1.2.6. Distributivity of multiplication over addition for rational numbers

To understand this, consider the rational numbers −3/4, 2/3, and −5/6.

Distributivity is a property that involves how multiplication interacts with addition. This property states that when you multiply a number by a sum, it is the same as multiplying each addend separately and then adding the results. In this case, we are looking at the rational numbers -3/4, 2/3, and -5/6 to demonstrate the property.

Think of distributivity like sharing a pizza among friends. If you have three friends and you buy a pizza that costs $10, you can either calculate the total cost by multiplying$10 by 3 (for all the pizzas) or by splitting the pizza first into 3 equal slices, counting how much each person gets, and then figuring the total cost based on the slices they took.

−3 × (2/4 + −5/6) = −3 × (8/24 + −20/24) = −3 × (−12/24).

In this step, we are applying the distributive property. We take −3 and multiply it by the sum of 2/4 and -5/6. The first step is to find a common denominator, which is 24 in this case, so we convert both fractions to have this denominator. After simplifying, we multiply −3 by the resulting fraction -12/24, which simplifies to -1/2.

Imagine you're distributing snacks among two groups of friends. You have 12 cookies (representing the total value of -12 as a negative because it’s a loss in terms of costs). Distributing them evenly among your friends demonstrates how you can break down the process—each friend gets cookies separately just like we distributed values in math.

Therefore, −3 × (2/4 + −5/6) = −3 × 2/4 + −3 × −5/6 = 1/2.

Finally, we compute separately the products after distributing the multiplication. We calculate the parts: -3 times 2/4 gives us a product of -3/2 and for -3 times -5/6 gives us positive results. Adding these results together gives us 1/2, concluding our property demonstration.

Returning to our pizza analogy, if you started with a negative amount (like owing cookies) and then share them evenly, your friends could potentially trade cookies back and forth, balancing each other's snack supply to finally see what’s left, just as we balance equations in math.

For all rational numbers a, b, and c, a × (b + c) = ab + ac and a × (b - c) = ab - ac.

The distributive property can be generalized. It means you can multiply any number by a sum or difference of other numbers and achieve the same result, allowing flexibility in computation. This formula helps simplify calculations and is foundational in algebra.

Imagine you have a job that pays you weekly based on hours worked. If you work overtime, instead of calculating the full amount each time, you can just calculate it based on the basic hours and multiply that figure by both the usual work and the overtime, thus confirming you’re using distributivity to arrive at your final paycheck.

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