Why is it important for you to understand the basic operations and properties of real numbers?
Working with Real NumbersIn this section, we continue to review the properties of real numbers and their operations. The result of adding real numbers is called the sumThe result of adding. and the result of subtracting is called the differenceThe result of subtracting.. Given any real numbers a, b, and c, we have the following properties of addition: Show
Given any real number a, a+0=0+a=a. Given any real number a, a+(−a) =(−a)+a=0. Given real numbers a, b and c, (a+b)+c=a+(b+c ). Given real numbers a and b, a+b=b+a. It is important to note that addition is commutative and subtraction is not. In other words, the order in which we add does not matter and will yield the same result. However, this is not true of subtraction. 5+10=10+515 =155−10≠10−5−5≠5 We use these properties, along with the double-negative property for real numbers, to perform more involved sequential operations. To simplify things, make it a general rule to first replace all sequential operations with either addition or subtraction and then perform each operation in order from left to right. Example 1Simplify: −10−(−10)+(−5). Solution: Replace the sequential operations and then perform them from left to right. −10−(−10)+(−5)=−10+10−5Replace−( −)withaddition(+).Replace+(−)withsubtraction(−) .=0−5=−5 Answer: −5 Adding or subtracting fractions requires a common denominatorA denominator that is shared by more than one fraction.. Assume the common denominator c is a nonzero integer and we have ac+bc=a+bcand ac−bc=a−bc Example 2Simplify: 29−1 15+845. Solution: First determine the least common multiple (LCM) of 9, 15, and 45. The least common multiple of all the denominators is called the least common denominatorThe least common multiple of a set of denominators. (LCD). We begin by listing the multiples of each given denominator: {9,18,27,36,45,54,63,72,81,90,…}Multiplesof9{15,30,45,60,75,90,…} Multiplesof15{45,90,135…}Mul tiplesof45 Here we see that the LCM(9, 15, 45) = 45. Multiply the numerator and the denominator of each fraction by values that result in equivalent fractions with the determined common denominator. 29−115+845=29⋅55−115⋅3 3+845=1045−345+845 Once we have equivalent fractions, with a common denominator, we can perform the operations on the numerators and write the result over the common denominator. =10−3+845= 1545 And then reduce if necessary, =15÷1545÷15=13 Answer: 13 Finding the LCM using lists of multiples, as described in the previous example, is often very cumbersome. For example, try making a list of multiples for 12 and 81. We can streamline the process of finding the LCM by using prime factors. 12=22⋅381=34 The least common multiple is the product of each prime factor raised to the highest power. In this case, LCM(12,81 )=22⋅34=324 Often we will find the need to translate English sentences involving addition and subtraction to mathematical statements. Below are some common translations. n+2Thesumofanumberand2.2−nThedifferenceof2andanumber.n−2 Here2issubtractedfromanumber. Example 3What is 8 subtracted from the sum of 3 and 12? Solution: We know that subtraction is not commutative; therefore, we must take care to subtract in the correct order. First, add 3 and 12 and then subtract 8 as follows: Perform the indicated operations. (3+12)− 8=(31⋅22+12)−8=(6+12)− 8=72−81⋅22=7−162=−92 Answer: −92 The result of multiplying real numbers is called the productThe result of multiplying. and the result of dividing is called the quotientThe result of dividing.. Given any real numbers a, b, and c, we have the following properties of multiplication:
Given any real number a, a⋅0=0⋅a=0. Given any real number a, a⋅1=1⋅a= a. Given any real numbers a, b and c, (a⋅b)⋅c=a⋅(b⋅c). Given any real numbers a and b, a⋅b=b⋅a. It is important to note that multiplication is commutative and division is not. In other words, the order in which we multiply does not matter and will yield the same result. However, this is not true of division. 5⋅10=10⋅550=50 5÷10≠10÷50.5≠2 We will use these properties to perform sequential operations involving multiplication and division. Recall that the product of a positive number and a negative number is negative. Also, the product of two negative numbers is positive. Example 4Multiply: 5(−3)(−2) (−4). Solution: Multiply two numbers at a time as follows: Answer: −120 Because multiplication is commutative, the order in which we multiply does not affect the final answer. However, when sequential operations involve multiplication and division, order does matter; hence we must work the operations from left to right to obtain a correct result. Example 5Simplify: 10÷(−2)(− 5). Solution: Perform the division first; otherwise the result will be incorrect. Notice that the order in which we multiply and divide does affect the result. Therefore, it is important to perform the operations of multiplication and division as they appear from left to right. Answer: 25 The product of two fractions is the fraction formed by the product of the numerators and the product of the denominators. In other words, to multiply fractions, multiply the numerators and multiply the denominators: ab ⋅cd=acbd Example 6Multiply: −45⋅2512. Solution: Multiply the numerators and multiply the denominators. Reduce by dividing out any common factors. −45⋅2512=− 4⋅255⋅12=−41⋅25551⋅123=−53 Answer: −53 Two real numbers whose product is 1 are called reciprocalsTwo real numbers whose product is 1.. Therefore, ab and ba are reciprocals because ab⋅ba=abab=1. For example, 23⋅32=66=1 Because their product is 1, 23 and 32 are reciprocals. Some other reciprocals are listed below: 58and85 7and17−45and−54 This definition is important because dividing fractions requires that you multiply the dividend by the reciprocal of the divisor. ab÷cd=ab cd⋅dcdc=ab⋅dc1=ab⋅ dc In general, ab÷cd=ab⋅dc=ad bc Example 7Simplify: 54÷35⋅12. Solution: Perform the multiplication and division from left to right. 