The long division algorithm, explained at last March 29, 2011

We’ll stop taking long division for granted in a minute, but first, let’s take a general look at division.  What is division?  Division is the shortcut for subtracting one value from another value repeatedly until we reach zero.  We say that “90 divided by 10 is equal to 9″ because 90-10-10-10-10-10-10-10-10-10 = 0.  Using the distributive property, we can rewrite this as 90 – 9(10) = 0.

 

If we then made steps to solve this equation, we’d get:  

 

90 – 10(9) = 0

     +10(9)    + 10(9)

90            =  10(9)      90 is “ten, nine times”.  Division is the inverse of multiplication.  

 

But division also works in situations where we would not eventually subtract to end with zero.  For example, we may end with the number 5.  “95 divided by 10 is equal to 9.5″ because 90-10-10-10-10-10-10-10-10-10-5, and that last 5 is one-half (.5) of the 10 we had been subtracting from 90.  This 5 is also known as a “remainder”.  The result of “95 divided by 10” can then be written as “nine with a remainder of 5”.  This remainder is a smaller number than the number being repeatedly subtracted, so does not quite fit in (“remains” outside). 

 

95 – 10(9) = 5

     +10(9)     + 10(9)

95            = 5 + 10(9)

95            = 10(9) + 5     [commutative property]

                                        95 is “ten nine times with five left over”               

 

However, subtracting repeatedly can be tedious for large numbers, such as “950 divided by 10”, so the long division algorithm was developed.  Formatting “950 divided by 10” to the algorithm, we would write: 

 

 

The sideways “L” is referred to as the “division bar” in this paper. 

 

Here, the decimal can be expanded to 950.00 because of the definition of our number system.  This also makes room for any possible “remainders”, or numbers less than 10 that are left over once we subtract all the 10’s from 950 that we can. 

 

To work this algorithm, we want to first ask ourselves, “how many 10’s can be subtracted from 9?”  We want to remember both: that our definition of division is “repeated subtraction” and that “multiplication is division’s inverse”.  10×9 is 90, not “9”.  We’d answer with “zero” and imagine a “0” above the 9 in 950.  Here, we’ll put in a zero, but usually we wouldn’t. 

 

 

Usually, when the number above the division bar is not zero, we would move to the second step of the algorithm.  But since our “0” is imaginary, we won’t move to step two quite yet. 

 

Starting the algorithm over, we’d then move in from the “9” and look at “95” and ask ourselves, “How many 10s cam be subtracted from 95?”  We’d answer with “9” and place it above the tens spot in 950.  

 

 

Our placement of the “9” actually represents a “90” as it is vertically aligned with the 950’s tens placement.  Had we placed this “9” over the 9 in 950, it would have represented a 900.  By placing our “9” over the 5 in 950, we’re saying that “ninety 10s can be subtracted from 950”, which we know to be true.  If you notice the placement of this “90”, you will see that its 9 is vertically aligned with the 950’s hundreds placement.  Does this make sense? (remember the definition of division as multiplication’s inverse!)

 

Continuing to step two of the algorithm, we write this “90” below the “95” to find the next remainder, remembering that this “90” actually represents a “900” because of its placement. 

 

 

We are left with “5”, which is actually a “50” because of its placement below the tens spot of the 950.  The “5” in the original 950 is in the tens spot, therefore our remainder of 5 is actually a remainder of 5×10, or 50. 

 

The algorithm then tells us to “bring down” the next digit in the original 950 and repeat the algorithm.  But what is the purpose of this “bringing down”?  We always need to keep place value intact and our “5” is really a “50”.  By bringing down the “0”, we accomplish this.  Also, remembering that our “9” is a 90 because of place value and that division is repeated subtraction, we have:

 

950 – 10(90), which yields 50 (not 5) 

 

We bring down the “0” from 950 and place it next to our 5 remainder to make a “50”, creating the 50 we already knew to be true. 

 

 

If that remainder of “5” had been a “0”, we’d be moving into the ones place.  Because it is a “5” and aligned in the tens spot of “950”, we’re still working within the tens.  We then ask ourselves, “How many 10s can be subtracted from 50?”, and continue on with the algorithm.

 

 

We end here by saying that “950 divided by 10 is 95”.  In the division algorithm, when we end with a “0”, we stop the algorithm because it means that the divisor (10) can be subtracted from the divisee “950” an even amount of times with no remainders.  But we can use this same algorithm when dividing any number by any other number. 

