1.2.3. Associativity

Recall the associativity of the four operations for whole numbers through this table:

In this chunk, we are discussing the associativity of addition for whole numbers. Associativity refers to the property where the way in which numbers are grouped in addition (or multiplication) does not change the result. For whole numbers, when adding three numbers, it doesn't matter how we group them; we will always get the same sum. For example, when evaluating (2 + 3) + 4 and 2 + (3 + 4), both will yield 9. However, subtraction is not associative, meaning grouping affects the outcome.

Think of adding fruits. If you have 2 apples, 3 oranges, and 4 bananas, it doesn't matter whether you first count the apples with oranges, or then add bananas. The total number of fruits remains the same. Conversely, if you were to take away fruits, the order in which you take them away can change how many remain (e.g., taking away 4 fruits from 5 is different from taking away 3 then 2).

Multiplication is associative. Is 7 × (2 × 5) = (7 × 2) × 5?

Is 4 × (6 × 0) = (4 × 6) × 0? For any three whole numbers a, b and c a × (b × c) = (a × b) × c.

This chunk focuses on the multiplication of whole numbers and highlights that multiplication is associative. This means that when multiplying three numbers together, we can regroup them without affecting the result. For instance, (3 × 2) × 4 = 6 × 4 = 24, and 3 × (2 × 4) = 3 × 8 = 24; hence, both expressions yield the same product. This property allows flexibility in calculations and simplifications.

Consider arranging boxes in configurations. If you have 2 rows of 3 boxes and 4 layers high, whether you calculate the boxes in layers or rows first, you will always end up with the same total boxes. Just like stacking the boxes does not change the total number.

Associativity of the four operations for integers can be seen from this table:

This chunk discusses the associativity of addition and multiplication for integers, explaining that both operations are associative similar to whole numbers. For example, rearranging how we add or multiply does not change the total. However, subtraction and division are not associative since changing grouping alters the outcome. For instance, 5 - (7 - 3) gives a different result compared to (5 - 7) - 3.

Imagine planning a trip with multiple stops. If you add the distances by regrouping the stops, the overall distance can remain unchanged (e.g., adding leg of the trip), but if you were to subtract certain distances and change the order, you'd likely end up with a different total distance remaining.

Refer to the addition of rational numbers:

−2 [3 (−5)] = −2 + (3 + (−5)) = (−2 + 3) + (−5) = 1 + (−5) = −4.

For any three rational numbers a, b and c, a + (b + c) = (a + b) + c.

This section shows how rational numbers also maintain the property of associativity for addition. No matter how we group the numbers, such as in the operation above, the result remains consistent. It's essential for calculations involving fractions, as we often add more than two fractions. This property ensures that we can compute sums in a flexible manner.

Think about sharing pizza slices. If you're sharing slices among friends, how you group the sharing does not change the number of slices eaten. Whether you combine groups of 2 and 3 first or share them together, the total number of slices each person receives remains the same.

Subtraction is not associative for rational numbers. Is −2 [−4 1] = −3? Check for yourself. Division is not associative for rational numbers.

In this part, we reiterate that subtraction and division do not hold the associative property in rational numbers. For instance, when you consider (3 - 2) - 1 or 3 - (2 - 1), you get different results based on how the numbers are grouped. Similarly, for division, dividing in different orders does not yield the same results, further solidifying that these operations lack associativity.

Imagine you have several tasks to complete based on a deadline. If you decide to subtract certain hours from your pool of time without careful consideration of the steps involved, changing the order can lead to remaining time discrepancies. For division, think of splitting a pie among friends; the way you decide to divide the pie can yield different portions to individuals based on the order in which you split it.

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