Unit 3: Order of Operations
We can use our four basic operations to solve so many types of mathematical and real-world problems, but we need to be careful when we combine or repeat them. The commutative and associative laws apply to certain operations but not others. We use the distributive law to guide calculations when we combine different operations. But we need to follow an order of operations, which we apply to any combination of operations we need.
In this unit, we learn how to use exponents and exponential notation to represent repeated multiplication. For example, repeatedly multiplying two against itself five times results in . We will pay careful attention to how negative numbers interact with our operations and clarify how to group different symbols (subjected to different operations) in the correct and clear order.
Completing this unit should take you approximately 4 hours.
3.1: Negative Numbers
Calculations and operations with negative numbers may seem strange at first compared to computations with positive ones. But there are plenty of real-world and mathematical scenarios that require them!
Temperature is a good example: the thermostat might read or degrees Celsius. Profits and losses offer another example. We say a business is in the red when it owes money. It is in the black when there is a surplus. For example, a business that has a tally of negative $500.00 (or ) is in the red by 500 dollars.
Our operations would be incomplete without negative numbers! The additive inverse property we discussed in Section 1.3 would no longer hold if negative numbers were not available. Also, we would not be able to cancel positive numbers.
3.2: Adding and Subtracting Negative Numbers
With practice, adding and subtracting negative numbers feels just like adding and subtracting positive ones. For example, you can addand in several ways. We can write it out as . You can think of negative quantities as debts or money owed. Combine a debt of $5 with a second debt of $3, and you obtain a total debt of $8. We can use other models to interpret and add negative numbers, such as a number line model.
3.3: Multiplying and Dividing Integers with Different Signs
Of course, we get new numbers when we multiply and divide old ones. The negative or positive sign of the result depends on the positive and negative signs of the numbers we started with. When you multiply or divide two positive numbers, the result is always positive. When you multiply or divide two negative numbers, the result is always positive. However, when you multiply or divide numbers with different signs (one positive and one negative), the result is always negative.
This chart summarizes these facts:
× positive negative ÷ positive negative positive positive negative positive positive negative negative negative positive negative negative positive
For example: , , , and .
Let's return to your aunt's coffee shop. She was already running a steady business, but her popularity really took off when she introduced her delicious breakfast sandwiches! One happy person posted a favorable review online, which caused two more people to stop by and try her food. Each of these two customers told two more people about their delicious breakfast. All four of these new people enjoyed their experience so much that they each recommended the shop to two more individuals.
And now we have questions! Just how many new customers will arrive in the next wave? And how many new customers will those new customers cause to arrive? At how many steps will this process repeat itself before the customers reach the 256 seating capacity? (Take a minute to do the math!)
We can organize this coffee shop phenomenon in several ways. For example, we can notate this as a chain of popularity, with symbols 1→ 2→ 4, meaning one review led to two new happy customers, which itself led to four new happy customers, and so on. The chain continues in the following pattern:
1→ 2→ 4→ 8→16To determine how many new customers will be stopping by your aunt's coffee shop, we simply multiply the previous number by two. If this chain of popularity keeps growing, we will arrive at the 256 seating capacity in eight steps.
1→ 2→ 4→ 8→ 16→ 32 → 64→ 128 → 256For the first step in this pattern (represented by the first arrow between one and two), we multiply by two one time. To get the number after the second arrow, we multiply by two twice, resulting in . The third, fourth, and fifth steps involve multiplying by two, three, four, and five times. During the last step, we arrive at 256: we have multiplied two by itself eight times. Rather than write out , we can use exponential notation to organize the symbols in a shorter, more informative way. We can write
The teeny, tiny eight in the superscript tells us how many times the two is multiplied against itself, while the number two itself tells us which number we are repeatedly multiplying. That teeny, tiny eight is called the exponent (and the two is referred to as the base). Check to see if the following equations also make sense:
, , ,
3.5: Order of Operations
It can be tricky to determine which operations you should carry out first when you are performing a calculation that has several different mathematical operations. Average Joe had this problem back in Unit 1! Thankfully, we can follow an easy set of rules called the Order of Operations. It is a lifesaver if you are working with a large, multi-step problem!