1.1 Review of Place Value

Transcription

1 1 1.1 Review of Place Value Our decimal number system is based upon powers of ten. In a given whole number, each digit has a place value, and each place value consists of a power of ten. Example 1 Identify the place value of each digit in the whole number 3,458,921,763. The place value of 3 on the far left is equal to 3 billions. The place value of 4 is equal to 4 hundred-millions. The place value of 5 is equal to 5 ten-millions. The place value of 8 is equal to 8 millions. The place value of 9 is equal to 9 hundred-thousands. The place value of 2 is equal to 2 ten-thousands. The place value of 1 is equal to 1 thousand. The place value of 7 is equal to 7 hundreds. The place value of 6 is equal to 6 tens. The place value of 3 on the far right is equal to 3 ones. Example 2 Identify the place values in the whole number 3,456,789,245,893. Then give the word name of this whole number. 3, 4 5 6, 7 8 9, 2 4 5, The word name of this number is three trillion, four hundred fifty-six billion, seven hundred eighty-nine million, two hundred forty-five thousand, eight hundred ninety-three.

2 2 Expanded Form The whole number 3,456,789,245,893 may be written as the sum 3,000,000,000, ,000,000, ,000,000, ,000,000, ,000, ,000, ,000, , , , This sum of the values of the place-value locations is known as the expanded form. Word Name of a Whole Number To write the word name of a given whole number, use the following procedure. PROCEDURE TO WRITE THE WORD NAME OF A WHOLE NUMBER 1. If the whole number does not already contain commas, then insert commas between each group of three digits, beginning at the far right of the whole number. 2. Starting at the left, write the word name of each group of digits that are separated by commas, and include the place-value name of the group. Note that the first group on the left may contain less than three digits. 3. If a group of three digits contains all zeros, do not include this group in the word name. 4. Include commas in the word name between each group of three digits. Note that the commas in the word name occur in the same locations as in the given whole number. Example 3 Write the word name for Begin by inserting commas between each group of three digits, beginning at the right is written as 340,789,003,320. The word name is three hundred forty billion, seven hundred eighty-nine million, three thousand, three hundred twenty. Example 4 Write the word name for 400,030,989. The word name is four hundred million, thirty thousand, nine hundred eightynine.

3 Example 5 Write the word name for 33,000, The word name is thirty-three million, three hundred forty-two. Note that the three zeroes representing the thousands group were not included in the word name. Place-Value Form of a Given Word Name The place-value form of a number is what we think of as the numeric form of a number. In order to write the place-value form, use the following procedure. PROCEDURE TO WRITE THE PLACE-VALUE FORM OF A WORD NAME 1. Write down the place-value form of each group of three digits and separate each group with a comma. Note that the leftmost group may contain fewer than three digits. 2. If a place-value group is not represented in the word name, be sure to represent this missing group with three zeroes in the place-value form. Example 6 Write the place-value form of three hundred four billion, forty-four thousand, two. The place-value form is 304,000,044,002. Note that there was not a millions group in the word name; therefore, a group of three zeroes was included in the place-value form. Example 7 Write the place-value form of eighty trillion, forty-four million, forty-two. The place-value form is 80,000,044,000,042. Note that there was neither a billions group nor a thousands group. A group of three zeroes was included in the place-value form in both the billions and the thousands locations. Also, the leftmost group contained only two digits.

4 1.2 Properties of Real Numbers 4 The number system that we use is called the set of real numbers. The set of real numbers satisfies certain properties called field properties. Many of the field properties of real numbers are used extensively in arithmetic and algebraic operations. We will look at some of these properties, and the ways in which they are used in arithmetic and algebra. THE COMMUTATIVE PROPERTY OF ADDITION The commutative property of addition states that two numbers or quantities may be added in any order. In other words, the order of addition may be switched. The following examples illustrate this property. Example: = Example: = Example: (3 + 4) + 5 = 5 + (3 + 4) Example: (62-34) + (27-12) = (27-12) + (62-34) Example: A + B = B + A where A and B represent real numbers. THE COMMUTATIVE PROPERTY OF MULTIPLICATION The commutative property of multiplication states that two numbers or quantities may be multiplied in either order. The following examples illustrate this property. Example: = Example: (2 + 16) 12 = 12 (2 + 16) Example: Example: A B = B A where A and B represent real numbers. 3 (A + 2) = (A + 2) 3 where A represents a real number. Example 1 Rewrite 45 (34-2) using the commutative property of multiplication. 45 (34-2) = (34-2) 45

