Subtraction on Numbers is Not Associative

Theorem

The operation of subtraction on the numbers is not associative.

That is, in general:

$a - \paren {b - c} \ne \paren {a - b} - c$

By definition of subtraction:

$\ds a - \paren {b - c}$

$=$

$\ds a + \paren {-\paren {b + \paren {-c} } }$

$\ds $

$=$

$\ds a + \paren {-b} + c$

$\ds \paren {a - b} - c$

$=$

$\ds \paren {a + \paren {-b} } + \paren {-c}$

$\ds $

$=$

$\ds a + \paren {-b} + \paren {-c}$

So we see that:

$a - \paren {b - c} = \paren {a - b} - c \iff c = 0$

and so in general:

$a - \paren {b - c} \ne \paren {a - b} - c$

$\blacksquare$

$\ds 5 - \paren {3 - 2}$

$=$

$\ds 5 - 1$

$\ds = 4$

$\ds \paren {5 - 3} - 2$

$=$

$\ds 2 - 2$

$\ds = 0$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.2$. Commutative and associative operations: Example $64$
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Example $2.1$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 28$. Associativity and commutativity
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): associative
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): associative