Remember my post about placeholders?
For starters, the decimal is a placeholder.
The zero before the decimal is a place-holding zero (also called leading zeros). Omitting these zeros changes the value of the decimal.
The zeros after the decimal are trailing zeros (or non significant zeros). We know that leading zeros are extraneous, unless they are between numbers that are not equal to zero. We can omit non significant zeros and the value of the decimal won't change.
Just as a whole number can be represented as a fraction by putting a 1 beneath it, such as
Where the decimal is placed is very important. One wrong move and you can increase/decrease a number exponentially from what it was supposed to be.
Here is our placeholder chart:
Thousands Hundreds Tens Ones Tenths Hundredths Thousandths Ten Thousandths
Let's place a number in this chart:
Thousands Hundreds Tens Ones Tenths Hundredths Thousandths Ten Thousandths
1 0 0 . 0
We placed our decimal point after the hundreds in this chart, making our number 100.0
If we multiply that number by 10, we will move the decimal one place to the right:
Thousands Hundreds Tens Ones Tenths Hundredths Thousandths Ten Thousandths
1 0 0 0 . 0
Our number is now 1000.0
What if we were to divide this number by 10,000?
We will move the decimal 4 places to the left, because 10,000 has 4 zeros in it:
Thousands Hundreds Tens Ones Tenths Hundredths Thousandths Ten Thousandths
0 . 1 0 0 0 0 0
Our number is now 0.10 We can exclude the other zeros that come after the zeros to the right of the decimal because they are leading zeros. We must not, however, take away the zero to the left because it is our place-holder. These zeros are significant.
I HATE it when these place-holder zeros are omitted. Some people prefer to write a number like 0.10 as simply .10 Keep in mind that the decimal is a little, tiny dot that can easily be overlooked. I might fail to see it and think the number was 10, not .10 Please, be kind and include that place-holder zero when you have one to the left of a decimal. Thanks!
Incidentally, 0.10 can be written as
If we were dividing 1,001 by 10,000 in our previous example, our placeholder chart would look like this:
Thousands Hundreds Tens Ones Tenths Hundredths Thousandths Ten Thousandths
0 . 1 0 0 1 0 0
In this case, the zeros in the ones and tenths place are significant because they are followed by a digit not equal to zero. We must include them. Thus, our answer is 0.1001 (we can omit the other zeros to the right because they are trailing zeros).
If we were to omit the zeros in between the first 1 and the second 1, our answer would be 0.11, which is not as accurate. This gets us into rounding.
As you may recall, whenever a digit to the right is 0,1,2,3, or 4, you will round down. Which means that you will drop every other digit to the right.
Whenever a digit to the right is 5,6,7,8, or 9, you will round up. Add a 1 to the digit in the first decimal place and then drop every digit to the right.
You might be asked to round to the nearest -th, which indicates the amount of decimal places you may be rounding:
Nearest tenth = 1 decimal place
Nearest hundredth = 2 decimal places
Nearest thousandth = 3 decimal places
Nearest ten thousandth = 4 decimal places
Nearest hundred thousandth = 5 decimal places
Nearest millionth = 6 decimal places
...and so on and so forth...(I doubt anyone will ask you to round to the nearest millionth anyway...)
Let's say you are asked to round 5.12345 to the nearest hundredth.
We know that the nearest hundredth is two decimal places to the right, so we will look at one decimal space to the right of that. In this case, that is the 3 in 5.12345. Since it is the number 3, we will round down. Therefore, 5.12345 rounded to the nearest hundredth is 5.12
Simple enough.
But don't forget that when you round up, you need to add 1 to the digit in the first decimal place.
What if we had to round 900.9999 to the nearest thousandth?
We look at the 9 four decimal places to the right (the spot just after the thousandth place), and add a 1. Thus, 900.9999 becomes 900.990. But we have essentially carried a 1, because by adding a 1 to the 9 three decimal places to the right, we have changed it to a 10. So then 900.990 becomes 900.90. Again, we are still carrying that 1 because we changed the 9 two decimal places to the right into a 10. So 900.90 becomes 900.0. Again, we have carried a 1. So 900.0 becomes 901.0. And finally, we are no longer carrying that 1.
What if we had had to round 900.9988 to the nearest thousandth instead?
