We’ve looked at how to add or subtract decimals. Now let’s move on to multiplication; we’ll look at three answers to the same sort of question.
What to do: Adding the decimal places
Here’s a question from 1999:
Multiplication How many decimal places should there be in this problem? 3.26 x 0.25
Karen appears to have learned enough about the topic to know that the key idea will be the placement of the decimal point in the answer.
Doctor Micah answered with a complete example of the process, leading up to the specific question:
Thanks for writing to Dr. Math. Decimals can be tricky, but once you learn the rules they aren't so bad. I assume you already know how to multiply two large numbers, such as 327 * 45. If you don't, please write back and we'll explain that also. If you do know how, then multiplying decimals is easy. Just forget about them and do the multiplication the way you normally would for two big numbers. Then add the number of decimal places in each of the numbers you started with. This is how many places from the right you should place the decimal in your result.
If you are not sure about multiplication in general, the last question in this post will dig into that, especially the extra zeroes you’ll see in the work below.
In Karen’s example, each number has two decimal places, so the answer will have four.
Here's an example:
4.25
x 1.53 Now, just leave out the
------ decimals and multiply normally.
425
x 153
-----
1275
21250
+ 42500
-------
65025 Now since 4.25 had two decimal
places and 1.53 had two decimal
places, the answer should have
2 + 2 = 4 decimal places. So:
4.25
x 1.53
-------
6.5025
In practice, we don’t have to actually remove the decimal points, but just ignore them, and then place a decimal point appropriately in the answer.
The only time it's tricky is when your answer ends with zeroes. You have to be sure not to drop the zeroes before you put in the decimal point:
0.45
x 2 Do the same thing; just forget about
----- the decimal and multiply.
45
x 2
----
90 Now, the answer should have
0 + 2 = 2 decimal places. So:
0.45
x 2
-----
0.90 AFTER you add the decimal point,
you can drop the ending zeroes. So:
0.45
x 2
-----
0.9
I hope you can do your own problem now. If you still have trouble, please feel free to write back to Dr. Math.
If you had dropped the zero at the end, you might have thought the answer was .09.
Note also that in this example, the multiplier, 2, had no decimal point at all, so there are no decimal places. We could have written it as “2.”, as we saw last time.
Why add the decimal places?
But why does that procedure work? How can we be sure it’s right? Here’s a question from 2001:
Counting Decimal Places When multiplying decimals why does placing the decimal in the answer by counting the number of decimal places in the problem and counting from the right the same number of places work? I have searched the Internet and even looked in my math book, but I still can't find anything. I asked one of the teachers at my school and he didn't know. My own math teacher won't help because she says I need to learn it on my own whether I get it or not. I would really like to show her up and prove I'm not a complete idiot. All the answers I've tried are apparently not good enough for her.
I answered:
Hi, Craig.
I tried searching our archives for this, hoping I could help you find the answer for yourself, but I didn't find a good complete answer. Here's how it works:
Suppose we are multiplying 1.23 by 4.5.
First, we can convert each number to a fraction by using all its digits as the numerator (with no decimal point), and using for a denominator a 1 followed by as many zeros as there were decimal places, so that 1.23 is 123/100.
You may want to think of it this way, where I repeatedly move the decimal point to the right to multiply by 10, until everything is a whole number:
1.23 12.3 123.
1.23 = ---- = ---- = ----
1.00 10.0 100.
(Remember, if you multiply the numerator and denominator of a fraction by the same number, you haven't changed the value.)
Note that even in this one step, we can either follow a rule, or think about why. That happens throughout math.
Writing both decimals as fractions, we have:
123 45 123*45 5535
1.23 * 4.5 = --- * -- = ------ = ---- = 5.535
100 10 100*10 1000
Now, when I multiplied the fractions, the denominator was the product
of two powers of ten, and turned out to be a 1 followed by the total
number of zeros in the two numbers, which is the total number of
decimal places in the two decimals! When I convert back to a decimal
at the end, that's how many decimal places I need.