5 5 THE ASSOCIATIVE PROPERTY OF ADDITION The associative property of addition states that when three numbers are added, parenthesis may be inserted around any two of the numbers and those two numbers may be added first. Example: = (2 + 4) + 7 = = 13 Also, = 2 + (4 + 7) = = 13 Example: = ( ) + 23 = = 69 Also, = 34 + ( ) = = 69 Example: X + Y + Z = (X + Y) + Z Also, X + Y + Z = X + (Y + Z) where X, Y, and Z represent real numbers. THE ASSOCIATIVE PROPERTY OF MULTIPLICATION The associative property of multiplication states that when three numbers or quantities are multiplied, parenthesis may be inserted around any two of the numbers and those two numbers may be multiplied first. Example: = (12 10) 5 = = 600 Also, = 12 (10 5) = = 600 Example: X Y Z = (X Y) Z Also, X Y Z = X (Y Z) where X, Y, and Z represent real numbers.

6 6 THE DISTRIBUTIVE PROPERTY OF MULTIPLICATION The distributive property of multiplication states that when multiplying by a sum or difference within parentheses, multiply by each number within the parentheses and then add or subtract these products. This property is demonstrated in the following examples. Example: 3 (4 + 5) = = = 27 Example: (7 2) 8 = = = 40 Example: 5 (3 2) = = = 5 Example: 10 ( ) = = = 270 Example: Example: 4 (A + B) = 4 A + 4 B where A and B represent real numbers. Rewrite 3 ( ) using the distributive property. 3 ( ) = The distributive property of multiplication is used in the process of multiplying whole numbers. Also, the distributive property is used extensively in algebraic operations. Example 2 Use the distributive property to rewrite 3 (A + B) where A and B represent real numbers. 3 (A + B) = 3 A + 3 B Example 3 Use the distributive property to rewrite (C D) 4. (C D) 4 = C 4 D 4 Multiplication of Whole Numbers Using the Distributive Property

7 7 When we multiply two whole numbers, we are really using the distributive property. In the following examples, the distributive property is used to multiply the two given whole numbers. Example 4 Multiply by using the distributive property of multiplication = (3 + 10) 45 = = = 585 Example 5 Multiply by using the distributive property of multiplication = (20 + 5) 40 = = = 1000 Example 6 Multiply by using the distributive property of multiplication = (20 + 2) 300 = = = 6600 MATH FACT The distributive property can be used to do multiplications mentally. This technique is useful in doing quick multiplications without a pen, paper, or calculator.

8 1.3 Operations with Whole Numbers Addition and Subtraction of Whole Numbers To add and subtract whole numbers, use the following procedure. 8 PROCEDURE TO ADD WHOLE NUMBERS 1. Line up the numbers vertically and add the columns of numbers from right to left. 2. Whenever a sum of digits equals or exceeds ten, carry 1 unit to the next place-value location for each group of ten. Example 1 Add 445, , , , = 18; therefore, 8 is written in the ones place, and 1 ten is 549,009 carried to the tens column. The sum of the tens column is 1 445, = 7. The sum of the hundreds column is = 8. The sum of the thousands column is = 22, and 2 1,002,878 is written in the thousands place, and 2 ten-thousands are carried to the ten-thousands column. The sum of the tenthousands column is = 10, and 0 is written in the ten-thousands place, and 1 is carried to the hundred-thousands column. The sum of the hundred-thousands column is = 10, and 0 is written in the hundred-thousands place, and 1 is carried to the millions place. The 1 in the millions place is carried down in the answer. PROCEDURE TO SUBTRACT WHOLE NUMBERS 1. Line up the numbers vertically and subtract the columns of numbers from right to left. 2. Whenever the upper number is smaller than the number being subtracted, borrow 1 from the upper number to the left. Since the number that you borrowed from is ten times as much, add 10 to the upper number. 3. If the number to the left is zero, borrow from a number farther to the left. For each 1 borrowed, add 10 to the placeholder immediately to the right. 4. To check your result, add the answer back to what was subtracted.

