Counting a Span Versus Finding a Difference between Two Numbers

Date: 06/03/2004 at 23:48:27
From: Linda
Subject: Subtracting related to years and dates

Why is it that the difference between year 2012 and year 2003 equals 
10 years but the difference between the numbers 2012 and 2003 equals 
9?  Why can't you use regular subtraction with dates? 

Date: 06/04/2004 at 09:46:22
From: Doctor Peterson
Subject: Re: Subtracting related to years and dates

Hi, Linda.

I wouldn't say that the difference between 2012 and 2003 is 10 years. 
Rather, there are ten years from 2003 THROUGH 2012. That is not the 
same as a difference!  The problem is that it is hard to define what 
you mean by "difference" in cases like this.

The time from Jan 1, 2003 to Jan 1, 2012 is 9 years, just as you would 
expect from subtraction.  The same is true for, say, June 4, 2003 to 
June 4, 2012.  This is a true difference in time.

It is only when you count inclusively -- asking for the number of 
years from 2003 to 2012 INCLUDING both those years in their entirety 
-- that you have to add 1 to the difference in order to get the right 
answer.  That's because you're adding all of 2012 to the time from Jan 
1, 2003 to Jan 1, 2012, which includes none of 2012.

The same sort of thing is true in other situations, such as counting 
the pages in a book.  If Chapter 5 starts on page 58 and ends on page 
72, and I want to know how many pages there are in that chapter, I 
add 1 to the end page (looking at the page where the NEXT chapter 
starts), and subtract 73-58 = 15.  That tells me that in reading from 
the first page of this chapter to the first page of the next chapter, 
I've advanced 15 pages, which are the pages of this chapter.

"Off-by-one errors" are one of the usual mistakes made by computer 
programmers, for the same reason.  You'd think counting would be 
first-grade stuff, but it can get very tricky sometimes!

If you have any further questions, feel free to write back.

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