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## Decimals

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**Decimals**• Introductions to Decimals • Adding & Subtracting Decimals • Multiplying Decimals & Circumference of a Circle • Dividing Decimals • Fractions, Decimals, & Order of Operations • Equations Containing Decimals**Like fractional notation, decimal notation is used to denote**a part of a whole. Numbers written in decimal notation are called decimal numbers, or simply decimals. The decimal 16.734 has three parts. 16.743 Whole- number part Decimal part Decimal point**1**1 1 10 1000 100,000 1 1 100 10,000 The position of each digit in a number determines its place value. hundred-thousandths hundreds tens tenths hundredths thousandths ten-thousandths ones 1 6 7 3 4 Place Value 100 10 1 decimal point**Notice that the value of each place is**of the value of the place to its left. 1 10 Martin-Gay, Prealgebra, 5ed**3**100 16.734 The digit 3 is in the hundredths place, so its value is 3 hundredths or . Martin-Gay, Prealgebra, 5ed**Writing (or Reading) a Decimal in Words**Step 1. Write the whole-number part in words. Step 2. Write “and” for the decimal point. Step 3. Write the decimal part in words as though it were a whole number, followed by the place value of the last digit. Martin-Gay, Prealgebra, 5ed**decimal part**whole-number part Writing a Decimal in Words Write the decimal 143.056 in words. 143.056 one hundred forty-three and fifty-six thousandths Martin-Gay, Prealgebra, 5ed**whole-number part**decimal decimal part A decimal written in words can be written in standard form by reversing the procedure. Writing Decimals in Standard Form Write one hundred six and five hundredths in standard form. one hundred six and five hundredths 5 must be in the hundredths place 106 . 05**thousandths place**Helpful Hint • When writing a decimal from words to decimal notation, make sure the last digit is in the correct place by inserting 0s after the decimal point if necessary. • For example, • three and fifty-four thousandths is 3.054 Martin-Gay, Prealgebra, 5ed**Writing Decimals as Fractions**Once you master writing and reading decimals correctly, then you write a decimal as a fraction using the fractions associated with the words you use when you read it. 0.9 is read “nine tenths” and written as a fraction as Martin-Gay, Prealgebra, 5ed**21**100 11 1000 twenty-one hundredths and written as a fraction as 0.21 is read as 0.011 is read as eleven thousandths and written as a fraction as Martin-Gay, Prealgebra, 5ed**37**29 = 0 . 37 = 0 . 029 100 1000 2 zeros 3 decimal places 3 zeros 2 decimal places Notice that the number of decimal places in a decimal number is the same as the number of zeros in the denominator of the equivalent fraction. We can use this fact to write decimals as fractions.**3**7 10 10 Comparing Decimals One way to compare decimals is to compare their graphs on a number line. Recall that for any two numbers on a number line, the number to the left is smaller and the number to the right is larger. To compare 0.3 and 0.7 look at their graphs. 0 0.3 0.7 1 0.3 < 0.7 or 0.7 > 0.3 Martin-Gay, Prealgebra, 5ed**Comparing decimals by comparing their graphs on a number**line can be time consuming, so we compare the size of decimals by comparing digits in corresponding places. Martin-Gay, Prealgebra, 5ed**Comparing Two Positive Decimals**Compare digits in the same places from left to right. When two digits are not equal, the number with the larger digit is the larger decimal. If necessary, insert 0s after the last digit to the right of the decimal point to continue comparing. Compare hundredths place digits 35.638 35.657 3 5 < 35.638 < 35.657 Martin-Gay, Prealgebra, 5ed**Helpful Hint**For any decimal, writing 0s after the last digit to the right of the decimal point does not change the value of the number. 