MSJC ~ San Jacinto Campus Math Center Workshop Series DECIMAL NUMBERS MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur

MSJC ~ San Jacinto Campus Math Center Workshop Series DECIMAL NUMBERS MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur

Introduction to Decimal Numbers A number written in decimal notation has 3 parts: Whole # part The decimal point Decimal part The position of the digit in the decimal number determines the digit’s value.

Place Value Chart Whole number part Decimal part Decimal point . 103 102 101 100 10-1 10-2 10-3 10-4 10-5 tens ones tenths thousands hundreds hundredths thousandths ten-thousandths Hundred-thousandths Whole number part Decimal part Decimal point

Writing a Decimal Number in Words Write the whole number part The decimal point is written “and” Write the decimal part as if it were a whole number Write the place value of the last non-zero digit Ex: Write 6.32 in words Six and thirty-two hundredths

Ex: Write 0.276 in words Ex: Write 10.0304 in words Zero and two hundred seventy-six thousandths Or two hundred seventy-six thousandths Ex: Write 10.0304 in words Ten and three hundred four Ten-thousandths

Writing Decimal Numbers in Standard Form Write the whole number part Replace “and” with a decimal point Write the decimal part so that the last non-zero digit is in the identified decimal place value Note: if there is no “and”, then the number has no whole number part.

Ex: Write in standard form “seven hundred sixty-two thousandths” Ex: Write in standard form “eight and three hundred four ten-thousandths” 8 3 0 4 . Ex: Write in standard form “seven hundred sixty-two thousandths” Note: no “and”  no whole part 7 6 2 0 .

Converting Decimal to Fractions To convert a decimal number to a fraction, read the decimal number correctly. Simplify, if necessary. Ex: Write 0.4 as a fraction 0.4 is read “four tenths”  Ex: Write 0.05 as a fraction 0.05 is read “five hundredths” 

0.007 is read “seven thousandths”  Ex: Write 0.007 as a fraction 0.007 is read “seven thousandths”  Note: the number of decimal places is the same as the number of zeros in the power of ten denominator Ex: Write 4.2 as a fractional number Note: there’s a whole and decimal part  Mixed number 4.2 is read “four and two tenths”  4

Your turn to try a few

Converting Fractions to Decimal Numbers (base 10 denominator) When the fraction has a power of 10 in the denominator, we read the fraction correctly to write it as a decimal number Ex: Write as a decimal number The fraction is read “three tenths” Note: no “and”  no whole part 0 . 3

. Ex: Write as a decimal number The fraction is read “twenty-seven hundredths” Note: no “and”  no whole part 2 7 0 . Ex: Write as a decimal number The mixed number is read “five and thirty-three thousandths” . 5 3 3

Converting fractions to decimals, take the numerator and divide by the denominator. If the fraction is a mixed number, put the whole number before the decimal. Rewrite as long division.

Ex: Write as a decimal number . 8 3 3 6 5 . 0 Place a bar over the part that repeats. 4 8 2 1 8 5/6 = 0.83 2 1 8 2 Is there an echo? This will repeat  repeating decimal number

Ex: Convert to a decimal Notice the mixed number – whole & fraction part  The decimal number will have a whole & decimal part The whole part is 2  2 . ________ Now convert the fraction 5/8 to determine the decimal part: . 6 2 5 2 5/8 = 2.625 8 5 . 0 4 8 2 1 6 4 4 0

Your turn to try a few

Rounding Decimal Numbers Rounding decimal numbers is similar to rounding whole numbers: Look at the digit to the right of the given place value to be rounded. If the digit to the right is > 5, then add 1 to the digit in the given place value and zero out all the digits to the right (“hit”). If the digit to the right is < 5, then keep the digit in the given place value and zero out all the digits to the right (“stay”).

Ex: Round 7.359 to the nearest tenths place Identify the place to be rounded to: Tenths Look one place to the right. What number is there? Compare the number to 5: 5 > 5  “hit” (add 1) 3 + 1 = 4 in the tenths place, zero out the rest 7.359 rounded to the nearest tenths place is 7.400 = 7.4

Ex: Round 22.68259 to the nearest hundredths place Identify the place to be rounded to: Hundredths Look one place to the right. What number is there? Compare the number to 5: 2 < 5  “stay” (keep) Keep the 8 and zero out the rest 22.68259 rounded to the nearest hundredths place is 22.68000 = 22.68

Ex: Round 1.639 to the nearest whole number Identify the place to be rounded to: ones Look one place to the right. What number is there? Compare the number to 5: 6 > 5  “hit” (add 1) 1 + 1 = 2 in the ones place, zero out the rest 1.639 rounded to the whole number is 2.000 = 2

Your turn to try a few

Decimal Addition & Subtraction To add and subtract decimal numbers, use a vertical arrangement lining up the decimal points (which in turn lines up the place values.) Ex: Add 16.113 + 15.21 + 2.0036 put in 0 place holders 16.113 15.21 + 2.0036 3 3 . 3 2 6 6

put in the decimal point 16 . 0000 - 9.6413 put in 0 place holders Ex: Subtract 24.024 – 19.61 1 1 3 1 24.024 put in 0 place holders - 19.61 4 . 4 1 4 Ex: Subtract 16 – 9.6413 1 9 9 9 5 1 put in the decimal point 16 . 0000 - 9.6413 put in 0 place holders 6 . 3 5 8 7

Your turn to try a few

Decimal Multiplication Decimal numbers are multiplied as if they were whole numbers. The decimal point is placed in the product so that the number of decimal places in the product is equal to the sum of the decimal places in the factors.

