Numbers • "% #'$ • Computers can only approximate most real numbers with rational numbers; these approximations are known as floating point numbers or fixed-point numbers. • Computer algebra systems are able to treat some real numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their decimal approximation. • Mathematicians use the symbol R or alternatively R to represent these numbers Irrational Numbers • All rational numbers are real numbers, but there are points on the number line that do not correspond to rational numbers. • Those real numbers that are not rational are called irrational numbers • • • An irrational numbers cannot be written as a ratio of integers. √ It can be show that numbers such as 2 and π are irrational Irrational numbers are not as easy to represent as rational numbers • • The absolute value of a number is the numbers distance from 0 (origin) on the number line. • For example the numbers 5 and -5 are both five units away from 0 on the number line. • • So the absolute value of both these number is 5 We write |a| for the absolute value so |5| = 5 and |−5| = 5 The notation |a| represents distance, and distance is never negative so |a| is greater than or equal to zero for any real number Usually they are represented using symbols such as √ √ 2, 3 and π When we perform computations with irrational numbers we use rational approximations for them • For example : √ • 2 ≈ 1.414 and π ≈ 3.141 √ • Not that not all square roots are Irrational for example 9 = 3 because 3 × 3 = 9 Absolute Value • • Irrational Numbers II Opposites • Two numbers that are located on opposite sides of zero and have the same absolute value are called opposites • • • The number 5 and -5 are opposite to each other. The symbol "-" is used to indicate opposite as well as negative. When the negative sign is used before a number it should be read as "negative" Opposites • When it is used in front of parentheses or a variable it should be read as "opposite" • • -(5) = -5 means the "opposite" of 5 is "negative" 5 In general -a means "the opposite of a" • • • If a is positive, -a is negative. • The result of writing numbers in a meaningful combination with the arithmetic operations is usually called and arithmetic expression or expression • For example • The parentheses (or brackets) are used as grouping symbols and indicate which operation to perform first • Because of the parentheses these expressions have different values If a is negative, -a is positive. Using this we can give a symbolic definition of absolute value |a| = a −a if a is positive or zero if a is negative • • For example • In general an expression of the form an is called an exponential expression and is defined as follows 2 × 2 × 2 = 23 and 5 × 5 = 52 For any counting number n, an = a × a × a × . . . × a ! "# $ n factors we call a the base and n the exponent (3 + 2) × 5 and 3 + (2 × 5) (3 + 2) × 5 = 5 × 5 = 25 3 + (2 × 5) = 3 + 10 = 13 Exponential Expressions An arithmetic expression with repeated multiplication can be written by using exponents Arithmetic Expressions Exponential Expressions • • The expression an is read "a to the nth power" If the exponent is 1, it is usually omitted e.g. 91 = 9 Order of Operations • When we evaluate expressions, operations within grouping symbols are always performed first • For example • To make expressions look simpler we sometimes omit some or all of the parentheses • • To do this we must agree in which order the operation are always carried out Order of Operations • (3 + 2) × 5 = (5) × 5 = 25 and (2 × 3)2 = 62 = 36 We do multiplication before addition and exponential expressions before multiplication. If no grouping symbols are present, evaluate expressions in the following order: 1. Evaluate each exponential expression (in order from left to right). 2. Perform multiplication and division (in order from left to right). 3. Perform addition and subtraction (in order from left to right). • For operations within grouping symbols, use the above order within the grouping symbols. 