Ignoring Zeros

Date: 09/03/98 at 17:41:22
From: Cole Krivoski
Subject: Math - Decimals

Here's the problem. I just want to know what to do. 

You have 0.76 and 0.760 and you have to find out which number is 
greater or less, or if they are equal. My teacher says they are equal. 

The thing I do not understand is the number 0.760. Is point 760 
thousandths right or wrong? So shouldn't 0.760 be greater? Or does the 
zero not count?

Date: 09/04/98 at 13:05:14
From: Doctor Peterson
Subject: Re: Math - Decimals

Hi, Cole. This is an important question! Basically the answer is, as 
you suggested, that the zero doesn't count; but it's important to 
understand why that particular zero can be ignored, but others can't.

There are several ways to explain the meaning of a decimal. One is to 
treat it as one big fraction, so that:

           76               760
   0.76 = ---  and 0.760 = ----
          100              1000

Now look closely, and you'll see that you can simplify 760/1000 by 
dividing the numerator and denominator by 10 to get 76/100. Does that 
look familiar? I've just shown that these numbers are equal.

Another way to look at decimals is by place value, just as you did for 
whole numbers. Then we can say that

                 1        1          1
    0.760 = 7 * -- + 6 * --- + 0 * ----
                10       100       1000

Now do you see that the zero doesn't add anything to the number? 
That's why it can be ignored. It's actually the same reason you can 
ignore the zero in 076, which just means no hundreds, just as this 
zero means no thousandths.

Now what about zeroes that aren't at the end? Look at the meaning of 
0.706:

                 1        1          1
    0.706 = 7 * -- + 0 * --- + 6 * ----
                10       100       1000

That's not the same as 0.76, because now the 0 does something: it 
changes the meaning of the 6 from 6 hundredths (in 0.76) to 6 
thousandths (in 0.706).

So the rule is: when there is a zero at the left side of a number (to 
the left of the decimal point), or at the right side of a number (to 
the right of the decimal point), you can ignore it. If a zero is 
between two non-zero digits, or between a digit and the decimal point, 
you have to pay attention to it.

I hope that helps.

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/