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The pre-calculus question reads --> state the various properties to easily and mentally compute:

$ 4765+(-896)+ (896+477)+(-4765+23)$

It is easy for me to see how the regrouping allows for easy mental math, so I would have said the associative property. I also see how $-896 + 896$ is the additive inverse property. However, the instructor indicated that this is problem also uses the commutative property and I don't see how this would apply here.

Can someone explain how this can be the commutative prop? Teacher just indicated that it is clear that those 3 properties were used.

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    $\begingroup$ You have $4765$ and $-4765$ separatedly, but you can cancel them anyway (as you previously did with $-896$ and $896$). $\endgroup$ – A. Salguero-Alarcón Feb 10 '17 at 22:39
  • $\begingroup$ You need commutivity to group the 4765's. $\endgroup$ – Kaynex Feb 10 '17 at 22:47
  • $\begingroup$ You need commutivity to move that -4765 next to that 4765. So of course it is useful. $\endgroup$ – fleablood Feb 10 '17 at 22:47
  • $\begingroup$ I'd simply say. Asciativity says we can regroup any way we want so we can drop and reintroduce the parentheses as we wish. Commutativity allows us to rearrange any way we want. So you can rearrange to (4765 -4765) + (896-896) + (477 +23) = 477 + 23 = 496. But I imagine the instruct wants you to isolate ever single step one by one. Which isn't hard but is tedious. $\endgroup$ – fleablood Feb 10 '17 at 22:52
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4765+(−896)+(896+477)+(−4765+23)

= 4765+ (−896)+ 896 + 477 + (−4765) + 23 [associative property -- we can re-group the additions in any combinations as long as we keep to addition of negatives, not subtraction]

= 4765+ ((−896)+ 896) + 477 + (−4765) + 23 [associative property again]

= 4765 + 0 + 477 + (−4765) + 23 [property of opposite or additive inverse of a real number]

= 4765 + 477 + (−4765) + 23 [property of zero as additive identity]

= 4765 + (−4765) + 477 + 23 [commutative property of addition, re-order a + b = b + a]

= (4765 + (−4765)) + 477 + 23 [associative property again]

= 0 + 477 + 23 [property of opposite or additive inverse again]

= 477 + 23 [property of zero as additive identity again]

= 400 + 70 + 7 + 20 + 3 [place value conventions of number system]

= 400 + 70 + 20 + 7 + 3 [commutative property again]

= 400 + 90 + 10 = 400 + 100 = 500 [addition facts and using properties of base ten system to "carry"]

This all may look insanely obvious to you. Spend a little time helping some kids who are having problems with arithmetic in Grades 1 to 3 and you will learn a new respect for the complexities and subtleties of numbers which you blissfully skim over every day, because you were lucky enough to master all of these skills when you were young.

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  • $\begingroup$ Thank you very much. This helps. $\endgroup$ – user163862 Feb 10 '17 at 23:19
  • $\begingroup$ You're very welcome. $\endgroup$ – victoria Feb 10 '17 at 23:21
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Commutative property allows for the moving of terms left and right. We would use the commutative property to move the $-4765$ term, and could even move the $896$ term so that it is even more clear. In essence, we would use the associate property to remove all parenthesis at the beginning, i.e., $$4765 + (-896) + (896+477) + (-4765 + 23) = \\4765 + (-896) + 896+477 + (-4765) + 23.$$ Then use the commutative property to reorder the terms, i.e., $$4765 + (-896) + 896+477 + (-4765) + 23 = \\ 4765 + (-4765) + 896 + (-896) + 477 + 23.$$ Then use the associative property to regroup the terms, i.e., $$4765 + (-4765) + 896 + (-896) + 477 + 23 = \\ (4765 +(-4765)) + (896+(-896))+(477+23).$$ Then use the additive identity to note the first two associated terms are zero and then the last is easily seen to be $500$, i.e., $$(4765 +(-4765)) + (896+(-896))+(477+23) = 0 + 0 + 500 = 500.$$

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  • $\begingroup$ Thank you very much for taking the time to clarify this. $\endgroup$ – user163862 Feb 10 '17 at 23:19
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Slow and pedantic:

$4765+(-896)+ (896+477)+(-4765+23)=$

$4765+(-896+896) + (477 +(-4765)) + 23= $ (Associativity)

$4765 + (-896+ 896) + (-4765 + 477) + 23 = $ (Commutivity)

$4765 + (-896+896) + (-4765) + (477 +23) = $(Associativity)

$4765 + (-4765) + (-896 + 896) + (477 + 23) = $(Commutivity)

$(4765 - 4765) + (-896 + 896) + (477+23) = $(Associativity)

$0 + 0 + (477+23) =$ (Additive Inverse)

$477 +23=$ (Additive Identity)

$(470 + 7) + (20 + 3)=$ (Associativity)

$470 + (7 + 20) + 3 = $(Associativity)

$470 (20 + 7) + 3 = $ (Commutativity)

$470 + 20 + (7+3) = $ (Associativity)

$470 + 20 + 10 = $ (Just plain ordinary arithmetic for f### sake)

$(400 + 70) + (20+10) = $ (Associativity)

$ (400 + 70) + 30 = $ (Just plain ordinary arithmetic for f### sake)

$400 + (70 + 30) = $(Associativity)

$400 + 100 = $(Just plain ordinary arithmetic for f### sake)

$500$ (Just plain ordinary arithmetic for f### sake)

Fast an furious.

Associativity means we can group and ungroup at will so

$4765+(−896)+(896+477)+(−4765+23)= 4765+(−896)+896+400 + 70 +7+(−4765)+20 +3=$

Commutativity means we can reorder at will so

$= 4765 - 4765 + 896 - 896 + 400 + 70 + 20 + 7+ 3 =$

And associativity again: $= (4765-4765) + (896 - 896) + (400 + (70 + (20 + (7+3)))$

Additive identity and inverses:

$= 0 + 0 + (400 + (70 + (20 + (7+3)))$

$=400 + (70 + (20 + (7+3))$

And finally, just ordinary arithmetic for f###s sake:

$=400 + (70 + (20 + 10)) = 400 +(70+30) = 400 + 100= 500$

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