9 Example 2 62 Beginning at the right, 9 is subtracted from 2. Since there 39 is not enough to subtract from, 1 is borrowed from 6 in the tens place. The 1 group of ten is added to 2, resulting in in the ones column and 5 in the tens column. Now, subtracting from right to left results in 12-9 = 3 in the ones place and 5-3 = 2 in the tens place. Check: Example Beginning at the right, 5 is subtracted from 7. In the next column to the left, borrowing is necessary. Since there is a zero in the hundreds place, 1 is borrowed from the 1 in the thousands place and is added to the hundreds place as 10 hundreds. Then, 1 is borrowed from the 10 hundreds and added to 3 in the tens place. Now, subtracting from right 1 to left results in 7 5 in the ones place, 13 6 in the tens Check: place, 9 4 in the hundreds place and zero in the thousands place. Procedure for Multiplication of Whole Numbers The procedure used for multiplication is actually an application of the distributive property. PROCEDURE FOR MULTIPLICATION OF WHOLE NUMBERS 1. Line up the numbers vertically. 2. Multiply each digit of the bottom number by each digit of the top number. Each of these products is called a partial product. Add up all of the partial products. Example = = = Sum of partial products = 4488

10 10 Example = = = = = = 18,000 23,115 Sum of Partial Products Note that in the previous examples, every partial product was written down. This multiplication may be done in a more condensed form by combining partial products as is shown here. 345 Here, is multiplied out to a product of 2415 and is multiplied out to obtain a product of 20,700. The two partial products are then added ,115 Procedure for Division of Whole Numbers The procedure for division of whole numbers involves four steps: divide, multiply, subtract, and carry down. The student may remember this sequence by remembering the first letters of the words in the phrase, Do Math, Stay Cool! One should be familiar with the terminology of division. The number that is being divided is called the dividend, the number that you are dividing by is called the divisor, and the result of division without the remainder is called the quotient. The remainder is not considered as part of the quotient and the remainder is written over the divisor as a fraction and added to the quotient. Thus, a typical division problem has the following components: Dividend Divisor = Quotient + Remainder / Divisor Quotient RemainderDivisor Divisor Dividend

11 11 The procedure to divide two whole numbers is shown on the following two pages. PROCEDURE TO DIVIDE TWO WHOLE NUMBERS 1. Divide the divisor into the leftmost group of digits of the dividend that is as large or larger than the divisor. Divide and place the quotient of this first division in its proper place-value location. Example: Perform the division Here, 31 is divided into 235, the group of digits that is big enough for 31 to divide into is 7 with a remainder. Write 7 in its proper location above the dividend. 2. Multiply this result of division by the divisor and place the result under the digits that you divided into. Thus, in the example, 7 is multiplied by 31 and the result is placed beneath In this example, 7 31 = Subtract the product obtained in the last step from the group of digits directly above it. In the example, 217 is subtracted from 235 to obtain a difference of Carry Down the digit to the right. In the example, 5 is carried down and added to 180, and the sum is

12 12 PROCEDURE TO DIVIDE TWO WHOLE NUMBERS - CONTINUED 5. Repeat the process: Divide Multiply Subtract Carry Down, starting with the division of the divisor into the difference obtained after subtracting. Repeat this process until the difference is smaller than the divisor Here, 185 divided by 31 results in 5. 5 multiplied by 31 equals r The result of subtracting 155 from 185 is 30. Since 30 is less than 31, 30 is the remainder. The remainder may be written as 30/31. The final answer is 75 r 30 or Example 6 Divide r Divide 313 by 52. The result, 6, is placed in the hundreds place-value location. Multiply the number 6 by 52 to obtain the product 312. Subtract the digits 312 from 313 to get the difference of 1. Carry down the digit 9 and combine with 1 to obtain 19. Repeat the fourstep process by dividing 52 into 19. Since 52 does not divide into 19, 0 is placed in the tens place. Multiply 0 by 52 to obtain 0. Subtract 0 from 19, carry down the 1, and then divide again. The last division results in 3 which is written in the ones place. Multiply 3 52, subtract this product from 191, leaving a remainder of 35. The final answer is 603 r 35 or

