How to mathematically prove that a/b/b = a/b^2?


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The expression a/b/b has two division operators in it, and the value of a/b/b depends on the order in which those division operations are performed, and the math jargon for the rules that determine which division operation, on which operands, gets done first, which gets done second,  in math jargon this is called the Order of Operations,
http://en.wikipedia.org/wiki/Order_of_operations

But I am getting ahead of myself.  I'm going to go back to that expression a/b/b, and write in some parentheses to indicate which division operation gets done first.

Suppose I put some parentheses around b/b, and do that operation first:

Step 1:   a/(b/b) = ? 
 
Step 2:  a/(b/b) = a/1,  using substitution (b/b)=1

Step 3:    = a/1 = a,  since a/1=a

But of course that's the wrong answer, according to the standard Order of Operations rules linked above, the same rules used by pretty much all modern machine-based calculators, including everything from cheap dollar store handheld calculators to heavy-duty number-crunching applications like MATLAB or Octave,
http://www.gnu.org/software/octave/
and also symbolic equation solvers like Maple and Mathematica,
and also programming languages, like C, C++, JavaScript, etc..

The correct way to do the operations for a/b/b is to do the division operation a/b first, like so:

Step 1:   (a/b)/b = ?

Step 2: = (a/b)/b = (a*(1/b))/b,  since x/b = x*(1/b)

Step 3: = (a*(1/b))/b = (a*(1/b))*(1/b) , since y/b = y*(1/b)

Step 4: = (a*(1/b))*(1/b) = a*(1/b)*(1/b),  using associative property of multiplication
(See: http://en.wikipedia.org/wiki/Multiplication#Properties)

Step 5: = a*(1/b)*(1/b) = a*((1/b)*(1/b) , using associative property of multiplication

Step 6: a*((1/b)*(1/b) = a*b^-2, using substitution (1/b)*(1/b) = b^-2


By the way, an easy way to remember the correct Order of Operations for something like a/b/b, is to imagine you were doing it on a cheap handheld calcucator, using literal numbers like 10/2/2  since cheap calculators don't do symbolic math like a/b/b (at the time of this writing).

First I press the [1] button , then the [0] button, entering decimal number "10" as the first (also leftmost, since English language is read from left to right) operand.

Then I press the division key [/]. The I press [2]. 

Then I press the division key [/] again.  Upon pressing this key, the calculator decides to do the first division operation I gave it, 10/2, and it writes that result to the display: 5

 Then I press the division key [/] again. Then I press [2].  Then, because I don't have any other operations to perform, I press the equals [=]  button, and the calculators display says: 2.5

In summary: 10/2/2 = 2.5 = 10/4 = 10*(2^-2), just like the sacred rules of the Order of Operations say it should.

iceng4 years ago
First definition option ....... a/(b/b) = a/1 = a

Second .....
(a/b)/b = (a ×1/b) × 1/b = a × (1/b × 1/b) = a × (1/ (b × b)) = a × (1/b^2 ) = a/b^2
kelseymh iceng4 years ago
But your "first definition option" is incorrect, according to standard arithmetic convention. Without parentheses, the two division operators have the same precedence, and therefore should be applied from left to right.
iceng kelseymh4 years ago
Ill pass that on to the HS instructor when I too pass.

In the meantime I agree most math operators do work left to right.
I say most because I have not seen all.

Another instructor taught me ;
when there is a chance of misinterpretation use parenthesis to be clear !

Are we teaching yet ?  ;-)

kelseymh iceng4 years ago
I definitely agree with that latter recommendation, and this case from the student is an excellent example.

Normally, with proper mathematical notation, the double division would be written with a horizontal bar, and both 'b's would appear below that bar, making the parentheses (now surrounding the product b*b) implicit.