54÷35⋅12=54⋅ 53⋅12=5⋅5⋅14⋅3⋅2= 2524 In algebra, it is often preferable to work with improper fractions. In this case, we leave the answer expressed as an improper fraction. Answer: 2524 Try this! Simplify: 12⋅34÷18. Answer: 3 Grouping Symbols and ExponentsIn a computation where more than one operation is involved, grouping symbols help tell us which operations to perform first. The grouping symbolsParentheses, brackets, braces, and the fraction bar are the common symbols used to group expressions and mathematical operations within a computation. commonly used in algebra are: ()Parentheses[]Brackets{}Braces Fractionbar All of the above grouping symbols, as well as absolute value, have the same order of precedence. Perform operations inside the innermost grouping symbol or absolute value first. Example 8Simplify: 2−(45−215). Solution: Perform the operations within the parentheses first. 2−(45−215)=2−(45⋅33 −215)=2−(1215−215)=2− (1015)=21⋅33−23=6−23 =43 Answer: 43 Example 9Simplify: 5−|4−(−3)||−3|−(5−7). Solution: The fraction bar groups the numerator and denominator. Hence, they should be simplified separately. 5−|4−(−3)||−3|−(5 −7)=5−|4+3||−3|−(−2)= 5−|7||−3|+2=5−73+2=−2 5=−25 Answer: −25 If a number is repeated as a factor numerous times, then we can write the product in a more compact form using exponential notationThe compact notation an used when a factor a is repeated n times.. For example, 5⋅5⋅5⋅5=5 4 The baseThe factor a in the exponential notation an. is the factor and the positive integer exponentThe positive integer n in the exponential notation an that indicates the number of times the base is used as a factor. indicates the number of times the base is repeated as a factor. In the above example, the base is 5 and the exponent is 4. Exponents are sometimes indicated with the caret (^) symbol found on the keyboard, 5^4 = 5*5*5*5. In general, if a is the base that is repeated as a factor n times, then When the exponent is 2 we call the result a squareThe result when the exponent of any real number is 2., and when the exponent is 3 we call the result a cubeThe result when the exponent of any real number is 3.. For example, 52=5⋅5=25“5squared”53 =5⋅5⋅5=125“5cubed” If the exponent is greater than 3, then the notation an is read, “a raised to the nth power.” The base can be any real number, (2.5)2=(2.5)( 2.5)=6.25(−23)3=(−23)(−23)(−2 3)=−827(−2)4=(−2)(−2)(−2) (−2)=16−24=−1⋅2⋅2⋅2⋅2=−16 Notice that the result of a negative base with an even exponent is positive. The result of a negative base with an odd exponent is negative. These facts are often confused when negative numbers are involved. Study the following four examples carefully:
The parentheses indicate that the negative number is to be used as the base. Example 10Calculate:
Solution: Here −13 is the base for both problems.
Answers:
Try this! Simplify:
Answers:
Order of OperationsWhen several operations are to be applied within a calculation, we must follow a specific order to ensure a single correct result.
Note that multiplication and division should be worked from left to right. Because of this, it is often reasonable to perform division before multiplication. Example 11Simplify: 53−24÷6⋅ 12+2. Solution: First, evaluate 53 and then perform multiplication and division as they appear from left to right. 53−24÷6⋅12+2=53−24÷6⋅12+2=125−24 ÷6⋅12+2=125−4⋅12+2=125−2+2=123+2=125 Multiplying first would have led to an incorrect result. Answer: 125 Example 12Simplify: −10−52+(−3)4. Solution: Take care to correctly identify the base when squaring. −10−52+(−3)4=−10−25+81=−35+81=46 Answer: 46 We are less likely to make a mistake if we work one operation at a time. Some problems may involve an absolute value, in which case we assign it the same order of precedence as parentheses. Example 13Simplify: 7−5|−22+(−3)2|. Solution: Begin by performing the operations within the absolute value first. 7−5|−22+(−3)2| =7−5|−4+9|=7−5|5|=7−5⋅5=7−25=−18 Subtracting 7−5 first will lead to incorrect results. Answer: −18 Try this! Simplify: −62−[−15−(− 2)3]−(−2)4. Answer: −45 Key Takeaways
Topic Exercises
Part A: Working with Real NumbersPerform the operations. Reduce all fractions to lowest terms. The formula d=|b−a| gives the distance between any two points on a number line. Determine the distance between the given numbers on a number line. Determine the reciprocal of the following. Perform the operations.
Part B: Grouping Symbols and ExponentsPerform the operations. Perform the operations.
Part C: Order of OperationsSimplify.
Part D: Discussion BoardAnswers
What are the properties of operations on real numbers?The following are the four main properties of real numbers:. Commutative property.. Associative property.. Distributive property.. Identity property.. What is the importance of properties of operations on the set of integers?These principles or properties help us to solve many equations. To recall, integers are any positive or negative numbers, including zero. Properties of these integers will help to simplify and answer a series of operations on integers quickly.
Why is it important to learn the properties of addition?The properties of addition are the set of rules that are used while adding two or more numbers. These properties are applicable to integers, fractions, decimals, and algebraic expressions. Using the properties of addition makes calculation easier and helps to solve complex problems in Math.
What properties of real numbers can be applied in the problem?To summarize, these are well-known properties that apply to all real numbers:. Additive identity.. Multiplicative identity.. Commutative property of addition.. Commutative property of multiplication.. Associative property of addition.. Associative property of multiplication.. Distributive property of multiplication.. |