DECIMAL NUMBERS

MSJC ~ San Jacinto CampusMath Center Workshop

SeriesJanice Levasseur

DECIMAL NUMBERS

MSJC ~ San Jacinto CampusMath Center Workshop

SeriesJanice Levasseur

Introduction to Decimal Numbers

• A number written in decimal notation has 3 parts:

Whole # partThe decimal pointDecimal part

• The position of the digit in the decimal number determines the digit’s value.

Place Value Chart

.

ones

tens

hund

reds

thou

sand

s

tent

hs

hund

redt

hs

thou

sand

ths

ten-

thou

sand

ths

Hun

dred

-tho

usan

dths

Whole number part Decimal part

Decimal point

100101102103 10-1 10-2 10-3 10-4 10-5

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

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 “eight and three hundred four ten-

thousandths”

8 . 3 0 40

Ex: Write in standard form “seven hundred sixty-two thousandths”

Note: no “and” no whole part

0 . 7 6 2

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”

1005

52

Ex: Write 0.05 as a fraction

0.05 is read “five hundredths”

104

201

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

10007

Ex: Write 4.2 as a fractional numberNote: there’s a whole and decimal part Mixed number

4.2 is read “four and two tenths”

4 514

102

• 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 number103

The fraction is read “three tenths”

Note: no “and” no whole part 0 . 3

Ex: Write as a decimal number10027

The fraction is read “twenty-seven hundredths”

Note: no “and” no whole part

0 . 2 7

Ex: Write as a decimal number100033

5

The mixed number is read “five and thirty-three thousandths”

5 . 3 30

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. n

dd n

Ex: Write as a decimal number65

6 5 . 0

8 .

4 8

2

0

0

3

1 8

2

0

0

3

1 8

2 Is there an echo?

This will repeat repeating decimal number

= 0.83

Place a bar over the part that repeats.

5/6

Ex: Convert to a decimal

Notice the mixed number – whole & fraction part

8 5 . 0

6 .

4 82

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:

0

0

2

1 6

4

0

0

5

4 0

852

= 2.6252 5/8

• 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

16.11315.21 2.0036

000

put in 0 place holders

+

6623.33

Ex: Subtract 24.024 – 19.61

24.02419.610

put in 0 place holders-

414.4

31

1 1

Ex: Subtract 16 – 9.6413

16 9.6413

. put in the decimal point

put in 0 place holders0000

-

5 9 9 91

7853.

1

6

• 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

48.0 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

1.2345 x 100 = 123.45

1.2345 x 1000 = 1234.5

1 zero move decimal point 1 place to the right

2 zeros move decimal point 2 places to the right

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 . 310. 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 . 0 . 21 x 100 = 21000

• 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

80 0. 5.. 5

.

Why does this work? Multiplication Property of One, “Magic One”

Consider the fraction representation of the division:

5.085.0

1010

55.8 Which is the equivalent

division we get after moving the decimal point.

x5.085.0

Ex: Divide 0.85 0.5

80 0. 5.. 5

.1

53 5

7

3 5

0

Ex: Set up the division 37.042 0.76

0 4 20 37. .. 76

.

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.

3 7 0 4 20 . 7 6 .

.

0

4

3 0 4

6 6 4

8

6 0 8

5 6 2

7

5 3 2

3 0 0

3

2 2 8

the answer to the division (i.e. the rounded quotient) is 48.7

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

10 12345.6

.1

1023

2

2034

3

3045

4

4056

5

506

0

0

6

12345.6 10 = 1234.56

Ex: Divide 12345.6 100

100 12345.6

.1

10023

2

20034

4

3

30045

5

4

40 05 6

6

5

5 0 06 0

0

0

6

0

0

12345.6 100 = 123.456

So what have we seen?

12345.6 10 = 1234.56

12345.6 100 = 123.456

1 zero move decimal point 1 place to the left

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.