In this case, it simplifies to 100/2 = 50. Now let’s try this:
4,000 / (40/10) / 5We do the division in parentheses first, getting
4,000/4/5Starting at the left, we get 4,000/4 = 1,000. Then dividing 1,000 by 5 gives us 200.
Here’s a challenge!
Here is a riddle that ought to get your brain running at full speed. Consider this infinite product:
1 × (−1)× 1 × (−1)× 1 × (−1)× 1 × (−1)× ···If we start multiplying from left to right, we get a sequence of products that come out like this, in order:
1,−1,−1, 1, 1, −1,−1, 1, ...The products switch back and forth between 1 and −1, endlessly. The final product therefore cannot
be defined. No integer can have two values “at the same time.” But suppose we try to apply the com-
mutative law to the original expression infinitely many times! We can then rearrange the factors to
get this:
(−1)× (−1)× 1 × 1 × (−1)× (−1)× 1 × 1 × ···Now imagine that we try to group the integers in pairs, infinitely many times, like this:
[(−1)× (−1)]× [1 × 1] × [(−1)× (−1)]× [1 × 1] × ···That gives us
1 × 1 × 1 × 1 × ···The sequence of products in this case is
1, 1, 1, 1, ...Now it seems as if the product of the whole thing is equal to 1! What’s going on?
Solution
We derived a contradiction here because we improperly used the commutative and grouping laws. First,
we tried to apply these rules to a product that is undefined in its most basic form. Second, we acted as if
these rules can be used in a single expression infinitely many times, without really knowing if we can get
away with such tricks. Evidently we can’t!
The Associative Law for Multiplication 77