Transcript 10 decimals

DECIMAL NUMBERS
MSJC ~ San Jacinto Campus
Math Center Workshop Series
Janice Levasseur
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.
tens
ones
Whole number part
Decimal point
10-2
10-3
10-4
Decimal part
10-5
Hundred-thousandths
hundreds
.
10-1
ten-thousandths
100
thousandths
101
hundredths
102
tenths
103
thousands
Place Value Chart
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 nonzero 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 tenthousandths”
8 . 0
3
0
4
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
4
2

0.4 is read “four tenths” 
10 5
Ex: Write 0.05 as a fraction
0.05 is read “five hundredths”
5
1


100 20
Ex: Write 0.007 as a fraction
0.007 is read “seven thousandths”
 7
1000
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
2
1
4.2 is read “four and two tenths”  4
4
10
5
• 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
3
Ex: Write as a decimal number
10
The fraction is read “three tenths”
Note: no “and”  no whole part
0.
3
Ex: Write as a decimal number 27
100
The fraction is read “twenty-seven hundredths”
Note: no “and”  no whole part
0.
2
7
33
Ex: Write as a decimal number 5
1000
The mixed number is read “five and thirty-three
thousandths”
5
.
0
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.
n
Rewrite
d
as long division.
d n
5
Ex: Write as a decimal number
6
6
.8
5 .0
4 8
2
1
3 3
0 0
0
8
2 0
18
2
Place a bar over the part
that repeats.
5/6 = 0.83
Is there an echo?
This will repeat
 repeating decimal number
5
Ex: Convert 2 to a decimal
8
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 0 0
4 8
2 0
1 6
4 0
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
+
16.113 0
15.21 00
2.0036
3 3 .3 2 6 6
put in 0 place holders
Ex: Subtract 24.024 – 19.61
1 13
-
1
24.024
19.610
put in 0 place holders
4 .4 1 4
Ex: Subtract 16 – 9.6413
15 9 9 9
1
16. 0000
- 9.6413
6 .3 5 8 7
put in the decimal point
put in 0 place holders
• 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
.0 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
1.2345 x 100 = 123.45
1 zero  move decimal
point 1 place to the right
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?
 Move the decimal point in 34.31
to the right 3 times
34 . 310.
 34.31 x 1000 = 34,310
3
Ex: Multiply 21 x 100
How many zeros are there in 100?
 Move the decimal point in 21
to the right 2 times
21 . 0 0 .
 21 x 100 = 2100
2
• 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
.
0 .5
0. 8 . 5
Why does this work?
Multiplication Property
of One, “Magic One”
Consider the fraction representation of the division:
0.85 0.85
10 8.5

x

Which is the equivalent
0 .5
0 .5
10
5
division we get after moving
the decimal point.
Ex: Divide 0.85  0.5
1.7
0 .5
0. 8 . 5
5
3 5
3 5
0
Ex: Set up the division
37.042  0.76
.
0 . 76
37. 0 4. 2
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.
4 8. 7 3
0 . 7 6 3 7 0 4. 2 0
30 4
6
6
6 4
0 8
5 6 2
5 3 2
3 0 0
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
12 3 4. 5 6
10 12345.6 0
10
23
20
34
30
45
40
56
50
60
12345.6
 10 = 1234.56
Ex: Divide 12345.6
 100
12 3 . 4 5 6
100 12345.6 0 0
100
234
200
345
300
45 6
40 0
560
50 0
60 0
12345.6
 100 = 123.456
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.