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July 1st, 2017, 07:41 AM   #1
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Definition of Division

Help me out with this intuition.


(124 + 104 + 84 + 64 + 44 + 24) / (62 + 52 + 42 + 32 + 22 + 12)

Why is the answer 2 and not 12.

If division is multiplication of the reciprocal and we pair a number from the
left with one from the right that should get us

2 + 2 + 2 +2 +2 +2 ?


124 X 1/62 + 104 x 1/52 ...
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July 1st, 2017, 09:09 AM   #2
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because


$\displaystyle \frac{{\left( {124 + 104 + 84 + 64 + 44 + 24} \right)}}{{\left( {62 + 52 + 42 + 32 + 22 + 12} \right)}} = \frac{{444}}{{222}}$


The rules are that we evaluate the contents of brackets before anything else.

So we must add up the contents of the brackets before we can perform the division.
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July 1st, 2017, 09:12 AM   #3
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Quote:
Originally Posted by studiot View Post
because


$\displaystyle \frac{{\left( {124 + 104 + 84 + 64 + 44 + 24} \right)}}{{\left( {62 + 52 + 42 + 32 + 22 + 12} \right)}} = \frac{{444}}{{222}}$


The rules are that we evaluate the contents of brackets before anything else.

So we must add up the contents of the brackets before we can perform the division.
Okay thanks, more newbie questions are sure to come!

if the brackets where not there then pairing is legitimate?
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July 1st, 2017, 09:19 AM   #4
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Quote:
Originally Posted by Rujaxso View Post
if the brackets where not there then pairing is legitimate?
if the brackets were not there:


124 + 104 + 84 + 64 + 44 + 24 / 62 + 52 + 42 + 32 + 22 + 12

then you would have to follow general simplification rule, doing the division first then rest of the addition.
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July 1st, 2017, 09:22 AM   #5
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Quote:
Originally Posted by Rujaxso View Post
Okay thanks, more newbie questions are sure to come!

if the brackets where not there then pairing is legitimate?
No there are no explicit rules that allow the sort of pairing you mean.

The rules for order of precedence are

Brackets
Order or powers
Division
Multiplication
Addition
Subtraction.

Often shortened to BODMAS

https://www.google.co.uk/search?hl=e...04.TI89T4X7jU0

In your case these rules would make the computation

124 + 104 + 84 + 64 + 44 + 0.38 + 52 + 42 + 32 + 22 + 12 = 580.38
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Last edited by studiot; July 1st, 2017 at 09:46 AM.
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July 1st, 2017, 09:35 AM   #6
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PEMDAS BODMAS must be a UK thing

I guess my idea of using the distribution over addition from multiplication by turning the division into multiplication because dividing is the same as multiplying by the reciprocal does not allow me to do so? Cause you can pair by distribution eg. A(B+C) = AB + AC


A + B + C + D / E + F + G + H = A + B + C + D X 1/e + 1/f + 1/g + 1/h

= A/E + A/F +A/G + A/H

Last edited by Rujaxso; July 1st, 2017 at 09:39 AM.
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July 1st, 2017, 09:56 AM   #7
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The rules allow you to write a sum of ratios


$\displaystyle \frac{A}{E} + \frac{B}{F} + \frac{C}{G} + \frac{D}{H} = \frac{{BCD + ACD + ABD + ABC}}{{EFGH}}$


I was going to ask you to see if you could write your expression using the rules and I see you had the same idea, but have not yet got it quite right.

I assume you have mistakenly meant to write these and not all As as above.

You can also include brackets, but it doesn't add anything new.


$\displaystyle \left( {\frac{A}{E}} \right) + \left( {\frac{B}{F}} \right) + \left( {\frac{C}{G}} \right) + \left( {\frac{D}{H}} \right)$
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July 1st, 2017, 09:56 AM   #8
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I need to start typing my stuff in using Latex or whatever you call it. =p

So if you plug the original numbers you get 12?

Last edited by Rujaxso; July 1st, 2017 at 10:08 AM.
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July 1st, 2017, 10:09 AM   #9
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Quote:
Originally Posted by Rujaxso View Post
PEMDAS BODMAS must be a UK thing

I guess my idea of using the distribution over addition from multiplication by turning the division into multiplication because dividing is the same as multiplying by the reciprocal does not allow me to do so? Cause you can pair by distribution eg. A(B+C) = AB + AC


A + B + C + D / E + F + G + H = A + B + C + D X 1/e + 1/f + 1/g + 1/h

= A/E + A/F +A/G + A/H
PEMDAS is the US version of the UK's BODMAS. Just a difference between American and British English.

It is not a mathematical principle, but a convention of mathematical notation to allow communication in an unambiguous way.

$(124 + 104 + 84 + 64 + 44 + 24)\ /\ (62 + 52 + 42 + 32 + 22 + 12) =\\ \dfrac{124 + 104 + 84 + 64 + 44 + 24}{62 + 52 + 42 + 32 + 22 + 12} = \dfrac{444}{222} = 2.$

What is enclosed in parentheses (brackets} represents a single number that must be evaluated before doing anything else because that is how mathematicians have agreed to communicate with each other. You could call a table a glimpf and understand what you mean, but people would not understand what you were talking about at all. Language is a social construct and so is mathematical notation.

If you wanted to say

$\dfrac{124}{62} + \dfrac{104}{52} + \dfrac{84}{42} + \dfrac{64}{32} + \dfrac{44}{22} + \dfrac{24}{12} =\\
2 + 2 + 2 + 2 + 2 + 2 = 12$

in in-line format you would write it as

$124/62 + 104/52 + 84/42 + 64/32 + 44/22 + 24/12 =\\
2 + 2 + 2 + 2 + 2 + 2 = 12$

because , in PEMDAS notation, D for division precedes A for addition. You do all divisions before you do any additions.
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Last edited by JeffM1; July 1st, 2017 at 10:11 AM.
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July 1st, 2017, 10:16 AM   #10
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$\displaystyle \frac{A+B+C+D}{E+F+G+H}\neq \frac{A}{E}+\frac{B}{F}+\frac{C}{G}+\frac{D}{H}$
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