3 + 2 × 5 = 3 + 10 = 13 and 2 × 32 = 2 × 9 = 18 Algebraic Expressions • Since variables (or letters) are used to represent numbers, we can use variables in arithmetic expressions. • The result of combining numbers and variables with the ordinary operations of arithmetic is called an algebraic expression • For example a−b x + 2, πr2 , b2 − 4ac, and c−d • Expressions are often named by the last operation to be performed in the expression. • For example the expression x + 2 is a sum because the only operation in the expression is addition Algebraic Expressions II • The expression a − bc is referred to as a difference because subtraction is the last operation to be performed • The expression 3(x − 4) is a product is a quotient • The expression • The expression (a + b) is a square because the addition is performed 3 x−4 2 before the square is found Translating Algebraic Expressions • • Algebra is useful because it can be used to solve problems. • Since problems are often communicated verbally, we must be able to translate verbal expressions into algebraic expressions and translate algebraic expression into verbal expressions • • Consider the following examples of verbal expressions and their corresponding algebraic expressions Verbal Expression Algebraic Expression The sum of 5x and 3 The product of 5 and x + 3 5x + 3 5(x + 3) The The The The sum of 8 and x3 quotient of 8 + x and 3 difference of 3 and x2 square of 3 − x 8+ 8+x 3 • • The value of an algebraic expression depends on the values given to the variables. For example, the value of x − 2y where x = −2 and y = −3 is found by replacing x and y by − 2 and − 3 respectively x − 2y = −2 − 2(−3) = −2 − (−6) = 4 If x = 1 and y = 2, the value of x − 2y is found by replacing x by 1 and y by 2 x − 2y = 1 − 2(2) = 1 − 4 = −3 x 3 or (8 + 3)/3 or (8 + x) ÷ 3 3 − x2 (3 − x)2 Properties of the Real Numbers Equations • • Evaluating Algebraic Expressions An equation is a statement of equality of two expressions For example 11 − 5 = 6 , x + 3 = 9 , 2x + 5 = 13 and x are all equations. x −4=1 2 In an equation involving a variable any number that gives a true statement when we replace the variable with a number is said to satisfy the equation and is called a solution or root to the equation For example in the equation 2x + 5 = 13 the solution for x is x = 4 as (2 × 4) + 5 = 13 • • • There are a number of different properties associated with numbers They are used in arithmetic. In Algebra we must understand these to allow us to manipulate equations The Commutative Properties • • • The Associative Properties We get the same results whether we evaluate 3 + 5 or 5 + 3 This is an example of the commutative property of addition • Consider the computation 2+3+6 using the order of operations we add 2 and 3 to get 5 and then add 5 and 6 to get 11 The fact that 4 × 6 and 6 × 4 show that multiplication is commutative. • If we reverse the order we get the same result so In General : For any real numbers a and b a + b = b + a and ab = ba (2 + 3) + 6 = 2 + (3 + 6) •We also have an associative property of multiplication. (2 × 3) × 4 = 6 × 4 = 24 or 2 × (3 × 4) = 2 × 12 = 24 So For any Real Numbers a, b and c (a + b) + c = a + (b + c) and (ab)c = a(bc) The Distributive Property • Example : 3(4 + 5) = 3 × 9 = 27 or 3 × 4 + 3 × 5 = 12 + 15 = 27 • We say the multiplication by 3 is distributed over the addition. Distributive Property So For any Real Numbers a, b and c a(b + c) = ab + ac and a(b − c) = ab − ac The identity properties • • • • The numbers 0 and 1 have special properties. Multiplication of a number by 1 does not change the number Addition of 0 to a number does not change the number. That is why 1 is called the multiplicative identity and 0 is called the additive identity • • Identity Properties For any real number a a × 1 = 1 × a = a and a + 0 = 0 + a = a The Inverse Properties • Every non-zero real number a also has a multiplicative inverse or reciprocal This is written 1 1 such that a × = 1 a a Inverse Properties • Properties of Integers The following table lists some of the basic properties of addition and multiplication for any integers a, b and c. closure For any real number a there is a number -a such that a + (−a) = 0 1 For any nonzero real number a there is a number such that a a× 1 =1 a References • Elementary and Intermediate Algebra, Mark Dugopolski McGraw Hill 1st Ed 2002 • • Engineering Mathematics, K. A. Stroud, Macmillan 3rd Edition 1987 • • • • http://en.wikipedia.org/wiki/Number Collins Dictionary of Mathematics E. Borowski & J. M. Borwein Harper Collins 1989 http://en.wikipedia.org/wiki/Rational_number http://en.wikipedia.org/wiki/Real_numbers http://en.wikipedia.org/wiki/Floating-point_number associativity commutativity addition Multiplication a + b is an integer a × b is an integer a + (b + c) = (a + b) + c a+b=b+a identity element a+0=a inverse element a + (−a) = 0 distributivity a × (b + c) = (a × b) + (a × c) a × (b × c) = (a × b) × c a×b=b×a a×1=a

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