13 13 Rounding To Round a whole number, use the procedure that is given here. PROCEDURE TO ROUND A NUMBER 1. Find the place-value location to be rounded. 2. Look at the digit to the right. If the digit to the right is greater than or equal to 5, then add one to the first digit. If the digit to the right is less than or equal to 4, then leave the first digit as it is. 3. Insert zero placeholders in all of the place-value locations to the right of the rounded digit. Example 7 Round 3,406,822 to the nearest thousand. Add 1 to 6 3,406,822 rounds to 3,407,000 Look at 8 The thousands place is 6. Look at the digit to the right which is 8. Since 8 is greater than or equal to 5, we add 1 to 6 and round 6 up to 7, then we insert zero placeholders to the right of the rounded digit. Example 8 Round 7,996,426 to the nearest ten thousand. Add 1 to 9 7,996,426 rounds to 8,000,000 Look at 6 The ten thousands place is 9. Look at the digit to the right which is 6. Since 6 is greater than or equal to 5, we add 1 to 9 and round 9 up to 10. This means that 1 is carried to the next column and added to 9 in the hundred thousands place. Another 1 is carried and added to the millions place, making 8 million. Zero placeholders are inserted to the right of 8.

14 14 Estimation The ability to estimate a sum, difference, product or quotient is perhaps more useful to the average person than almost any other math skill for the following reasons. Estimation is used to do quick approximations when use of pencil and paper or a calculator is not practical. Think of all the times you have been shopping and you have wondered whether that item on sale was really a good deal. Beforehand estimation of an answer calculated with pencil and paper or a calculator allows a person to detect any error that may have been made. By estimating the answer first, we know what the answer should approximately be equal to. To estimate a sum, difference, product, or quotient, the following strategy usually works well. PROCEDURE FOR ESTIMATION 1. Round each number to its largest place. 2. Perform operations on the rounded numbers. Example 9 Estimate the division rounds to 500 and 51 rounds to 50. The estimate is = 10. The actual answer is 9 r 38 or Example 10 Estimate the sum and compare the estimate to the actual answer. 47 rounds to 50, 69 rounds to 70, and 92 rounds to 90. The estimate is = 210. The actual answer is 208.

15 15 Example 11 Estimate the product , and compare the estimate to the actual answer. 102 rounds to 100, and 29 rounds to 30. The estimate is = The actual answer is Example 12 Estimate the division , and compare the estimate to the actual answer rounds to 2000 and 995 rounds to The estimate is = The actual answer is 2 r 219 or

16 Exponents and Order of Operations Definition of an Exponent An exponent is a superscript following another number called the base. The exponent indicates that the base is multiplied by itself by the number of times indicated by the exponent. 3 Example 1 4 = = 64 In this example, the base is 4 and the exponent is 3. The exponent of 3 indicates that 4 is multiplied by itself 3 times. 5 Example 2 2 = = 32 6 Example 3 10 = = 1,000,000 9 Example 4 10 = = 1,000,000,000 3 Example 5 Evaluate = (2 2 2) + 42 = = Example 6 Evaluate = ( ) + (10 10 ) = = 1100 Exponents and Powers of Ten As you may have noticed in the previous examples involving powers of 10, when 10 is raised to a power, the result consists of a digit of 1 followed by zero placeholders. The number of zeroes is equal to the value of the exponent on ten. 11 Example 7 10 = = 100,000,000,000 Eleven Zeroes Notice that the result of multiplying out the eleven factors of ten consists of a digit of 1 followed by 11 zero placeholders.

17 17 5 Example = ,000 = 3,400,000 Multiplication and Division by Powers of Ten When multiplying or dividing a decimal by a power of ten, the decimal point is moved to the right or the left by the number of places equal to the value of the exponent on ten. Note that the decimal point must be inserted to the right of the ones place-value location of whole numbers before this property can be applied. 5 5 Example = = PLACES 8 Example = PLACES 8 8 Example = = PLACES Remember the following two rules: MULTIPLICATION BY A POWER OF TEN When multiplying by a power of ten, move the decimal point to the right the number of places indicated by the exponent. Note that for whole numbers the decimal point must be inserted to the right of the ones place. 6 6 Example: = = 45,000,000

18 18 DIVISION BY A POWER OF TEN When dividing by a power of ten, move the decimal point to the left the number of places indicated by the exponent. Note that for whole numbers the decimal point must be inserted to the right of the ones place-value location. Order of Operations 7 7 Example: = = Multiple arithmetic operations are performed in a precise order consistent with the accepted order of operations. For example, the expression is correctly evaluated to yield a result of 14. However, if the operations in are performed from left to right, then the incorrect result is 18 since is 12, 12 2 = 6, = 9, and 9 2 is equal to 18. MATH FACT Correct use of order of operations is essential in the use of mathematical formulas. Proper use of order of operations is critical in manipulating algebraic expressions. The correct order of operation is given here. ORDER OF OPERATIONS 1. Perform all operations within parentheses first. 2. Perform exponent operations before multiplying, dividing, adding or subtracting. 3. Divide and multiply from left to right in the expression before adding or subtracting. 4. Last, subtract and add from left to right in the expression. 3 Example 12 Evaluate

