RWM101: Foundations of Real World Math

The commutative law of addition and multiplication tells us that the order you use to add or multiply numbers does not matter. In other words, $3+2=5$, and $2+3=5$. The same is true of multiplication: $2 \times 3 = 6$, and $3 \times 2 = 6$. You obtain the same result regardless of the order you use to add or multiply numbers.

Note that the commutative property only holds for addition and multiplication. It does not work for subtraction or division. For example, $10-2=8$, but $2-10=-8$. In subtraction, the results are not the same when you change the order of the numbers. Likewise in division: $20/2=10$, but $2/20=0.10$. Again, the results are not the same when you change the order of the numbers in the division.

But familiar, old, real numbers are not the only kinds of entities we add and multiply in mathematics. For example, you will learn that matrix multiplication is not commutative when you study matrices in linear algebra and vector calculus. Similarly, we use a more-advanced, exotic kind of number called a quaternion for all sorts of tasks, such as robotics, 3-D animation, and video game design. Quaternion multiplication is noncommutative.

While quaternions and matrices are beyond the scope of this course, they provide an important lesson: rearrangement can affect useful kinds of multiplication. Sometimes order matters! The fact that real number addition and multiplication are commutative warrants an appreciative and careful understanding of the property.

Just after closing, you accidentally knock over the cash register at your aunt's coffee shop, and all of the day's money spills out onto the floor. You carefully put the five-dollar bills into one pile, all of the one-dollar bills into another pile, and all of the quarters, dimes, nickels, and pennies into their own piles. Then you add up the total. Thankfully, you check your math, and it matches the day's receipt: $1,500.00 exactly. Would you have tallied up this same amount if you had placed both five- and one-dollar bills into a single pile and grouped all of the coins into another separate pile? Would having a different grouping alter their count?

Here is another puzzling story: Average Joe loves math. He is known far and wide for his love of averaging. Whenever you hand him two numbers (say 2 and 10), Average Joe will add them and then divide by 2 (to give 6). If you were to hand him the numbers 5 and 15, he would happily average them to produce the number 10. But what happens if one day Joe's friend Above-Average Sharon hands him three numbers, say 2, 10, and 14? Will Average Joe first compute the average of 2 and 10, yielding 6, and then compute the average of 6 and 8? Or will he first average 10 and 14, yielding 12, and then compute the average of 12 and 2? Does it matter which grouping Average Joe uses to compute his averages?

In our first story, the change in grouping will not affect the final result, but it does in our second Average Joe story! Certain mathematical combinations or operations depend on how the parts or numbers are grouped, and sometimes they do not.

The associative property is the adjective we use to describe the operations that are unaffected by grouping. More specifically, the associative law of addition and multiplication tells us that no matter how we group or "associate" the numbers we add or multiply, the outcome remains the same.

This may sound awfully similar to the commutative property, but it expresses a different (although similarly useful) property. In arithmetic calculations, we often place parentheses around a set of numbers to indicate we are associating or grouping them together. We always carry out the calculation in parentheses first. For example, $5+(2+1)=5+3=8$. When we calculate this, we first calculate $2+1=3$. Then, we add 5 to get 8. We can also write $(5+2)+1=8$. Here, we first calculate $5+2=7$ and then add 1 to get a sum of 8. We get the same result regardless of how we group the numbers.

This same law works for multiplication. When we can compute $(2\times 2)\times 3=12$. We can change the grouping and write $2\times(2\times 3)=12$. The placement of the grouping does not change the answer.

In math, the additive identity is the name for a special number that does nothing when it is added to any other number. For real numbers, the number 0 plays this special role; this means that 0 is the additive identity for real numbers. In other words, the identity property of addition simply states that there is an additive identity called 0. This is probably a familiar fact for most of you. For example, $0+5=5$ or $0+(-7)=-7$.

It may seem strange to pay so much attention to such a simple or lazy number like 0. What is the big deal? It does nothing in addition. Why make such a fuss or give it a technical name?

As it turns out, paying careful attention to 0's special status early on will help you understand more complicated, unfamiliar numbers and objects. How about other operations? Do they feature a do nothing identity element? If we had not paid enough attention to 0 early on, this question would not have even occurred to us! But now we can ask, "does Average Joe's averaging operation have an identity" element?

The inverse property of addition states that every real number has a special companion that we call its additive inverse. We define this additive inverse in relation to our additive identity, 0. Basically, the sum of a number and its additive equals 0, or the additive inverse of a number and its negative is 0. The fact that $5+(-5)=0$ tells us that the number $-5$ is the additive inverse of the number $5$.

Note that there are no additive inverses if we only use positive numbers (the natural numbers)! We must use the larger, richer collection of integers to use our new property.

You can think about this inverse property as mathematical cancel culture, where we regard our additive identity, 0, as a do-nothing or neutral number. The existence of additive inverses simply tells us we can undo, neutralize, or cancel every number. We can undo or cancel $10$ by adding $-10$; we can cancel $-42$ by adding $-(-42)=42$. Here is a fun question to ponder: What is the additive inverse of $0$?