Each decimal place in the two factors resulted in a ten in the denominator; so we are dividing by that same total when we place the decimal point at the end.
We could more formally express it this way (if Craig has learned about negative exponents): $$1.23\times4.5=(123\times10^{-2})\times(45\times10^{-1})=(123\times45)\times10^{-2+-1}=5535\times10^{-3}=5.535$$
The addition of exponents adds the numbers of decimal places.
Why not just line up the decimal points?
Here’s a similar question from 2007:
Multiplying Decimal Numbers and Placing the Decimal Point When I multiply something like 24.5 x 36.7, I get the decimal point in the wrong place. 24.5 36.7 -------- 171.5 1470.0 7350.0 ------- 8991.5 answer is 899.15 - get decimal point wrong ....
Tel appears to have treated this as an addition, where the decimal points are lined up. To do it right, it helps to understand the difference.
Doctor Ian answered, again using a different example:
Hi Tel,
I like to work with integers, so I usually change decimals to fractions when I want to multiply or divide:
2.57 * 2.63
257 263 257 * 263
= --- * --- = ---------
100 100 10,000
We write a decimal as a fraction by, in effect, treating it as \(\frac{2.57}{1}\) and multiplying numerator and denominator by 100 (1 followed by as many zeros as there are decimal places) to eliminate the decimals and create a fraction of whole numbers. This is essentially the same as the way we read a decimal as a fraction.
Now I can do the multiplication on top without worrying about decimal points:
257
* 263
-----
771 <- 3 * 257
15420 <- 60 * 257
51400 <- 200 * 257
-----
67591
to get
67,591
------
10,000
And that means the same thing as
6.7591
So the reason we have four decimal places in the answer is that the two places in each factor produce 100 for each denominator; and when we multiply these, we add the number of zeros.
As a quick check, note that I've got 2 point something * 2 point something so the answer should be a little bigger than 4. This can be a good way to make sure you didn't slide a decimal point too far, or not far enough! If I end up with something like 67, I can see right away that's too big; similarly, something like 0.67 would be too small. Anyway, the moral is that being able to go back and forth between decimal and fraction notations is a big advantage, both in being able to explain WHY various decimal operations work the way they do, and in being able to AVOID lots of complicated operations entirely, by working with integers. Does that make sense? Try your problem this way, and please let me know how it works out for you.
This sort of check is particularly important when you do calculations with a calculator, where one of the easiest mistakes to make is to misplace the decimal point. By considering only the whole parts, we can estimate the correct answer and catch huge errors.
Tel replied, asking how this approach relates to what he was taught (like what we showed above):
Got it - a good way to simplify. Thanks. I am sure I used to do it with the decimal point and putting a zero, then 2 zero's - etc... or am I making it up?
Doctor Ian’s suggestion is the thinking that lies behind the method Tel (and the rest of us) learned. He responded:
Hi Tel, The usual way of doing it is to 1. Count up the total number of decimal places. 2. Forget about them, and multiply as if you just have integers. 3. 'Put back' the number of decimal places you found in step (1). Do you see why it amounts to the same thing? We're doing step (1) when we convert to fraction notation and multiply the denominators. We're doing step (2) when we multiply the numerators. We're doing step (3) when we convert back to decimal notation. Once you know what's going on, it's quicker to do the multiplication without explicitly constructing the fractions. But the fraction method is always there as a backup.
So counting decimal places is the usual “what”; thinking about denominators is the “why” in the background, which you could do if you were unsure.
Tel closed:
Even clearer - many thanks indeed!
Why do the decimal places add up? Four reasons
Here’s a question from 2002:
Number of Places in Products of Decimals When you multiply decimals, why is there always the same number of decimal places in the answer as in the problem? Why couldn't there be more or fewer than there are in the problem? For example, if you have a number with two decimal places (like 35.47) times another one with two decimal place (like 57.91), why do there HAVE to be four decimal places in the answer?