Ex: Multiply 1.2 x 0.04 Think 12 x 4  12 x 4 = 48 1.2 has 1 decimal place 0.04 has 2 decimal places Therefore the product of 1.2 and 0.04 will have 1 + 2 = 3 decimal places 48 .  1.2 x 0.04 = 0.048

Ex: Multiply 3.1 x 1.45 Think 31 x 145  31 x 145 =4495 3.1 has 1 decimal place 1.45 has 2 decimal places Therefore the product of 3.1 and 1.45 will have 1 + 2 = 3 decimal places 4 4 9 5 .  3.1 x 1.45 = 4.495

Multiply by Powers of 10 When multiplying by 10, 100, 1000, … Move the decimal in the number to the right as many times as there are zeros. 2.345 times 10, move the decimal one place to the right, 23.45

Ex: Multiply 1.2345 x 10 Think 12345 x 10  12345 x 10 = 123450 1.2345 has 4 decimal place 10 has 0 decimal places Therefore the product of 1.2345 and 10 will have 4 + 0 = 4 decimal places 123450 .  1.2345 x 10 = 12.3450 = 12.345

Ex: Multiply 1.2345 x 100 Think 12345 x 100  12345 x 100 = 1234500 1.2345 has 4 decimal place 100 has 0 decimal places Therefore the product of 1.2345 and 100 will have 4 + 0 = 4 decimal places 1234500 .  1.2345 x 100 = 123.4500 = 123.45

Ex: Multiply 1.2345 x 1000 Think 12345 x 1000  12345 x 1000 = 12345000 1.2345 has 4 decimal place 1000 has 0 decimal places Therefore the product of 1.2345 and 1000 will have 4 + 0 = 4 decimal places 12345000 .  1.2345 x 1000 = 1234.5000 = 1234.5

So what have we seen? 1.2345 x 10 = 12.345 1 zero  move decimal point 1 place to the right 1.2345 x 100 = 123.45 2 zeros  move decimal point 2 places to the right 1.2345 x 1000 = 1234.5 3 zeros  move decimal point 3 places to the right To multiply a decimal number by a power of 10, move the decimal point to the right the same number of places as there are zeros.

Ex: Multiply 34.31 x 1000 How many zeros are there in 1000? 3  Move the decimal point in 34.31 to the right 3 times 34 . 31 .  34.31 x 1000 = 34,310

Ex: Multiply 21 x 100 How many zeros are there in 100? 2  Move the decimal point in 21 to the right 2 times 21 . .  21 x 100 = 2100

Your turn to try a few

Decimal Division To divide decimal numbers, 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. Divide like with whole numbers.

Ex: Set up the division of 0.85 0.5 . 5 . 8 5 Why does this work? Multiplication Property of One, “Magic One” Consider the fraction representation of the division: Which is the equivalent division we get after moving the decimal point.

Ex: Divide 0.85 0.5 1 7 . . . 5 . 8 5 5 3 5 3 5

Ex: Set up the division 37.042 0.76 . . 76 37 . 0 4 2 .

Ex: Divide 37.042 0.76, round to the nearest tenth. When dividing decimals, we usually have to round the quotient to a specified place value. Ex: Divide 37.042 0.76, round to the nearest tenth.  the answer to the division (i.e. the rounded quotient) is 48.7 4 8 . 7 3 . 7 6 3 7 4 . 2 3 0 4 6 6 4 6 0 8 5 6 2 5 3 2 3 0 2 2 8

Divide by Powers of 10 When dividing by 10, 100, 1000, … Move the decimal in the number to the left as many times as there are zeros. 76.89 divided 10, move the decimal one place to the left, 7.689

Ex: Divide 12345.6 10 1 2 3 4 . 5 6 10 12345.6 10 2 3 20 12345.6 10 = 1234.56 3 4 30 4 5 40 5 6 50 6

Ex: Divide 12345.6 100 1 2 3 . 4 5 6 100 12345.6 100 23 4 12345.6 100 = 123.456 200 34 5 300 45 6 40 0 5 6 5 0 0 6 0

So what have we seen? 12345.6 10 = 1234.56 1 zero  move decimal point 1 place to the left 12345.6 100 = 123.456 2 zeros  move decimal point 2 places to the left To divide a decimal number by a power of 10, move the decimal point to the left the same number of places as there are zeros.