Recently a teacher (Hi, Edite!) asked for help teaching how to divide decimals; in particular, she wanted to be able to provide a deeper understanding of the process, giving a good reason for what we do. Here I want to start a long-delayed series on operations with decimals, doing exactly this for all four basic operations. We’ll start with addition and subtraction, then cover multiplication next time, and get to division last.
Adding decimals
We’ll start with addition, using this question from 1998:
Adding Decimals I can't figure this out! 894.56 + 4563.5 --------
The way it’s typed suggests a likely issue!
What to do: Line them up
Two of us answered. The first was Doctor Barrus:
Hi, Jennifer!
Adding decimals isn't that different from adding whole numbers. There are just a few extra steps. I'm going to work a problem that's similar to yours, and then I hope you'll see how to do yours.
Okay! Here's what we do. Say I have the problem:
992.536
+ 473.9
-------
?
1. The first thing to do is to line up the decimal points in a straight vertical line. So I would rewrite my problem as
992.536
+ 473.9
---------
with all the decimal points in a line.
As we’ll see later, this puts all the ones (units) in a column, all the tenths in another, and so on.
2. Next, add as many zeroes as you need to "fill in the blanks" in the decimal parts. For example, I would make my problem
992.536
+ 473.900
---------
See how I filled in the empty spaces from step 1? This works because 473.9 = 473.900, just the way 1 = 1.0. Adding zeroes to the end of the decimal part of a number doesn't change the number.
With experience, you don’t need to actually write the zeroes; you can just no add anything in those columns. But this does make things clearer.
3. Now add normally. The decimal point stays in that straight line, but the numbers just add following the normal rules, like carrying and stuff. So in my problem, I'd have
992.536
+ 473.900
---------
1466.436
I added this by just following the same steps I would to add
992536
+ 473900
--------- ,
and I put the decimal point in the answer underneath all the other decimal points.
That’s the process; next, we’ll see in more detail what it means.
Why: Modeling it with money
Later, Doctor Mateo answered, doing the original problem with more explanation:
Hello Jennifer,
To add (or subtract decimals) the first thing you should do is line up the decimal points.
894.56
+ 4563.5
----------
It is important for you to line up the decimals vertically the way I did above so that you can keep track of the place-value position of the digits.
We want to add ones to ones, tenths to tenths, and so on.
You could think about it like this. Suppose you have a WHOLE LOT of coins: pennies, nickels, dimes, and quarters. There are lots of ways to count the money, but what is probably the fastest way to do it since you have a whole lot of change to count? Probably separating the coins into stacks of pennies, stacks of nickels, stacks of dimes, and stacks of quarters, and then counting up the value of each stack. If you have enough of a certain coin you might even collect it in a coin holder.
Putting all the coins of each kind into a stack saves us from having to add different values together (or from accidentally counting a dime as a penny).
So when you add (or subtract) decimals, we like to put our digits into "stacks" of similar values too.
If you remember, we like to put the digits into columns or "bundles" of powers of tens just the way we would put dimes with other dimes. So in this example we place the numbers into their place-value holder:
thousands // hundreds // tens // ones // . // tenths // hundredths
// 8 // 9 // 4 // . // 5 // 6
+ 4 // 5 // 6 // 3 // . // 5 //
We discussed place value concepts in Place Value: Whole Numbers and in Place Value: Decimals.
After you line up the decimal points, you might want to put zeros in the empty spaces as space-holders, like this:
894.56
+ 4563.50
----------
One of the reasons we place zeros in the empty spaces after the decimal point is that we can have a record of the number of "coins" in that position. If you have no hundredths then you record a "0" to show that you had no hundredths to count.
Now you can do regular addition. Just make sure you bring the decimal point down in the same position it is in the problem.
Have fun adding.
Finishing up:
11 1 894.56 + 4563.50 ---------- 5458.06
By putting the decimal point in the same column, we say that the sum of the tenths is a number of tenths!
Mixing whole numbers with decimals
What if some numbers don’t have a decimal point? Here’s a question from 2002:
Hidden Decimal Points I get confused when subtracting 4 from 24.98. I keep coming up with 24.94. Could you explain why whole numbers do not show their decimal point?
Michael presumably did this, just lining up the ends of the numbers:
24.98
- 4
-------
24.94
I answered:
Hi, Michael. The first thing to do is to keep in mind, as you are aware, that the decimal point of a whole number is hidden to the right of the number. Why? Because in a whole number like 24, the rightmost digit is always the ones place; when we add tenths and hundredths, and so on, to the right of it, we need to mark where the ones place is. So we use a decimal point to separate the whole part from the fractional part, and the units place is just to the left. Whole numbers don't show it, simply because it's not necessary, and we don't like to waste ink. (It can also look confusing, as in my next sentence.)
The decimal point at the end does look odd in a sentence:
But we CAN show it; here we want to subtract 4. from 24.98. When you subtract whole numbers, you line up corresponding places by lining up the right side, so that you subtract ones from ones, tens from tens, and so on. With decimals, you do the same thing, but you do it by lining up the decimal points. That means we have to subtract 24.98 - 4. ------- Now we are subtracting 4 ones from 4 ones, not 4 ones from 8 hundredths, which wouldn't make sense.
Fraction or whole, what we are really doing is lining up the place values, so that the “ones” are in one column, whether that is indicated by a decimal point or not. But writing it definitely helps.
To make it clearer, we can now fill in the empty places with zeros if it makes you feel better:
24.98
- 4.00
-------
Now just do the subtraction as if the decimal points weren't there:
24.98
- 4.00
-------
20.98
There's the answer. The only trick is lining up the decimal points, as if they were holes you put a pin through to hold everything in place. Where the pin pokes through the bottom, you put a decimal point in the answer.