19 19 Raise 2 to the third power first. Then, multiply 5 8. Then add = = = 43 Example 13 Evaluate 2 (3 + 6) Perform the operations within the parentheses first. Then, multiply. 2 (3 + 6) = 2 9 = 18 2 Example 14 Evaluate 3 + ( ) 8 2 First, begin evaluating the expression within the parentheses. Since the multiplication operation takes priority over the addition operation, the product 2 3 = 6 is obtained first. Then the sum is obtained ( ) 8 2 = (4 + 6) 8 2 = (10) Now, the exponent operation is performed, 10 = (10) 8 2 = Since division takes priority over addition or subtraction, 8 2 is replaced with =

20 20 Finally, the addition and subtraction operations are performed from left to right to obtain the final answer of = = Example 15 Evaluate [(4 3) + 2] In this example, there are two sets of parentheses, one of which is nested within the other. When evaluating expressions containing nested parentheses, do operations within the innermost parentheses first. Thus, begin the calculations by subtracting [(4 3) + 2] = [(1) + 2] 5 Perform the exponent operation 1 = = 1. Then add this result to [(1) + 2] = [ 1 + 2] = [ 3 ] = Now there are two exponent operations to be performed. Perform these from 3 2 left to right and replace 2 with 8 and replace 3 with = Evaluate 3 8 = 24. Then add and subtract from left to right to obtain the final result of = =

21 = 19 Example 16 Evaluate ( ) 2. (4 3 2 ) In this example, the fraction bar indicates division. This expression is equivalent to [( ) ] (4 + 3 ). Thus, the expression in the numerator must be evaluated, and the expression in the denominator must be evaluated. The two results must then be divided. = = = = ( ) 2 (4 3 2 ) (1 4 3) 2 (4 3 2 ) (1 12) 2 (4 3 2 ) (13) 2 (4 3 2 ) 169 (4 3 2 ) The exponent operation, 2 2 = 4 is performed first. Four is multiplied by three. One is added to twelve. 13 is squared to yield = 169. Now, all of the operations in the numerator are completed. 169 = Operations are now performed in the denominator. (4 9) The exponent operation, 3 3 = 9 is performed first. 169 = All operations in both the numerator and the (13) denominator are complete. The final operation consists of dividing the numerator by the denominator. = = 13

22 22 ORDER OF OPERATIONS AND YOUR CALCULATOR Scientific and graphing calculators will perform operations in their correct order and return the correct result. Example: can be input as it is given here and the calculator will automatically perform the operations in the proper order and a correct result of 21 will be returned on both a scientific and a graphing calculator. Non-scientific calculators will often return an incorrect answer if numbers and operations are input from left to right. Example: If is input, the non-scientific calculator will perform operations from left to right with no preference for multiplication. The calculator will first add to obtain 7, then multiply 7 5 to obtain 35, and then subtract 2 to return an incorrect result of 33. Evaluation of Formulas Many real-world applications of mathematics involve use of formulas. A formula consists of a mathematical expression consisting of letters or symbols which are combined with mathematical operations. PROCEDURE TO EVALUATE A FORMULA Replace the letters or symbols with their numerical equivalents. Example 17 The formula for the area of a rectangle is Area = L W where L represents length and W represents width. If the length of a rectangle is 40 inches and the width is 22 inches, what is the area?

23 23 By replacing L with 40 and W with 22, the area is equal to = 880 square inches. Example 18 The formula for the surface area of a rectangular solid is Surface Area = 2 L H + 2 L W + 2 W H where L represents the length, H represents the height, and W represents the width. If L = 8 inches, W = 5 inches, and H = 3 inches, what is the surface area of the rectangular solid? To evaluate this formula, replace L, H, and W with their numerical equivalents. Surface Area = 2 L H + 2 L W + 2 W H = = = 158 square inches Note that the multiplication operations were done before the additions. In all formulas, the correct order of operations must be used. MATH FACT The correct order of operations must be followed when evaluating formulas.