Place Value
Period
Powers of Ten
Decimal
Decimal Point
Fraction
Percent
Tenths
Hundredths
Thousandths
Ten Thousandths
Millionths
Ten Thousandths
•
•
•
•
•
•
Standard Form
Written Form
Expanded Form
Pictures
Fractions
Percent
In our place value system, the value of a digit depends on its
place, or position, in the number.
Each place has a value of 10 times the place to its right.
A number in standard form is separated into groups of three
digits using spaces (USA uses commas). Each of these
groups is called a period.
Not every number is a whole number. Our decimal system lets
us write numbers of all types and sizes, using a symbol called
the decimal point.
●
●
●
●
Millionths
●
Hundred
Thousandths
●
Ten Thousandths
Thousandths
Hundredths
Tenths
●
Ones
Tens
Hundreds
Thousands
Ten Thousands
Hundred Thousands
Millions
Ten Millions
Hundred Millions
Billions
Smaller
Bigger
Numbers Get
Numbers Get
Units/Ones
Thousands
Millions
Millionths
1
Hundred
Thousandths
1
Ten
Thousandths
•
Thousandths
Hundredths
Tenths
Decimal
Ones
1
•
•
0.1
0.01
0.001
0.0001
0.00001
0.000001
1
10
1
100
1
1 000
1
10 000
1
100 000
1
1 000 000
•
1
10
1
102
1
103
1
104
1
105
1
106
As you move right from the decimal point, each place
value is divided by 10.
125 . 578
hundreds tens ones
tenths hundredths thousandths
Example:
Read 38.7425
Solution:
Step 1: Values to the left of the decimal point are
greater than one.
38 means 3 tens and 8 ones.
Step 2: The word name of the decimal is determined by
the place value of the digit in the last place on the right.
The last digit (5) is in the ten-thousandth place.
38.7425 is read as thirty-eight and seven thousand four
hundred twenty-five ten thousandths
Millions
Thousands
Hundreds
Tenths
Hundredths
Thousandths
0
•
4
5
1
•
1
2
•
5
One
1.
Ten
Hundred
Thousand
Ten
Thousand
Hundred
Thousand
Million
Ten Million
Hundred
Million
1
•
Read the entire number first, then add the last right digit’s place value to the end
E.g “four hundred fifty one” thousandths
2. When there is a whole number too, read the whole number first then add “AND” the
decimal number as above
e.g. “ one hundred fifty AND twelve hundredths
*** Notice Decimal Number Names end in “–ths” ***
The value of a digit is determined by its place value.
Number
Place Value (of
underlined digit)
Value of the
digit (as a
decimal)
Value of the
digit (as a
fraction)
3 .145
Ones
3
3
3. 145
Tenths
0.1
1
10
3.1 45
Hundredths
0.04
4
100
3.14 5
Thousandths
0.005
5
1000
When the decimal point of a number is not shown (for example,
in whole numbers), then it is assumed to be at the end of the
number on the right hand side
Example :
321 = 321.
4 = 4.
A decimal number is a number that has digits before and
after a decimal point. The decimal point is placed after the
ones digit.
3.145
Example :
Each digit in a decimal number has a place value depending
on its position.
Tens
3
Ones
.
Decimal
point
Tenths
Hundredths
1
4
5
Thousandths
• Read the whole number before the decimal
without the use of “and.”
• Write that number down.
• At the decimal point use the word “and”
• Read the entire number after the decimal as if it
were a whole number.
• Add the name of the place value of the last digit
on the right hand side (after the decimal of
course)
Example: Write the following
decimal in words 8 243.67
Eight thousand, two hundred fortythree AND sixty-seven hundredths
Example: Write the following decimal number in
standard form:
two hundred six and fifty-four ten-thousandths
5 4__
206 .__
0 __
0 __
The word “ten-thousandths” indicates that we
need four decimal places.
