Decimal Numbers in Mathematics
INTRODUCTION 3.0
In this lesson we will formally define decimal numbers and operations on decimal numbers. We will look the form of
decimal numbers, define how to describe decimal numbers, and perform operations such as addition, subtraction,
and multiplication. Finally, we will cover how to convert decimals to percentages and percentages to decimals.
Lecture 3.1: Decimals
Decimals are used frequently in our everyday lives. Some common examples might be when we say that a price of a
pack of gum is $1.69, that the weight of a bag of apples is 1.8 pounds, or that a restaurant is 5.2 miles away. A
number written as a decimal has two parts: The whole part, which is written in front of the decimal point, and the
fractional part, which is written after the decimal point.
The numbers to the right of the decimal point are arranged in decimal places. The first decimal place after the
decimal point is the tenths place. For example, the number 6.1 is read out loud as six and one tenth. The name of the
decimal places, moving from left to right, are shown below:
Lecture 3.2: Addition of Decimals
Adding decimals is similar to adding whole numbers: We add up the digits from right to left, carrying when necessary.
There is only one additional step when dealing with decimals, and that is to carry the decimal point to our result.
For example, suppose that we want to add 2.41 and 1.9. The first step is to write these numbers vertically, aligning
their decimals points:
You may want to add extra zeros to the right-hand side of your decimals so they have the same number of decimal
places. (Adding a zero to the right side of a decimal does not change its value.)
Next, fill in the decimal point for our answer by "dropping" it directly below the other decimal points:
Then, we add each digit, starting with the column on the right and working our way to the left:
Thus, the sum of 2.41 and 1.9 is 4.31.
Lecture 3.3: Subtraction of Decimals
To subtract decimals, we follow a procedure that's similar to subtracting whole numbers. Suppose that we want to
subtract 3.1 from 11.65. First, we write the numbers vertically, aligning their decimal points:
At this point, you may want to add extra zeros to the right-hand side of your decimals so that they have the same
number of decimal places. (Adding zeros to the right side of a decimal does not change its value.)
Next, fill in the decimal point for our answer by "dropping" it directly below the other decimal points:
Then subtract each digit, starting with the column on the right and working our way to the left:
Thus, the difference between 11.65 and 3.1 is 8.35.
Lecture 3.4: Multiplication of Decimals
To find the product of two decimals, such as 2.02 and 0.6, we multiply the same way that we multiply whole numbers.
The only catch is that we need to keep track of the decimal point. The first step is to write the number vertically,
aligning their right-hand sides:
Then, we momentarily ignore the decimal points and multiply as if they were whole numbers:
Finally, we will place the decimal point in our result. To do this, we add the number of decimal places in the two
numbers. The number 2.02 has two decimal places to the right of the decimal point and the number 0.6 has one
number to the right of the decimal point. This gives us a total of 3 decimal places in our product.
Thus, the product of 2.02 and 0.6 is 1.212.
Lecture 3.5: Division of Decimals
Dividing decimals requires careful attention to placement of the decimal point. Before performing division, we want to
ensure that the problem is set up appropriately. Let's consider as an example.
First, write this as a long division problem using the given decimal places:
Whenever we are dividing decimals, we always want a whole number outside of the division sign. In this case, we
have the number 0.2 outside of the division sign. How do we convert this to a whole number? It's easy: Move the
decimal point to the right one place. The rule is that whatever you do outside of the division sign you must also do to
the inside of the division sign, so if you move the decimal point one position to the right on the outside, you must also
move the decimal point one position to the right on the inside:
Now we can think of the quotient as the following:
Next, we are ready to perform division. The first step is to locate the position of the decimal point in our answer. To do
this, we add a decimal point directly above the decimal point in the number under the division sign:
Now we can start dividing, in the same fashion we divide whole numbers.
To continue to divide the decimal, we need to add and carry down a zero under the division sign:
We can keep adding zeros as long as we have a remainder or until you have gone to sufficient decimal places.
Remember, adding zeros to the right-hand side of the decimal does not change its value.
Thus, the quotient of 0.41 and 0.2 is 2.05.
Lecture 3.6: Changing Decimals to Fractions
To convert a number from a decimal to a fraction is straightforward. Let's convert the decimal 0.623 to a fraction. The
first step is to rewrite the fractional part in the numerator:
To determine the denominator, count the number of decimal places, which are to the right of the decimal point. In our
example, we have 3 decimal places. Your denominator will be the number 1 followed by as many zeros as you have
decimal places. Since we have 3 decimal places, we'll add three zeros:
Thus, the decimal 0.623 is equivalent to the fraction 623/1000, or, in words, "six hundred twenty three thousandths."
Conclusion: none
References: none