algebra life 2

Associativity

A lot of algebraic structures, like the addition and multiplication of real numbers or the rational numbers have the associative property, which is often taken for granted.

It basically says that given a binary operation ∗ , is associative if for all three elements x, y, and z in a given set S, (x ∗ y)  ∗ z = x ∗ (y ∗ z). Then the products can be unambigiously written as x  ∗ y  ∗ z.

From this, the general associative law follows in that for any number of elements, their products can be unambiguiously written without any parentheses.

This can be proven by mathematical induction, but it is not the easiest one to formulate.

As a non-math major, it took me some time to absorb this fact.

Non-associativity

Without the associative property, as the number of elements increase, the number of possible products increase quickly (look at the Catalan numbers for a formula).

Operations that don’t have associativity include subtraction, division and exponention.

For example, we can see that (2/3)/5 does not equal 2/(3/5). The first equates to 2/15 and the second equates to 10/3.

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