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You already know how to write a fraction. Decimals are another way of writing a fraction.
In order to create a decimal first print a period (.), known as a decimal point, and write a series of digits either before or after this period.
Examples of decimals include
In this section we will learn exactly what is meant by these numbers.
Remember when we did base 10 representation of a whole number? Well decimal notation is similar, only here we'll be reading from left to right, rather than right to left and each column will decrease by a power of 1/10 (one-tenth), rather than increase by a power of ten.
Let's examine
4 is the first digit to the right of the decimal point.
The digit to the immediate right of the decimal sits in the one tenth column.
Thus
Now let's look at 0.46
We already know that the first digit stands for
The second digit to the right of the decimal point (6) sits in the
Therefore the digit 6 stands for
So
Now let's look at
Finally we'll get a chance to work on
As you can see once all of the digits are accounted for, the equivalent fraction has the entire decimal number as it's numerator (without the decimal point).
Change 1.37 into a mixed fraction.
Change 34.333 into a mixed fraction.
0.333 is the same as .333.
You'll get your chance to try this out on your own, but first I have to talk about nothing.
Change 0.0012 into a fraction.
Change 0.340 into a fraction.
Change 0.0102 into a fraction.
Change 0.700 into a fraction.
Now you will get your chance to shine.
We already know that the first digit stands for
The third digit to the right of the decimal point (2) sits in the
Therefore the digit 2 stands for
So
We already know that the first digit stands for
The fourth digit to the right of the decimal point (1) sits in the
Therefore the digit 1 stands for
So
Now count the number of digits in the decimal. The denominator is 10 raised to that number (or a "1" followed by that many zeros).
Don't worry about reducing these fractions down to lowest terms (at least not yet). We'll get to that on the next page of this chapter.
Now to let's take another look at the above examples.
0.4 will have 4 has it's numerator.
0.4 has one decimal place, so the denominator will be
0.46 will have 46 has it's numerator.
0.46 has two decimal places, so the denominator will be
0.462 will have 462 has it's numerator.
0.462 has three decimal places, so the denominator will be
0.4621 will have 4621 has it's numerator.
0.4621 has four decimal places, so the denominator will be
Let's look at two more examples before moving on to our next point.
37 will be the numerator.
0.37 has two decimal places, so the denominator will be
The fraction expression of
333 will be the numerator.
0.333 has three decimal places, so the denominator will be
The fraction expression of
What do you think numbers like 0.9912 and 0.56123 would look like as fractions?
All of the decimals we have seen thus far have a zero before the decimal point. What about numbers like 1.37 or 34.333?
Well the numbers on the left side of the decimal point constitute whole numbers, and the numbers on the right side of the decimal point constitute a proper fraction. Taken together the entire decimal represents a mixed fraction.
We'll now see this in the next two examples.
Well 1.37 actually means
We know
Therefore
Well 34.333 actually means
We know
Therefore
If a decimal has no number whatsoever on the left side of the decimal point, it is assumed that the whole number part is zero.
.4 is the same as 0.4.
0.46 is the same as .46.
0.462 is the same as .462.
And .4621 is the same as 0.4621.
Yes, I'm going to talk about zero.
What is the difference between a number like 0.5 and 0.50?
The answer is absolutely nothing!
In fact you can attach as many zeros as you would like to the end of a decimal and it wouldn't make the slightest difference!!
Example:
Example:
Another example of this involves whole numbers. You can extend any whole number simply by attaching a decimal point to it's tail end and adding as many zeros as you want.
Example: 50 is the same number as
Example: 2 is the same as
Example:
One warning however: make sure you put your zeros at the end of the decimal, not the beginning!
There is a big difference between
If you're confused think of it in terms of dollars and cents: would you rather have five cents of fifty cents?
We will now work out four more examples.
12 will be the numerator.
To figure out the denominator, we first have to know how many decimal
places are in 0.0012.
Since the zeros occur immediately after the decimal point, and not at the very end of the number, we are forced to count them.
Thus 0.0012 has four decimal places.
Here the zero occurs at the end of the number. Therefore we can ignore it.
34 will be the numerator.
To figure out the denominator, we first have to know how many decimal
places are in 0.34.
0.34 has two decimal places.
102 will be the numerator.
To figure out the denominator, we first have to know how many decimal
places are in 0.0102.
Since the zero occurs immediately after the decimal point, and not at the very end of the number, we are forced to count it.
So 0.0102 has 4 decimal places.
Here the zeros occurs at the end of the number.
We can therefore ignore them.
0.700 = 0.7
7 will be the numerator.
To figure out the denominator, we first have to know how many decimal
places are in 0.7.
0.7 has one decimal place.
Press the "generate a problem" button to get started.
You will then see a statement like "0.34 = 34/___"
It is your job to enter the denominator.
Once you enter your answer, click the "check my answer" button.
The computer will tell you whether or not your answer was correct.
You will receive further instructions if your answer was incorrect.
After you complete a problem you can generate another problem by first pressing the "reset" button and then once again clicking the "generate a problem" button. Continue doing problems until you feel confident in your ability to understand decimals.
At that point you have the following options:
Text, images, and source code c2003 Martin Selditch