When we clean it up, the answer is 206.0054
0.58
Fifty eight hundredths
0.854
Eight hundred fifty four thousandths
12.5
1.777
0.0005
100.10
0.351
23.6
Twelve and five tenths
One and seven hundred seventy seven
thousandths
Five ten thousandths
One hundred and ten hundredths
seven thousand
Three hundred fifty one thousandths
Twenty three and six tenths
Five hundredths
0.05
Twenty three thousandths
0.023
Thirty nine and six tenths
39.6
Thirty and one ten thousandths
Seven tenths
nine thousand and one hundredth
30.0001
0.7
9 000.01
Eighty and five hundred four one thousandths
80.541
Twenty one ten thousandths
0.0021
• Similar to writing whole numbers in
expanded form.
• Write the number that appears before
the decimal point in expanded form.
• For decimals, place a zero in the ones
place.
• Also, substitute zeroes for all spaces
after the decimal point that come before
the digit that you are working with.
For Example: Write the following decimal 13.361 in expanded form.
There are two ways to write 13.361 in expanded form.
1. Use the decimals
2. Use the fractions
13.361 = 10 + 3 + 0.3 + 0.06 + 0.001
13.361 = 10 + 3 + 3 + 6 + 1
10 100 1000
Decimal numbers can also be represented by pictures using the
base 10 blocks
Unit = 1
Rod = 10
Flat = 100
Cube = 1000
Using the following pictures write a fraction and decimal for the grey area.
There are 100 units in this
flat
50 are grey
So the fraction is 50
100
and the decimal is 0.5
Using the following pictures write a fraction and decimal for the coloured area.
There are 100 units in this
flat
50 are coloured
So the fraction is 75
100
and the decimal is 0.75
Using the following pictures write a fraction and decimal for the coloured area.
There are 100 units in this
flat
6 are coloured
So the fraction is 6
100
and the decimal is 0.06
Basic Rules:
1.
Add as many zeros LEFT of digits that are BEFORE a decimal.
2.
Add as many zeros as you want RIGHT of the digits AFTER the decimal.
Equivalent Decimals
Examples:
0.5 = 0.50 = 0.500 = 0.500000
2.4 = 02.40 = 2.400 = 0002.4000.
034 is the same as 34. The number still has no hundredths, 3 tens and 4 ones.
1.5 is the same as 1.50. The number still has 1 one, 5 tenths, and no hundredths.
000032.456000 = 032.456 = 32.456 = 32.4560000
These all have 3 tens,, 2 ones, 4 tenths, 5 hundredths and 6 thousandths.
< (less than)
≤ (less than or equal to)
> (greater than)
= (equals)
≥ (greater than or equal to)
Basic Steps
•
Compare the whole number parts first.
•
Start and move LEFT TO RIGHT
•
Compare the next most significant digit of each number in the same place.
•
If they are equal, move onto the next place to the right.
** Think of it as PAC –MAN, his mouth is ALWAYS open towards the LARGER number.
Examples:
a). Compare 1123 and 1126
b). Compare 567 and 497
c). Compare 1 and 0.002
d). Compare 0.402 and 0.412,
e). Compare 120.65 and 34.999
f). Compare 12.345 and 12.097.
1123 < 1126
567 > 497
1 > 0.002
0.402 < 0.412
120.65 > 34.999
12.345 > 12.09.
BASIC RULES
1.
If the number to the RIGHT of the Rounding Number is 5 or more (5, 6, 7 8 or 9) then the
LEFT (Rounding Number) number goes UP by ONE place.
2. If the number to the RIGHT of the Rounding Number is 4 or less (4, 3, 2, 1 or 0) then the LEFT
(Rounding Number) number remains the SAME.
3. Every number AFTER the Rounding Number becomes 0.
4. Everything BEFORE the Rounding Number remains the same.
5. If the Rounding Number is a 9 and it goes up by one, then a 0 is placed in the Rounding Number
Place and 1 is moved to the next place to the LEFT. (same as carrying in adding)
Examples
Because ...
