Conjecture About Squares of Consecutive NumbersDate: 08/28/2001 at 22:06:58 From: Bryan Subject: Squares of Consecutive Numbers I noticed last year that when any two rational numbers whose absolute values are 1 apart are squared, the difference between the two resultant numbers is equal to the sum of the two original numbers. Example: 4^2 - 3^2 = 4 + 3 4.3^2 - 3.3^2 = 4.3 + 3.3 (with a negative number use absolute value) (-3)^2 - 2^2 = |-3| + |2| [(-3)^2] - [(-2)^2] = |-3| + |-2| or (-3) + (-2) since both numbers are negative I was unable to find anything about it in the archives, possibly because I have no idea what to call it. Date: 08/29/2001 at 09:06:05 From: Doctor Rick Subject: Re: Squares of Consecutive Numbers Hi, Bryan, thanks for writing to Ask Dr. Math. I don't know that it has a name, but it can be proved in general by algebra. Let the first number be n; then the second number is (n+1). The difference of their squares is |(n+1)^2 - n^2| |n^2 + 2n + 1 - n^2| |2n + 1| |n + (n+1)| which is the sum of the two numbers. Note that this is true not just for rational numbers, but for all numbers. I used the definition of "difference" as the greater number minus the lesser, which is the absolute value of (either number minus the other). Note that there is one difference between my result and your statement: you put the absolute values around each number separately: (n+1)^2 - n^2 = |n| + |n+1| I put the absolute values around the sum: (n+1)^2 - n^2 = |n + (n+1)| These are the same unless the numbers have different signs. Let's try n = -1/4 and n+1 = 3/4: |(-1/4)^2 - (3/4)^2| = |1/16 - 9/16| = |-8/16| = 8/16 = 1/2 |n| + |n+1| = |-1/4| + |3/4| = 1/4 + 3/4 = 1 so your statement isn't true in this case. On the other hand, |n + (n+1)| = |-1/4 + 3/4| = |1/2| = 1/2 The difference of the squares is equal to the absolute value of the sum, not the sum of the absolute values. You were very observant to notice the pattern that the difference of the squares of consecutive numbers is the sum of the numbers. You had a good idea when you thought about extending the pattern to include negative numbers. You found a way that worked for the numbers you checked. Good work! Your work was what we call "induction." You noticed a bunch of facts and made a conjecture: a pattern that seemed to fit the facts. You tried adding new facts - negative numbers - and found that you had to change your pattern slightly to account for all the new facts you tried. You were still using induction - an important part of math. I have shown you the next step that a mathematician takes. The next step is "deduction": you try to PROVE that the pattern works for ALL numbers, using the rules of numbers. Just testing the pattern on lots of numbers doesn't prove that it works for ALL numbers. In fact, I have shown that your pattern doesn't work for n between -1 and 0. Using algebra to try to prove your conjecture, I ended up proving a modified theorem. Now we KNOW that this theorem is true for ALL numbers. It's no longer a conjecture. Thanks again for writing. This has been an interesting exploration of the way math works, and I hope it is helpful to you. Keep up the good work! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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