Decimal
Numbers that fall in between integers; expresses a part-to-whole relationship in terms of place value.
Example:
1.2 is a decimal.
The integers 1 and 2 are not decimals.
An integer written as 1.0, however, is considered a decimal.
Digit
There are ten digits that make up all numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The three-digit number 412 consists of the digits 4, 1, and 2.
Place Value
Every digit in a given number has a particular place value. The place value depends upon the digits location relative to the decimal point.
| 5 |
6 |
7 |
8 |
9 |
1 |
0 |
2 |
3 |
. |
8 |
3 |
4 |
H U N D R E D
M I L L I O N S |
T E N
M I L L I O N S |
O N E
M I L L I O N S |
H U N D R E D
T H O U S A N D S |
T E N
T H O U S A N D S |
T H O U S A N D S |
H U N D R E D S |
T E N S |
O N E S |
|
T E N T H S |
H U N D R E T H S |
T H O U S A N D T H S |
Place Value and Powers of 10
Place values decrease from left to right by powers of 10.
Table:
| Words |
thousands |
hundreds |
tens |
ones |
tenths |
hundredths |
thousandths |
| Numbers |
1000 |
100 |
10 |
1 |
0.1 |
0.01 |
0.001 |
| Powers of 10 |
10^3 |
10^2 |
10^1 |
10^0 |
10^-1 |
10^-2 |
10^-3 |
Rounding
Simplifying a number to a certain place value. Drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep. If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep.
Example:
9.1278 rounded to the nearest tenth = 9.1, since the dropped 2 is less than 5.
9.1278 rounded to the nearest hundredth = 9.13, since the dropped 7 is greater than (or equal to) 5.
9.1278 rounded to the nearest thousandth = 9.128, since the dropped 8 is greater than (or equal to) 5.
Adding or Subtracting Decimals
Write the problem vertically and line up the decimal points. Add any necessary zeroes to the right side of any numbers in order to make the numbers the same length.
Example:
71.2
+184.99
256.19
Multiplying Decimals
Drop the decimal points and multiply normally (as you would multiply whole numbers). At the end, count the total number of digits to the right of the decimal in the original numbers. Insert the same number of decimal places into the answer.
Example:
0.7 * 3 = ?
First, multiply normally: 7*3 = 21
Then count the decimals represented in the original numbers; in this case, we have one decimal among the original numbers.
Insert one decimal into the answer, 21, to come up with 2.1
Dividing Decimals (Dividend)
If there is a decimal in the dividend (the inner number), bring the decimal point up to the answer and then divide normally.
Example:
3.09
4 | 12.36
12
03
0
36
Dividing Decimals (Divisor)
If there is a decimal in the divisor (the outer number), first shift the decimals an equal number of places in both the divisor and the dividend until the divisor is an integer. Then follow the instructions above.
Re-writing Decimals Using Powers of 10
Decimals can be re-written in terms of powers of 10 or vice versa.
Example:
0.00624 = 6 * 10^-3.
Terminating Decimals
Decimals that terminate, or end, at some point; decimals that do not go on forever. 13.2 is a terminating decimal. 13.2 does not terminate. π also does not terminate.