3.1416 rounded to hundredths is 3.14
... the next digit (1) is less than 5
1.2635 rounded to tenths is 1.3
... the next digit (6) is 5 or more
1.2635 rounded to 3 decimal places is 1.264
... the next digit (5) is 5 or more
134.9 rounded to tens is 130
... the next digit (4) is less than 5
12,690 rounded to thousands is 13,000
... the next digit (6) is 5 or more
1.239 rounded to units is 1
... the next digit (2) is less than 5
Basic Steps:
•
First, line up the decimal points
•
When one number has more decimal places than another, use 0's as place holders to give
them the same number of decimal places.
•
Add.
Example #1:
76.69 + 51.37
1) Line up the decimal points:
76.69
+51.37
2). Then add.
Example #2 :
1)
Line up the decimal points:
2) Then add.
76.69
+51.37
128.06
12.924 + 3.6
12.924
+ 3.600
12.924
+ 3.600
16.524
Basic Steps:
1). Line up the decimal points on all the numbers
2). When one number has more decimal places than another, use 0's as place holders to give
them the same number of decimal places
3). Subtract.
Example:
18.2 - 6.008
1) Line up the decimal points.
18.2
- 6.008
2) Add extra 0's, using the fact that 18.2 = 18.200
3) Subtract.
18.200
- 6.008
18.200
- 6.008
12.192
Basic Steps.
1.
Line up the NUMBERS, not the decimals.
2.
Multiply the same way as you would with whole numbers.
3.
After multiplying, add the numbers of digits to the RIGHT of the decimal point in both factors.
4.
This is how many places the decimal will move to the LEFT of the last right digit.
Example 1:
1. Multiply
Example 2:
4.032 × 4
1. Multiply
6.74 × 9.063
2. Line up Numbers
4.032
x
4
2. Line up Numbers
3. Multiply
4.032
x
4
16128
3. Multiply
4. Count the Number of Decimal Places
in both Numbers.
The decimal moves 3 digits from the right:
4.032
x
4
16.128
6.74
x 9.063
6.74
x 9.063
2022
4044
0000
+6066
.
6108462
4. Count the Number of Decimal Places
in both Numbers.
The decimal moves 5 digits from the right:
6.74
x 9.063
61.88462
Basic Steps.
1.
2.
When dividing the decimal goes STRAIGHT UP from where it is in the dividend.
Divide the same way as whole numbers.
Example 1:
Example 2:
1.
Divide
15.567 ÷ 3
2.
______
3 | 15.567
1. Divide
2.
241.8 ÷ 22
______
22 | 241.8
3. Place the decimal straight up
from the dividend
3. Place the decimal straight up
from the dividend
4. Solve
4. Solve
5.189
3 | 15.567
-15
05
-3
26
-24
27
-27
0R
10.99
22 | 241.800
-22
21
-0
218
-198
200
-198
2R
Basic
1.
2.
3.
4.
5.
Steps.
First, it is easier to make the Divisor a whole number.
Move the decimal in the Divisor as many places to the RIGHT as to create a whole Number.
Move the decimal in the Dividend the same number of places to the RIGHT.
Next, the decimal goes STRAIGHT UP from where it is in the dividend.
Divide the same way as whole numbers.
Example 1:
Example 2:
1.
Divide
24.808 ÷ .3
2.
______
0.3 | 24.808
3.
Move the Decimal in the Divisor
and the Dividend 1 places to
the RIGHT to make the Divisor
a Whole Number
3. Move the Decimal in the Divisor
and the Dividend 2 places to
the RIGHT to make the Divisor
a Whole Number
4.
The decimal goes straight up now
4. The decimal goes straight up now
5.
Solve
5. Solve
82.69
3 | 248.08
-24
08
- 6
20
-18
28
-27
1R
1. Divide
2.
250.85 ÷ 0.25
______
0.25 | 250.85
1003.4
25 | 25085.0
-25
00
- 0
08
-0
85
-75
100
-100
0R