8.5 = 8.50 = 8.500, and so on When a whole number is written as a decimal, the decimal point is placed to the right of the ones digit. 15 = 15.0 = 15.00, and so on Martin-Gay, Prealgebra, 5ed**Rounding Decimals**We round the decimal part of a decimal number in nearly the same way as we round whole numbers. The only difference is that we drop digits to the right of the rounding place, instead of replacing these digits by 0s. For example, 63.782 rounded to the nearest hundredth is 63.78 Martin-Gay, Prealgebra, 5ed**Rounding Decimals To a Place Value to the Right of the**Decimal Point Step 1. Locate the digit to the right of the given place value. Step 2. If this digit is 5 or greater, add 1 to the digit in the given place value and drop all digits to the right. If this digit is less than 5, drop all digits to the right of the given place. Martin-Gay, Prealgebra, 5ed**tenths place**digit to the right Rounding Decimals to a Place Value Round 326.4386 to the nearest tenth. Locate the digit to the right of the tenths place. 326.4386 Since the digit to the right is less than 5, drop it and all digits to its right. 326.4386 rounded to the nearest tenths is 326.4 Martin-Gay, Prealgebra, 5ed**Adding and Subtracting Decimals**Section 5.2**Adding or Subtracting Decimals**Step 1. Write the decimals so that the decimal points line up vertically. Step 2. Add or subtract as for whole numbers. Step 3. Place the decimal point in the sum or difference so that it lines up vertically with the decimal points in the problem. 22 Martin-Gay, Prealgebra, 5ed**85.00- 13.26**Helpful Hint Recall that 0s may be inserted to the right of the decimal point after the last digit without changing the value of the decimal. This may be used to help line up place values when adding or subtracting decimals. 85 - 13.26 becomes two 0s inserted 71.74 23**Helpful Hint**Don’t forget that the decimal point in a whole number is after the last digit. 24 Martin-Gay, Prealgebra, 5ed**Estimating Operations on Decimals**Estimating sums, differences, products, and quotients of decimal numbers is an important skill whether you use a calculator or perform decimal operations by hand. Martin-Gay, Prealgebra, 5ed**Estimating When Adding Decimals**Add 23.8 + 32.1. Exact Estimate rounds to rounds to This is a reasonable answer. Martin-Gay, Prealgebra, 5ed**Helpful Hint**When rounding to check a calculation, you may want to round the numbers to a place value of your choosing so that your estimates are easy to compute mentally. Martin-Gay, Prealgebra, 5ed**Evaluating with Decimals**Evaluate x + y for x = 5.5 and y = 2.8. Replacexwith5.5andywith 2.8 inx + y. x+y = ( ) + ( ) 5.5 2.8 = 8.3 28 Martin-Gay, Prealgebra, 5ed**Multiplying Decimals and Circumference of a Circle**Section 5.3**21**7 3 x 1000 10 100 Multiplying Decimals Multiplying decimals is similar to multiplying whole numbers. The difference is that we place a decimal point in the product. 0.7 x 0.03 = = = 0.021 1 decimal place 2 decimal places 3 decimal places 30 Martin-Gay, Prealgebra, 5ed**Multiplying Decimals**Step 1. Multiply the decimals as though they were whole numbers. Step 2. The decimal point in the product is placed so the number of decimal places in the product is equal to the sumof the number of decimal places in the factors. 31 Martin-Gay, Prealgebra, 5ed**Estimating when Multiplying Decimals**Multiply 32.3 x 1.9. Exact Estimate rounds to rounds to This is a reasonable answer. Martin-Gay, Prealgebra, 5ed**Multiplying Decimals by Powers of 10**There are some patterns that occur when we multiply a number by a power of ten, such as 10, 100, 1000, 10,000, and so on. 33 Martin-Gay, Prealgebra, 5ed**76.543 x 10 = 765.43**76.543 x 100 = 7654.3 76.543 x 100,000 = 7,654,300 Multiplying Decimals by Powers of 10 Decimal point moved 1 place to the right. 1 zero Decimal point moved 2 places to the right. 2 zeros Decimal point moved 5 places to the right. 5 zeros The decimal point is moved the same number of places as there are zeros in the power of 10. 