We would again look four decimal places to the right, this time at the eight, and add a 1. Thus, 900.9988 would become 900.999
What if we had had to round 900.992 to the nearest thousandth instead?
900.992 would be rounded down to 900.990
Adding and Subtracting Decimals
When adding or subtracting decimals, all you need to do really is make sure that their decimal points line up. Adding trailing zeros can help make sure that the numbers are lined up and avoid mistakes. You can always eliminate them later
Let's say you had this equation: 0.005 + 1.832 - 5.923 + 0.67459
Line them up:
0.005
+1.832
- 5.923
+0.67459
Add trailing zeros:
0.00500
+1.83200
+0.67459
________
2.51159
2.51159 ***tip: when subtracting decimals, put the larger number first and then just add a negative
-5.92300 to the final result***
________
-3.41141
Multiplying and Dividing Decimals
It's more difficult to simply line up decimal points when you are multiplying/dividing, so don't.
When multiplying, just keep track of how many digits to the right of the decimal point you have in each number and then move the decimal to the left by the same amount of digits when you are finished.
Multiply as normal.
Multiply 0.55 x 1.23456
0.55 has 2 digits to the right of the decimal point
1.23456 has 5 digits to the right of the decimal point
Total digits to the right of the decimal point = 7
123456
x 55
_______
617280
617280
_______
6790080
Starting from the right, count 7 digits
0.6790080
aaand you have your answer!
When dividing, you can move the digits of one number to the right until it is a whole number. Just make sure that you move the decimal point of the other number by the same number of digits.
Divide 6.543 by 0.21
If we move the decimal point of 0.21 2 digits to the right = 21.0
Move the decimal point of 6.543 2 digits to the right as well = 654.3
(Please bear with my improvisation for the long division sign)
31.1571
21 [ 654.3000 <---don't forget that if you need to add more digits to an equation that you can keep
-63 adding on zeros to the dividend!
24
-21
33
-21
120
-105
150
-147
30
-21
12
And we have our answer: 31.1571
Conversion between Fractions and Percents
Remember in the beginning of this post I mentioned that decimals, fractions and percents are similar?
They are pretty much all ways of expressing a part of a whole. Which means that we can convert from decimals to percentages to fractions back to decimals again!
Decimals to Fractions
To change decimals into fractions, you want to determine where the decimal point is placed in our decimal (which will become the numerator) and then match that place with your denominator. To do this, we move the decimal point to the right for both the numerator and the denominator. Then simplify, if necessary.
Let's try this with 0.25
Feeling adventurous enough to attempt a larger decimal?
How about converting 0.123456789 to a fraction?
Wow. Hopefully you get it by now lol.
Fractions to Decimals
Since fractions basically mean to divide one number by another (the numerator by the denominator), we can find our decimals by dividing the fractions.
Let's go back to our result of
0.25
4[1.00 <---remember that we can keep adding zeros if we want to generate more digits!
-0
10
- 8
20
-20
0
What if we were converting
Do the same thing, divide 3 by 1:
0.333333
3[1.000000
-0
10
-9
10
-9
10
-9
10
-9
10
-9
10
-9
10
Okay...this can go on forever!
This type of decimal, a decimal whose digits repeat themselves forever, is known as a repeating decimal.
A decimal whose digits are not infinite (thereby finite) are called terminating decimals.
Anyway, we obviously can't write out all the digits of a repeating decimal such as this, so instead we put a bar over the digit to indicate that it is a repeating decimal.
So our answer to converting
__
is 0.333
Decimals to Percentages
Since 1 is equal to 100% (more on that later), we simply move any decimal point two places to the right to convert it to a percentage. You can literally treat it as if % = 1/100 or 100 * % = 1 because that is actually what it means.
Let's try this with 0.25
Move the decimal point of 0.25 two places to the right = 025.0 = 25.0 = 25%
What about 1.25?
Move the decimal point of 1.25 two places to the right = 125.0 = 125%
What about 1.2525?
Move the decimal point of 1.2525 two places to the right = 125.25 = 125.25%
Easy, right?
Great!
Oh, and as always, there is lots to practice
We need to hurry up and finish with this basic math so we can move on to bigger and better things! There's a whole world that awaits us and it's filled with numbers! Numbers that seem less scary with every passing post (I hope).
No comments:
Post a Comment