Decimals and Decimal Operations

1 Introduction to Decimals Reading and Writing Decimals : Note: There is a relationship between fractions and numbers written in Decimal notation. Three-tenths 310 1 zero 1 Decimal place Three-hundredths 3100 2 zeros 2 Decimal places Three-thousandths 31000 3 zeros 3 Decimal placesA number written in Decimal notation has three parts. 351 . 7089 Whole-number part Decimal Point Decimal Part A number written in Decimal notation is often called simply a Decimal . The position of a digit in a Decimal determines the digit s place value. In the Decimal 6,584, , the position of the digit 2 determines that its place value is thousandths.

2 When writing a Decimal in words, write the Decimal part as if it were a whole number; then name the place value of the last digit. Six thousand four hundred eighty-one ten-thousandths Five hundred forty-nine and two hundred thirty-eight thousandths (The Decimal point is read as and ) To write a Decimal in standard form, zeros may have to be inserted after the Decimal point so that the last digit is in the given place-value position. Five and thirty-eight hundredths (8 is in the hundredths place.) Nineteen and four thousandths (Insert two zeros so that the 4 is in the thousandths place.) Seventy-one ten-thousandths (Insert two zeros so that 1 is in the ten-thousandths place.)

3 Example 1 Write in words. Solution Three hundred seven and four thousand twenty-seven ten-thousandths Example 2 Write six hundred seven and seven hundred eight hundred-thousandths in standard form. Solution Rounding Decimals : Rounding Decimals is similar to rounding whole numbers except that the digits to the right of the given place value are dropped instead of being replaced by zeros. If the digit to the right of the given place value is less than 5, drop the digit and all digits to the right. If the digit to the right of the given place value is greater than or equal to 5, increase the number in the given place value by 1 and drop all digits to its right.

4 Round to the nearest hundredth. rounded to the nearest hundredth is Example 3 Round to the nearest ten-thousandth. Solution Example 4 Round to the nearest hundred-thousandth. Solution Addition and Subtraction of Decimals To add or subtract Decimals , write the numbers so that the Decimal points are on a vertical line, placing zeros as place holders if necessary. Add or subtract as for whole numbers, and write the Decimal point in the sum or difference directly below the Decimal points in the addends or subtrahends. Add: Subtract: Subtract: Note: By placing the Decimal points on a vertical line, we make sure that digits of the same place value are added or subtracted.

5 Multiplication of Decimals To multiply Decimals , multiply the numbers as in whole numbers. Write the Decimal point in the product so that the number of Decimal places in the product is the sum of the Decimal places in the factors. Multiply: Given place value 9 > 5 Increase 7 by 1 and drop all digits to the right of 7. Given place value 5 = 5 Given place value 1 < 5 1284 642 Multiply: Note: Two zeros must be inserted between the 2 and the Decimal point so that there are 5 Decimal places in the product. Multiplying by Powers of 10: To multiply a Decimal by a power of 10 (10, 100, 1000.)

6 , move the Decimal point to the right the same number of places as there are zeros in the power of 10. 10 100 1000 10,000 38,925. 100,000 389,250. 10 10 10 10 38,925. 10 389,250. Note: If a power of 10 is written in exponential notation, the exponent indicates how many places to move the Decimal point. Division of Decimals To divide Decimals , move the Decimal point in the divisor to the right to make the divisor a whole number. Move the Decimal point in the dividend the same number of places to the right. Place the Decimal point in the quotient directly over the Decimal point in the dividend, and then divide as in whole numbers.

7 Divide: Move the Decimal point 2 places to the right in the divisor and then in the dividend. Place the Decimal point in the quotient. 1300 227 5 227 5 0 Moving the Decimal point the same number of Decimal places in the divisor and dividend does not change the value of the quotient, because this process is the same as multiplying the numerator and denominator of a fraction by the same number. In the example above, .. Dividing by Powers of 10: To divide by powers of 10 (10, 100, 1000, ..), move the Decimal point to the left the same number of places as there are zeros in the power of 10.

8 10 10 100 10 1 Decimal place 2 Decimal places 3 Decimal places 3 Decimal places 2 Decimal places 5 Decimal places 1000 10 10,000 10 Note: If the power of 10 is written in exponential notation, the exponent indicates how many places to move the Decimal point. Comparing and Converting Fractions and Decimals Convert Fractions to Decimals : Every fraction can be written as a Decimal . To write a fraction as a Decimal , divide the numerator of the fraction by the denominator. The quotient can be rounded to the desired place value. Convert to Decimal . rounded to the nearest hundredth is rounded to the nearest thousandth is rounded to the nearest ten-thousandth is Convert 3 to a Decimal .

9 Round to the nearest thousandth. 3 3 rounded to the nearest thousandth is Convert Decimals to Fractions: To convert a Decimal to a fraction, remove the Decimal point and place the Decimal part over a denominator equal to the place value of the last digit in the Decimal . Reduce if possible. 47100 2751000 1140 745100 7920 1623100 1623 100 503 1100 16 Identify Order Relations Between 2 Decimals or a Decimal & a Fraction: Decimals , like whole numbers and fractions, can be graphed as points on the number line. The number line can be used to show the order of Decimals . A Decimal that appears to the right of a given number is greater than the given number.

10 A Decimal that appears to the left of a given number is less than the given number. Note: 3, , and represent the same number. Find the order relation between and 38 38 hundredths thousandths hundredths hundredths Practice Problems 1. Find the quotient of and 2. Find the sum of , , , and 3. Place the correct symbol, or , between the two numbers. 4. Write in words. 5. Round to the nearest hundred-thousandth. 6. Convert 2 to a Decimal . Round to the nearest hundredth. 7. Convert to a fraction. 8. Add: 9. Write thirty-four and twenty-five thousandths in standard form.