Much like the identity property of addition (see section 1.3), the identity property of multiplication states that there is a number that serves as the multiplicative identity that does nothing when it is multiplied against any other number. What special number behaves in this way? Two will not work since, for example, $2\times 3=6$ and $3$ was not left alone in this multiplication. It multiplied to become $6$! Can you also see why $-1$ and $5$ fail to be multiplicative identities? Take a minute to explore and practice multiplying various numbers before reading on, and you will likely stumble across that one special, lonely number that works.

As you may have figured out, the number $1$ has this magical, do nothing property. For example, $1\times 3=3$ and $(-5)\times 1=-5$. In short, the multiplicative identity property states that if you multiply any number by $1$, the answer is simply the number you started with.

And as you may have also guessed, the reason for emphasizing $1$'s special status as a do-nothing multiplicative identity is the same as that for zero's special status as a do-nothing additive identity: it can be useful to pay attention to these special rules and objects, particularly when using other, more abstract or new mathematical operations.

The inverse property of multiplication tells us that almost every real number has a multiplicative inverse. Since we treat the multiplicative identity, $1$, as a neutral element, we can cancel numbers (multiplicatively). For example, the multiplicative inverse of $2$ is the number $\frac{1}{2}$; this follows since $2\times \frac{1}{2}=1$.

Note that we would not be able to access multiplicative inverses like $\frac{1}{2}$ if we only use integers (fractions like $\frac{1}{2}$ are not whole numbers)! We need fractions or rational numbers to be able to cancel numbers multiplicatively.

To cancel a number $a$ multiplicatively, we always multiply by $1/a$. While it is correct to call $\frac{1}{a}$ the multiplicative inverse of $a$, we also call it the reciprocal of $a$ (just like how we call the additive inverse of a number its negative). Unlike additive inverses, not every real number has a multiplicative inverse: zero is the one special number we cannot cancel (multiplicatively). The reason we cannot invert $0$ (multiplicatively) involves the familiar rule you cannot divide by $0$ (see section 1.8 below).

Photo by Scott Beale

Although zero serves as the additive identity for our numbers – literally leaving them all alone – it has a more destructive impact when used to multiply. Multiplying any number by $0$ always results in $0$. This is why some mathematicians call $0$ the annihilator: all other numbers or objects are annihilated when multiplied by $0$. For example, both $15\times0=0$ and $0\times (-33)=0$. While mathematicians accept this aspect of the additive identity as fact, there are good, everyday, real-world reasons for this rule, too.

For example, let's say your coffee shop manager tells an employee, Every time a customer orders a large coffee, we make $\$2.00$. When 45 customers order large coffees, multiplication tells us the shop earned $2\times 45=90$ dollars. However, if zero customers come in that day, clearly, the shop has earned $\$0.00$: $2\times 0=0$ in large coffee sales. We can generalize and adjust this common experience to see that $(any\:number)\times 0=0$.

Because division does not enjoy the commutative property, dividing into $0$ and dividing by $0$ are two different questions. One of these computations is easy to carry out, while the other is impossible. Can you guess which is which before you read more?

Let's begin by considering division by zero. Many mathematics educators model the operation of division using shared pizzas or pies. If two students show up to a party where there are 10 pizzas, then they can each receive an equal number of pizzas, namely $\frac{10}{2}=5$ pizzas each; if 80 students show up to that same party, each student will receive $\frac{10}{80}=\frac{1}{8}$ of a single pizza, or one (small) slice each.

Thinking about our pizza party example suggests why the answer to this division by zero question is so strange: imagine we have 10 pizzas to share among the students at a party, only now zero students show up! How much pizza does each student receive? A reasonable reply is that the question does not make any sense since no students are present. There is no answer or number we can offer in response to this question!

The mathematical rule for division by $0$ reads like this:

Whenever we divide a number by $0$, the answer is undefined.

For example, $\frac{5}{0}$ = undefined. The technical response is, We cannot assign a meaningful value or answer to the expression $\frac{5}{0}$.

Our picture of arithmetic is almost finished. We have our real numbers and operations for combining them ($+$a, $-$, $\times$, $\div$), and we have learned about some properties these operations enjoy. But our last puzzle piece asks: How do these operations interact with one another?

Addition and subtraction interact in understandable ways: these are inverse operations. The same is true for multiplication and division. But how do addition and multiplication interact?

Our answer is encoded in the distributive law for numbers. This property tells us how to distribute a multiplication across a sum (we write the sum in parentheses). For example, we can use the distributive property to rewrite $2\times (3+5)$ as $(2\times3)+(2\times5)$. The answer is the same, and writing it this way makes it easier to simplify large calculations and figure out the answer without having to write it down (mental math).

Here is an abstract statement of the distributive law:

$a\times(b+c)=(a\times b)+(a\times c)$

As its name suggests, we are distributing the multiplied number $a$ to each number in the sum.