That’s an interesting analogy.
Sometimes we line up the decimal points, sometimes we don’t
Finally, here is a question from 2003, comparing addition and subtraction to multiplication and division:
Lining Up Decimal Points Why do addition and subtraction have to be lined up by the decimals but division and multiplication don't have to be?
This will get us into topics we’ll cover more fully next time; if you don’t want spoilers, stop when we move beyond addition.
Adding and subtracting
Doctor Ian answered, starting with the positive story about addition:
Hi Rashonda,
That's a really good question! It's easy to just take it for granted, without ever stopping to think about it.
Suppose I have
3 cherries, 2 apples, 3 pears
and you have
4 apples, 1 pear, 3 watermelons
and we want to know what we have together. Could we do this?
3 cherries 2 apples 3 pears
4 apples 1 pear 3 watermelons
---------- -------- -------------
7 3 6
That doesn't make any sense at all, does it? We'd have to make sure that we're adding the same kinds of things:
3 cherries 2 apples 3 pears
4 apples 1 pear 3 watermelons
---------- -------- ------- -------------
3 cherries 6 apples 4 pears 3 watermelons
This is the same as the money analogy above; but fruits don’t have “place value”, so we have to put them in the same order, making up a set of piles. With money, or with numbers, there is a natural order.
This is also what's going on when we add decimals. When we write a number like '123.4', that's really just a very compact way of writing the sum
1 * 100 + 2 * 10 + 3 * 1 + 4 * 1/10
And when we write a number like 5.432, we're just writing the sum
5 * 1 + 4 * 1/10 + 3 * 1/100 + 2 * 1/1000
Now, following the example of the fruit, can we do this?
1 * 100 2 * 10 3 * 1 4 * 1/10
5 * 1 4 * 1/10 3 * 1/100 2 * 1/1000
------- -------- --------- --------
6 6 6 6
Does that make any more sense than adding cherries to apples? No! So we need to line up terms where the digits have the same meaning:
1 * 100 2 * 10 3 * 1 4 * 1/10
5 * 1 4 * 1/10 3 * 1/100 2 * 1/1000
------- ------ ----- -------- --------- ----------
1 * 100 2 * 10 8 * 1 8 * 1/10 3 * 1/100 2 * 1/100
But how do we do this? By lining up the decimal points! So lining up the decimal points is just an easy way of making sure that we're not adding apples to pears, so to speak.
Does that make sense?
So we don’t add, as in the first example,
1 2 3.4
+ 5.4 3 2
---------
6 6 6 6
(where would the decimal point even go?) but rather
1 2 3.4
+ 5.4 3 2
-------------
1 2 8.8 3 2
Now we’re adding apples to apples.
Hidden in that explanation is a deeper mathematical explanation. When we add, say, the tenths column in that example, we are adding $$(4\times\frac{1}{10})+(4\times\frac{1}{10})=(8\times\frac{1}{10})$$ This amounts to “combining like terms” as if the place values were a variable, or factoring out the common factor \(\frac{1}{10}\). That’s why the sum of tenths is a number of tenths! We’ll see below how this doesn’t happen with the other operations, so that lining up place values doesn’t help.
What follows is a preview of the next two posts, so stop reading if you don’t want a spoiler:
Multiplying
Now we move on to the negative story: what makes the other operations different.
Okay, so much for addition and subtraction. What about multiplication and division? Let's consider the product
3.21
* 23.4
-------
Now, what we'd _really_ like is to get rid of those decimal points entirely, or at least temporarily. We can use a trick to do that. Note that
3.21 = 32.1 * 1/10 = 321 * 1/100
and
23.4 = 234 * 1/10
So
3.21 * 23.4
= (321 * 1/100) * 23.4 'Save' two decimal places
= (321 * 1/100) * (234 * 1/10) 'Save' one decimal place
= (321 * 234) * (1/100 * 1/10)
= (321 * 234) * 1/1000
= 75114 * 1/1000
= 75.114 'Restore' three decimal places
Notice that when you multiply two digits, you don’t get a “like digit” with the same place value; instead, you combine the two place values (by multiplying). The product of tenths and hundredths has a new place value, thousandths.
Moreover, in multiplication, we don’t combine all the digits in one column; we multiply every digit in one number with every digit in the other. So we can pull out all the place information about each factor, and combine them in the final answer:
So we can just forget about the decimal points while we're doing the multiplication, so long as we know where to put the decimal point when we're done. In practice, we do this:
3.21 <-- 'save' two decimal places
* 23.4 <-- 'save' one decimal place
-------
321
* 234
-------
75114 * 1/1000 <-- 'restore' three decimal places
to get 75.114
So in a sense, the answer to the question "Why don't we have to line up the decimal points when we multiply?" is that we remove them until we're done. You can't line up what isn't there.
Or, if you prefer, you don’t need to line up what doesn’t matter yet. We’ll see more about this next time.
Dividing
What about division?
_____
3.21 ) 23.4
We do a slightly different trick here, which is essentially the same thing we do when we find equivalent fractions:
23.4 23.4 100
---- = ---- * ---
3.21 3.21 100
2340
= ----
321
So now we have
________
321 ) 2340.0
Once again, a little extra work up front lets us get rid of the decimal point in the divisor. So once again, there's nothing to line up.
Here he multiplied each number by 100, which eliminates the decimal point in the divisor, and then just divide by the resulting whole number. (But in a sense we do line up some decimal points – in the dividend and the quotient.)
We’ll see all this in more detail in two weeks.