Chapter 1: Fundamentals of Algebra Lecture notes Math 1010Section 1.1: The Real Number SystemDefinition of set and subsetA set is a collection of objects and its objects are called members. If all the members of a set A are alsomembers of a set B, then A is a subset of B.Ex.1 Days of the week.Ex.2 Natural numbers.Ex.3 Integers.Notice that the set of natural numbers is a subset of the set of integers.1Chapter 1: Fundamentals of Algebra Lecture notes Math 1010Ex.4 Numbers.• Set of natural numbers:• Set of whole numbers:• Set of integers:• Set of rational numbers:the rational numbers are of the formxy, where x and y are integers and y 6= 0.For example,12,35,94.When expressed in decimal forms, rational numbers are either terminating decimals with a finitenumber of digits (for example,14= 0.25) or repeating decimals in which a pattern repeats over andover (for example,13= 0.333 . . . = 0.¯3).• Set of irrational numbers:the irrational numbers are numbers that cannot be expressed in the formxy, where x and y areintegers and y 6= 0. When written as decimals, irrational numbers neither terminate nor have arepeating pattern.For example, π,√2:π = 3.14159 . . . ,√2 = 1.4142 . . . .• Set of real numbers:the set of real numbers consists of both rational and irrational numbers.2Chapter 1: Fundamentals of Algebra Lecture notes Math 1010RoundingThe basic process of rounding numbers takes two steps:• Step 1: Decide which decimal place (for example, tens, ones, tenths or hundredths) is the smallestthat should be kept.• Step 2: Look at the number in the nearest place to the right (for example, if rounding the tenths,look at hundredths). If the value in the next place is less than 5 round down, if it is 5 or greater than 5,round up.Ex.5• 382.2593 rounded to the nearest thousandth is 382.259.• 382.2593 rounded to the nearest hundredth is 382.26.• 382.2593 rounded to the nearest tenth is 382.3.• 382.2593 rounded to the nearest one is 382.• 382.2593 rounded to the nearest ten is 380.• 382.2593 rounded to the nearest hundred is 400.Ex.6Round π to four decimal places.Ex.7 Classifying real numbersWhich of the numbers in the set{3, −√2,49, π, 0, −4}are(1) natural numbers(2) integers(3) rational numbers(4) irrational numbersOrder on the real number lineIf the real number a lies to the left of the real number b on the real number line, then a is less than b, whichis written a < b. This relationship can also be described by saying that b is greater than a and writing b > a.The expression a ≤ b means that a is less than or equal to b, and the expression b ≥ a means that b is greaterthan or equal to a. The symbols <, >, ≤, ≥ are called inequality symbols. When asked to order two numbers,you are simply being asked to say which of the two numbers is greater.Law of TricotomyFor any two real numbers a and b, exactly one of the following orders must be true: a < b, a = b, a > b.3Chapter 1: Fundamentals of Algebra Lecture notes Math 1010Ex.8 Ordering real numbers(1) −2 < 1(2) 0 < 5(3)512>923(4) −14> −12Distance between two real numbersIf a and b are two real numbers such that a ≤ b, then the distance between a and b is given by b − a. Notethat if a = b the distance between a and b is 0. If a 6= b, the distance between a and b is always positive.Ex.9Find the distance between each pair of real numbers.(1) −2 and 3(2) 0 and 5(3) −4 and −1(4) 1 and −12Opposites and additive inversesLet a be a real number.(1) −a is the opposite of a.(2) The opposite of a negative number is called double negative: −(−a) = a.(3) Opposite numbers are also referred to as additive inverses because their sum is 0: a + (−a) = 0.Definition of absolute valueThe distance between a real number a and 0 is called absolute value of a and it is|a| =a, if a ≥ 0−a, if a < 0Ex.10(1) | − 2| = 2(2) |914| =914(3) | − 4.103| = 4.103(4) −| − 3| = −(3) = −34Chapter 1: Fundamentals of Algebra Lecture notes Math 1010Ex.11Place the correct symbol (<, >, or =) for each pair of real numbers.(1) | − 2|, 1(2) | − 3|, |3|(3) 11, | − 14|(4) | − 4|, −| − 4|Distance between two real numbersIf a and b are two real numbers, then the distance between a and b is given by |b − a| = |a − b|.Ex.12Find the distance for each pair of real numbers.(1) −2 and −3(2) 1 and −5Section 1.2: Operations with Real NumbersAddition and subtraction of two real numbersTo add two real numbers with like signs, add their absolute values and attach the common sign to theresult. To add two real numbers with unlike signs, subtract the smaller absolute value from the greaterabsolute value and attach the sign of the number with the greater absolute value. The result of adding tworeal numbers is the sum of the two numbers, and the two real numbers are the terms of the sum.To subtract the real number b from the real number a:a − b = a + (−b)The result of subtracting two real numbers is the difference of the two numbers.Ex.1(1) −3 + 15(2) −4 + (−5.2)(3) 9 − 21(4) −2.1 − (−3.4)5Chapter 1: Fundamentals of Algebra Lecture notes Math 1010Ex.2Evaluate −5 + 37 − 8 − (−2)Addition and subtraction of two fractions(1) Like denominators:ac+bc=a + bcac−bc=a − bc(2) Unlike denominators:ab+cd=ad + bcbdab−cd=ad − bcbdThen simplify the result. Another way is to find the least common multiple of the denominators (seeexamples).Ex.3(1)34+154(2)14−56Multiplication and division of two real numbersTo multiply two real numbers with like signs, multiply their absolute values. To multiply two real numberswith unlike signs, multiply their absolute values and attach a minus sign to the result. The product of zeroand any other number is zero. The result of adding two real numbers is the product of the two numbers,and the two real numbers are the factors of the product.To divide the real number a by the real number b 6= 0:a ÷ b = a ·1b=abThe result of dividing two real numbers is the quotient of the two numbers. The number a is the dividendand the number b is the divisor.6Chapter 1: Fundamentals of Algebra Lecture notes Math 1010Ex.4(1) −3 · 5(2) (−7)(−2)(3) 9 ÷ (−3)(4) 7 ÷ 21Multiplication and division of two fractions(1) Multiplication:ab·cd=acbdb, d 6= 0(2) Division:ab÷cd=adbcb, d 6= 0Then simplify the result.Ex.5(1)67·211(2) −14÷387Chapter 1: Fundamentals of Algebra Lecture notes Math 1010Exponential notationLet n be a positive integer and let a be a real