Adding a Zero When Multiplying a Decimal

Date: 06/21/2001 at 07:17:01
From: Andrew 
Subject: Adding the 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?

Date: 06/21/2001 at 10:40:05
From: Doctor Rick
Subject: Re: Adding the zero when multiplying a decimal

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.

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.

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

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.

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.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/