Place Value

Place value refers to a digit’s worth in a number. Oftentimes, place value is split
up into columns, and we refer to them as the ones’ column, the tens’ column, the
hundreds’ column, the thousands’ column, the ten thousands’ column, and so on. These
columns look like this:

In this example, the 1 is in the ones’ column, so we know there is one 1. The 2
is in the tens’ column, so we know there are 2 groups of 10, or 20 total. Here is
another diagram to help you visualize what we mean:

You can see that here, we have 2 tens, which equals 20. Every time we put a digit
in a place value column, you have that many groups of the value specified. So, if
you have 2 in the tens’ column, you have two groups of ten, which equals twenty.
If you have a 3 in the hundreds’ column (like our first example) it means you have
three groups of 100, which equals 300.

Place value is very important when
adding and
subtracting numbers. In order to add and subtract correctly, you must
make sure the columns of each number are correctly aligned. If you incorrectly line
up the columns, your answer will be incorrect as well. You should work on lining
up the columns, like this:

This is the correct way to set up an addition or subtraction problem. When lining
up the columns, always start with the ones’ column, because sometimes one number
will have more digits than another number, but they will both have a digit in the
ones’ column filled. Therefore, you might get a problem that looks like this: 359
+ 24 =

Then, you would have to line it up. Remember to start with the ones column, which
would look like this:

Notice that in this example, the ones’ and tens’ columns both lined up, and the
hundreds column only had 1 digit in it (from the first number). This is perfectly
fine! Sometimes, as the place values columns get higher, you won’t have a digit
in every single space. That’s alright, as long as your columns all line up starting
at the ones’ place value.

When you do the addition for these two numbers, you end up with 383; this is the
correct answer.

Here is an example of the wrong way to line up this addition problem—it would look
like:

Notice the empty space in the ones’ column in the bottom number. Clearly, this space
should not be here, because the 4 of the 24 is the ones’ digit, thus it belongs
in the ones’ column.

If you did the addition of this problem, lined up incorrectly, you would get 559,
which is not even close to the correct answer.

You might ask yourself the same question that is in the diagram above—what should
go in this space? If you’re asking yourself that question, you know that the problem
is incorrectly lined up, and you should shift the digits in the bottom number to
the right one place value so that the ones’ and tens’ columns align.

Thus, when it’s correctly aligned, it looks like this:

Now, here are a few practice problems to make sure you can correctly identify which
number is in which place value.

Here is your first example number: 13,497

Read the following questions, and type your answer into the box. Click on “Check
Answer” to show the correct answers.

Let’s try that one more time. Here is your example number: 39,582

Read the following questions, and type your answer into the box. Click on “Check
Answer” to show the correct answers.

Decimal Place Values

Just as whole numbers have place values, decimal places (digits after the decimal
in a number) also have place values. When looking at lining up decimal places, start
at the decimal point, and work from left to right. There is no “oneths” column.
The first place is the tenths column. Then hundredths, then thousandths, and so
on. You can see how these columns are named in the diagram below.

It is also important to line up decimal places when adding and subtracting numbers.
With whole numbers, we had to line up the ones’ column. However, if both numbers
have decimal points, you should focus on lining up the decimal point instead. Realistically,
this will also line up the ones’ column; however, if you ever have a number without
a ones’ digit, you will still be able to line up the decimal places in order to
do the addition or subtraction, like this:

Notice that the decimal places are lined up, which means that the tenths and hundredths
columns are also lined up. There is no ones’ digit in the bottom number, but if
there were these place values would also be aligned. This is how all decimal addition
and subtraction should be lined up.

Make sure that your decimal addition and subtraction does NOT look like this:

Notice that in this example, the decimal point falls in two different columns, making
it impossible to add them together. When this happens, you need to re-align the
numbers so that the problem looks like the first example:

Exceptions to the rule:

There are certain times that you do not need to line up the decimals—these are during
decimal multiplication and division. During decimal multiplication, you would line
up the digits furthest to the right, no matter how many place values there are in
the number. Decimal multiplication would look like this:

Notice that the decimal points are not lined up; however, the digits at the end
of both numbers are lined up. This is because, with multiplication, we multiply
each bottom digit by each top digit separately, add, and then place the decimal
point in the proper position. For a review on how to do this, please review how
to
multiply decimals.

The other exception to the rule concerning decimal place values is during division.
You do not need to worry about lining up decimal places when doing long division
because division is set up far differently from addition, subtraction, or multiplication.
For a review of doing decimal division, please review how to divide with
decimals.