34 Martin-Gay, Prealgebra, 5ed**Multiplying by Powers of 10 such as 10, 100, 1000 or 10,000,**. . . Move the decimal point to theright thesame number of places as there arezerosin the power of 10. Multiply: 3.4305 x 100 Since there are two zeros in 100, move the decimal place two places to the right. 3.4305 x 100 = 3.4305 = 343.05 35 Martin-Gay, Prealgebra, 5ed**Multiplying by Powers of 10 such as 0.1, 0.01, 0.001,**0.0001, . . . Move the decimal point to theleft thesame number of places as there aredecimal placesin the power of 10. Multiply: 8.57 x 0.01 Since there are two decimal places in 0.01, move the decimal place two places to the left. 8.57 x 0.01 = 008.57 = 0.0857 Notice that zeros had to be inserted. 36**Finding the Circumference of a Circle**The distance around a polygon is called itsperimeter. The distance around a circle is called thecircumference. This distance depends on theradiusor thediameter of the circle. 37 Martin-Gay, Prealgebra, 5ed**Circumference of a Circle**r d Circumference= 2·p·radius or Circumference =p·diameter C= 2prorC=pd 38 Martin-Gay, Prealgebra, 5ed**22**7 p The symbolpis the Greek letter pi, pronounced “pie.” It is a constant between 3 and 4. A decimal approximation forpis 3.14. A fraction approximation forpis. 39 Martin-Gay, Prealgebra, 5ed**4 inches**Find the circumference of a circle whose radius is 4 inches. C= 2pr= 2p·4= 8pinches 8p inches is the exact circumference of this circle. If we replace with the approximation 3.14, C= 8 8(3.14) = 25.12 inches. 25.12 inches is the approximate circumference of the circle. 40**Dividing Decimals**Section 5.4**quotient**divisor dividend 52.92 63 Division of decimal numbers is similar to division of whole numbers. The only difference is the placement of a decimal point in thequotient. If the divisor is a whole number, divide as for whole numbers; then place the decimal point in the quotient directly above the decimal point in the dividend. 0 8 4 -504 25 2 -252 0 42**divisor**dividend 6 3 . 52 . 92 63 . 529 . 2 63 52 9.2 If the divisor isnota whole number, we need to move the decimal point to the right until the divisor is a whole number before we divide. 8 4 - 504 25 2 -252 0 43 Martin-Gay, Prealgebra, 5ed**Dividing by a Decimal**Step 1. Move the decimal point in the divisor to the right until the divisor is a whole number. Step 2. Move the decimal point in the dividend to the right the same numberofplaces as the decimal point was moved in Step 1. Step 3. Divide. Place the decimal point in the quotient directly over the moved decimal point in the dividend. 44 Martin-Gay, Prealgebra, 5ed**Estimating When Dividing Decimals**Divide 258.3 ÷ 2.8 Exact Estimate rounds to This is a reasonable answer. Martin-Gay, Prealgebra, 5ed**456.2**= 45 . 62 10 1 zero 456.2 = 0 . 4562 1 , 000 3 zeros There are patterns that occur when dividing by powers of 10, such as 10, 100, 1000, and so on. The decimal point moved 1 place to the left. The decimal point moved 3 places to the left. The pattern suggests the following rule. 46 Martin-Gay, Prealgebra, 5ed**Notice that this is the same pattern as multiplying by**powers of 10 such as 0.1, 0.01, or 0.001. Because dividing by a power of 10 such as 100 is the same as multiplying by its reciprocal , or 0.01. To divide by a number is the same as multiplying by its reciprocal. Dividing Decimals by Powers of 10 such as 10, 100, or 1000, . . . Move the decimal point of the dividend to theleftthe same number of places as there are zerosin the power of 10. 47**Section 5.5**Fractions, Decimals, and Order of Operations**To write a fraction as a decimal, divide the numerator by**the denominator. Writing Fractions as Decimals 49 Martin-Gay, Prealgebra, 5ed**To compare decimals and fractions, write the fraction as an**equivalent decimal. Comparing Fractions and Decimals Compare 0.125 and Therefore, 0.125 < 0.25 50 Martin-Gay, Prealgebra, 5ed