Doctor Rick answered at multiple levels, starting with the fraction approach:
Hi, Nicholas.
I could give several different answers to your question.
First: You can regard your example as multiplying 35 47/100 by 59 91/100, or 3547/100 by 5991/100. The denominator will be 100*100, or 10,000. A number of 10,000ths is a number with 4 decimal places. When you're doing the multiplication in the usual way, you ignore the decimal points until the end. Then you count off as many decimal places as there are in the factors combined, and put the decimal point there.
3 5.4 7
* 5 9.9 1
---------
3 5 4 7
3 1 9 2 3
3 1 9 2 3
1 7 7 3 5
---------------
2 1 2 5.0 0 7 7
What you've done is to multiply 3547 by 5991, then divide by 10,000 (when you put in the decimal point).
But it can be a little more subtle, when you think about the final answer:
Second: In a sense, there don't HAVE to be 4 decimal places in the answer. Think again about multiplying decimals: the first answer we get, by multiplying the numerators and multiplying the denominators, may not be in lowest terms; if not, you can reduce the denominator: it might end up being 1000 or even 100 instead of 10,000, which means only 3 or 2 decimals in the answer. Here's an example:
35.25 * 59.92 = 2112.1800
I wrote that with four decimal places in the answer, but the number is the same as 2112.18, which has only two decimal places.
So we put four decimal places in the initial answer, but we don’t need to write them in the final answer. But when that happens, you need to be careful:
Third: When you do this multiplication the usual way, you'll see the zeros at the end, and they must be counted among the 4 decimal places. You can't drop them before you place the decimal point.
3 5.2 5
* 5 9.9 2
---------
7 0 5 0
3 1 7 2 5
3 1 7 2 5
1 7 6 2 5
---------------
2 1 1 2.1 8 0 0
You can't ignore the zeros because you're doing the same thing as before: multiplying the numerators and dividing by 10,000. When you ignore the decimal places, you get 3525 * 5992 = 21121800 - those zeros make a big difference! You're doing this:
35.25 * 69.92 = 3525/100 * 5992/100
3525 * 5992 21121800 211218
= ----------- = -------- = ------ = 2112.18
100 * 100 10000 100
So if you removed two digits at the end before placing the decimal point, you’d also have to count back two fewer places when you do so.
Fourth: In my second answer I said that the answer might not need four decimal places, because 2112.1800 is the same as 2112.18. But for some purposes that isn't true. Sometimes the number of digits we write in an answer tells people how accurate we believe the answer to be. For instance, if I say a line is 34.12 meters long, it implies that the measurement is accurate to the nearest hundredth of a meter. If I only knew the distance to the nearest tenth of a meter, then that final 2 would be only a wild guess and I shouldn't say it. So, if I say the distance is 34.10 meters, this is different from saying that it is 34.1 meters: the extra zero tells people that I believe the answer is correct to the nearest hundredth of a meter, not just the nearest tenth of a meter. If I know the sides of a rectangle are 35.25 meters and 59.92 meters to the nearest hundredth of a meter, then I know the area of that rectangle is 2112.1800 to the nearest 10,000th of a meter. If I said the product was 2112.18, I'd be claiming less precision than I actually have. So in this sense, if I get zeros at the end of my answer, I should leave them there and stick with four decimal places.
I think the point here was simply that zeros on the end, though mathematically they don’t change the value of a number, do imply its precision. Unfortunately, the last paragraph was written a little hastily. Notice what happens if we add an extra uncertain digit to each number, marked as “?”, and carry out the multiplication:
3 5.2 5 ?
* 5 9.9 2 ?
-----------
7 0 5 0 ?
3 1 7 2 5 ?
3 1 7 2 5 ?
1 7 6 2 5 ?
-----------------
2 1 1 2.1 ? ? ? ?
