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Mar 2, 2013

Bio:Still study in school, but spend most of my time building and inventing new things like machines and robots from scratch found around the house and coming up with new ideas, and I am a completely DIY ...read more »

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active| newest | oldesta/b/bhas two division operators in it, and the value ofa/b/bdepends on theorderin 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,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,according to the standard Order of Operations ruleshttp://www.gnu.org/software/octave/

and also symbolic equation solvers like Maple and Mathematica,

and also programming languages, like C, C++, JavaScript, etc..

The

to do the operations forcorrect waya/b/bis to do the division operation a/bfirst, 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 likea/b/b, is to imagine you were doing it on a cheap handheld calcucator, using literal numbers like10/2/2sincecheap calculatorsdon't do symbolic math likea/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 thefirstdivision operation I gave it,10/2, and it writes that result to the display:5Then 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:2.5In summary:

10/2/2 = 2.5 = 10/4 = 10*(2^-2), just like the sacred rules of the Order of Operations say it should.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}precedence, and therefore should be applied from left to right.In the meantime I

agreemost 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 ?

;-)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.