Question 1201401: Which of the following are true for all integers a,b, and c?
I.(a+b)+c=a+(b+c)
II.(a/b)/c=a/(b/c)
III.(a+b)/c=a+(b/c)
Found 3 solutions by josgarithmetic, ikleyn, math_tutor2020:
Answer by josgarithmetic(39616)   (Show Source):
Answer by ikleyn(52776)  (Show Source):
You can put this solution on YOUR website! .
Formula (I) is associative law of addition.
It is true for all integer numbers; for all rational numbers;
for all real numbers and for all complex numbers.
Formula (III) is not any general rule, so it not valid "for all integers a, b, and c".
Answer by math_tutor2020(3816) (Show Source):
You can put this solution on YOUR website!
Statement I is true for all integers a,b,c
It's also true for any real numbers a,b,c
For more information, search out "associative law of addition".
The basic idea is that we can add numbers in any grouping order we want. The order doesn't matter.
Examples:
(1+2)+3 = 1+(2+3) since both sides evaluate to 6
(4+7)+2 = 4+(7+2) since both sides evaluate to 13
An almost identical property would be the commutative property of addition. That property is a+b = b+a.
Statement II is NOT true for all integers a,b,c
We can construct a counterexample to see why.
a = 1
b = 2
c = 4
LHS = left hand side = (a/b)/c = (1/2)/4 = 0.5/4 = 0.125 exactly
RHS = right hand side = a/(b/c) = 1/(2/4) = 1/(0.5) = 2
LHS = RHS is not the case since 0.125 = 2 is false
Statement III is also not true in general
Counterexample:
a = 1
b = 2
c = 3
LHS = (a+b)/c = (1+2)/3 = 3/3 = 1
RHS = a + (b/c) = 1 + (2/3) = 1.667 approximately
LHS = RHS is false since 1 = 1.667 is false.
Conclusion: Statement I is the only true statement of the trio.
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