It turns out that not only the digits after the zeros are unknown, but also the last four digits. Only the first five digits of the answer are known, and even the fifth is not really certain. This is because, when we multiply two numbers with four significant digits, only four significant digits of the answer can be trusted. So keeping those zeros would actually claim more precision than you really have, rather than representing it accurately. We should give our answer not as 2112.1800 or 2112.18, but as 2112.
But the rest of what he said is valid.
I hope you can see that there is a good reason why a product of decimals has as many decimal places as the two factors combined.
What about the multiplication itself?
Let’s look at one more question, which is about the whole process, not just the decimal point. This is from 2001:
Adding a Zero When Multiplying a Decimal Hi, I don't understand why you add a zero when multiplying decimals, e.g. 2.3 x 1.4 ----- 9 2 2 3 0 ----- 3.2 2 why add the zero when multiplying the 1? But when you multiply 2.3 x 1.44 you add 2 zeros when you come to multiply by the 1. Why is this? Don't 1 and 1 have the same value?
This time, the question is not about the decimal point, but about the zeros!
The work for the second problem looks like this:
2.3
x 1.4 4
-------
9 2
9 2 0
2 3 0 0
-------
3.2 1 2
This time, there are two zeros after the “23”.
Doctor Rick answered:
Hi, Andrew.
Let's start by reviewing what happens when we multiply whole numbers. You probably do it one of these two ways:
23 23
x 14 x 14
---- ----
92 92
230 23
---- ----
322 322
The only difference is that, in the method on the right, we don't bother to write the zero.
Andrew’s own work is done the first way; we’ve used both ways above.
Here is what's happening in the whole-number case. We can write 14 as 1*10 + 4. Then the product can be written (using the distributive property -- I am assuming you have seen this by now): 14 * 23 = (1*10 + 4)*23 = 1*10*23 + 4*23 = (1*23)*10 + 4*23 The first partial product is 4*23 = 92. The second partial product is (1*23)*10 = 230. That's where the 10 comes from in this case: the 1 is ten TENS, so the 23 is 23 TENS, or 23*10.
So after multiplying 23 by 1, we stick a zero on the end to multiply it by 10, representing the place value of the 1.
How about the decimals?
Now let's take another look at your problem. The only difference between your example and mine is that, in yours, each factor is divided by 10, and therefore the product is divided by 10*10 = 100. I'll put in some more decimal points to make clear what the partial products REALLY mean. (We normally omit these decimal points, just as I was taught not to write that zero, because we don't need to think about decimal points until the final product.)
2.3
x 1.4
-----
.9 2
2.3 0
-----
3.2 2
Here, $$.4\times2.3=\frac{4}{10}\times\frac{23}{10}=\frac{92}{100}=.92$$ and $$1\times2.3=2.3=2.30$$
The problem can be written
1.4 * 2.3 = (1 + 4/10)*2.3
= 1*2.3 + (4*2.3)/10
The first partial product is (4*2.3)/10, or 9.2/10 = 0.92. Since this needs two decimal places, we shift the decimal point of the partial products left (as I have done) to make room for them.
The second partial product is just 1*2.3 = 2.3, so we shouldn't need to shift it - but since we have shifted the decimal point left, the 2.3 has to be shifted left along with its decimal point. That's where the zero (factor of 10) comes from.
As he said, we don’t usually write the decimal points in the partial products, but it’s interesting that the zero we add lines up the decimal points for the coming addition!
Do you see now why you shift the 2.3 left ONE place when the multiplier is 1.4, and TWO places when the multiplier is 1.44? In the latter case, we have this:
2.3
x 1.4 4
-------
.0 9 2
.9 2 0
2.3 0 0
-------
3.2 1 2
The decimal point of the partial products is shifted TWO places left because of the two decimal digits in the multiplier 1.44. This makes room for the three decimal places of 0.04*2.3 = 0.092. Therefore the partial product 1*2.3 is shifted two places left, to stay with the decimal point. The partial product (4*2.3)/10 is shifted only one place left; and the partial product (4*2.3)/100 is not shifted.
Next time, we’ll conclude with division, the most interesting of all.