Subtraction on Numbers is Not Associative

Theorem

The operation of subtraction on the numbers is not associative.

That is, in general:

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

Proof

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:

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

$\blacksquare$

Examples

$5$ minus $3$ minus $2$

\(\ds 5 - \paren {3 - 2}\)

\(=\)

\(\ds 5 - 1\)

\(\ds = 4\)

\(\ds \paren {5 - 3} - 2\)

\(=\)

\(\ds 2 - 2\)

\(\ds = 0\)

Sources

• 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