Does Associative Property Apply to Subtraction and Division Too?

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SUMMARY

The associative property applies to addition and multiplication, as demonstrated by the equations (a+b)+c = a+(b+c) and (ab)c = a(bc). However, it does not apply to subtraction and division. For example, (3-2)-1 results in 0, while 3-(2-1) results in 2, illustrating the inconsistency. Similarly, (2/1)/2 equals 1, while 2/(1/2) equals 4, confirming that the associative property is not valid for these operations.

PREREQUISITES
  • Understanding of basic arithmetic operations: addition, subtraction, multiplication, and division
  • Familiarity with the associative property in mathematics
  • Knowledge of additive and multiplicative inverses
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of arithmetic operations in depth
  • Explore the concept of additive and multiplicative inverses
  • Research the implications of non-associative operations in mathematics
  • Examine examples of non-associative operations in real-world applications
USEFUL FOR

Students, educators, and anyone interested in understanding the fundamental properties of arithmetic operations, particularly in the context of subtraction and division.

bballwaterboy
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I show that the assoc. property applies to addition and multiplication in my book:

(a+b)+c = a+(b+c)
(ab)c = a(bc)

But what about subtraction and division?
 
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What is (3-2)-1? What is 3-(2-1)?
What is (2/1)/2? What is 2/(1/2)?
 
##0-(0-1) = 1 \neq -1 = (0-0)-1##.
##1/(1/2) = 2 \neq \frac12 = (1/1)/2##.
 
jbriggs444 said:
What is (3-2)-1? What is 3-(2-1)?
What is (2/1)/2? What is 2/(1/2)?

:-p Got it! Answer = No, Associative Property does not apply to subtraction and division.
 
It is precisely because the associative law does not apply to subtraction and division that we do NOT think of them as separate operations. Instead "a subtract b" is "a plus the additive inverse of a" and "a divided by b" is "a times the multiplicative inverse of b".
 

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