 

Comments welcome!

 

 

Dividing by Zero Blows up the Universe! June 15, 2010

“Because the universe will blow up,” was the usual answer I got when my teachers tried to explain why we couldn’t divide by zero.  From a young age, I was a sort of anti-Pythagorean in that I believed people created numbers, not that the universe was ruled by them.  So why then did we create the divide-by-zero bomb? 

 

The best way I’ve found to describe why dividing by zero will destroy everything is to go back to translating fractions.  What does “1/2” really mean?  “1/2” translates to “1 out of 2” or “I have one piece of candy out of the two pieces on the table, so I have half of what is on the table.  My sister is a good sharer.”

 

Now try this with “0/2”.  This translates to “zero out of 2” or “I have zero pieces of the two that are on the table.  My sister’s cheap!”

 

Both of these situations are real.  You can have one piece of candy out of two.  You can have none of the pieces of candy.  Even if the fraction is an improper fraction, like “3/2”, certainly you can’t have three out of two pieces of candy; this makes no sense at all.  But then we remember that improper fractions can be written into mixed fractions, so “3/2” becomes “1 and ½”, and we sure can have 1 and a half of the pieces of candy on the table [leaving our cheap sister with just ½!  Haha!]!

 

So then comes “2/0”, which would translate to “2 out of zero” or “I have two pieces of candy out of the zero that are on the table.”  HUH??  This obviously doesn’t make sense! Despite what Little Orphan Annie and Jay-Z may lead us to believe, you can’t make something out of nothing.  It’s just basic physics. 

 

Once a student begins learning about slope and functions, the impossibility of “2/0” becomes even more obvious.  Let’s think of a graph that measures your height against your age.  “2/0” represents a rise (y-value or “height”) of 2 and a run (x-value or “time”) of 0.  This is to say that, for example, at time 0 you are 2 feet tall.  Ok, so maybe you were born 2 feet tall.  That’s possible.  Now let’s move up from coordinate (0, 2).  The slope of “2/0” tells us to move up 2 and over 0.  We move up two spaces to 4 feet tall and over to… over to nothing!  We stay at zero!  So a slope of “2/0” says that you can be 2 feet and 4 feet tall at the same point in time.  This is impossible!

To the Zero! [power]

“Any number to the zero power is equal to 1,” my teachers would say. “Why?” the students would ask. “Because.” the teachers would declare. And this would usually end it. Sometimes a persistent student would again ask, “Why?” to which he’d get the slightly more creative answer, “That’s just the way it is.” Because of the mystery surrounding the zero power as a kid, I always got the feeling that there was something supernatural about it; my teachers held the key and didn’t want to share.  

 

It wasn’t until graduate school that I finally found out why a number to the zero power always equaled one and how to explain this once mysterious phenomenon.

 

Now as a math teacher, I take the time to explain the zero power. It’s not mysterious, it’s just division. Whenever a question about the zero power comes up, I stop, go to a side board, and ask “What’s 2^3? How about 2^2?” When the kids answer “8” and “4”, I then ask, “Ok, what’s “2^1?” I create a table of their answers on the board:

 

    2^3             2^2              2^1

     8                 4                   2

 

We then step back and look at the pattern.

 

Soon the students see that the common difference between numbers is 2. “We divide by 2 to get to the next number,” they say. I then go on to ask, “Then what’s 2^0?”

 

Of course most students will answer 0 at first. I’ve realized that it’s a natural reaction to answer “zero” whenever hearing “zero” and “multiply” or “divide” within the same lesson. Other students will put 2^0 into the calculator and answer “1”, but that’s what I did as a student and it wasn’t good enough. I always wanted to know why.

 

I direct all students back to the table we created to find the next term “2^0”.

 

    2^3             2^2              2^1             2^0

      8                  4                 2                ??

 

“If we divide by 2 each time to get to the next term, what’s 2 divided by 2?” I ask. The students will answer, with an “oh, obviously!” tone, “1.”

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Another way to think of this concept is through exponent rules:

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To divide terms with exponets, we subtract the exponents, here giving us a 0 exponent.  If we then back up and look at the original fraction, we have a number over itself, which is equal to 1.  Therefore, any number to the 0 power is equal to 1.

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