by Jason Marshall
Math is known for precision. Answers to problems are usually black-and-white—right-or-wrong—right? It’s true that math is usually extremely precise, but ambiguity does occasionally creep in. Just like English, the language of math isn’t always exactly…well…exact. In this article, we’ll talk about two specific examples of this ambiguity—the first is interesting but ultimately benign, whereas the second could definitely get you into a bit of trouble.
The Mathematics of Money
We’ve talked a lot about positive and negative integers, and how to add and subtract them by visualizing stepping along the number line. Though this interpretation is helpful, it’s not unique. At the end of the last article I asked you to contemplate how financial transactions like deposits, withdrawals, and debts can be used to help you understand what you’re doing when adding and subtracting positive and negative numbers. How does it work? Here’s the gist.
Imagine you open an account with an initial balance of $0. Depositing money into the account is identical to adding a positive number to the balance, and withdrawing money is identical to adding a negative value (or equivalently, to subtracting that value). For example, when you physically deposit $20 into your new account, you’ve mathematically added positive 20. And if you then physically withdraw $5, you’ve mathematically added -5 (or subtracted 5).
Math and Calculating Debt
Okay, how about debts? Let’s say the entirety of your life’s savings is contained in the $100 you have in your pocket, and you borrow $20 from a friend. Does that mean your net worth is now $120? No, remember you borrowed that $20 and you have to pay it back—so you have a $20 debt. As we talked about in the article on negative integers, this debt can be represented by a negative number—in this case -$20. So your net worth is $100 + $20 + (-$20) = $100 + $20 - $20 = $100. In other words, your net worth hasn’t changed.
Now, what happens if your friend is amazingly generous and tells you not to worry about paying back the loan? Well, since debts are included in our calculation of your net worth by adding negative numbers, it follows that forgiven debts are included by subtracting negative numbers. So, if your friend in our example forgave the $20 debt you owed, your net worth would be expressed as $100 + $20 + (-$20) - (-$20) = $100 + $20 - $20 + $20 = $120. Your net worth increased since your friend gave you $20!
Solve Math Problems by Thinking About Money
So, whenever you’re given a problem about adding and subtracting positive and negative integers, you can think about it in terms of monetary transactions. For instance, let’s say you’re confronted by the problem 100 + 20 + (-20) - (-20). You can solve it by imagining you’ve instead been asked $100 + $20 + (-$20) - (-$20), and then applying the exact same line of reasoning about borrowing money from a generous friend that we used before.
You Can Solve Math Problems in Different Ways
But wait a minute. We’ve spent several articles talking about adding and subtracting integers by visualizing walking along the number line. And now I’m telling you to think of all this in terms of monetary transactions instead? How can both work? Well, imagine you start with $0 in your pocket, then add $100, then add another $20, and so on. Sound familiar? It should: That process is exactly analogous to starting at zero on the number line, walking 100 steps in the positive direction, then another twenty, and so on.
The important point is that adding and subtracting positive and negative numbers has many possible interpretations (all of which are equally valid). As we discussed in the very first Math Dude article, each of these interpretations simply represents a different application of the underlying abstract mathematical concept. The good news is this type of ambiguity won’t get you into trouble when solving problems—it just gives you options about how to think about them. The next type of ambiguity is not so kind, however. It can definitely cause you some grief if you’re not careful.
Test Taking Tip: Pay Attention to Definitions in Math!
A few readers have asked about my inclusion of zero in the set of natural numbers. Is zero actually a natural number? As per the theme of this article, the answer is ambiguous: It depends. How can that be possible? Math is precise, right? Well, usually, but not always. In this case, there are two conventions commonly used to define the set of natural numbers: one includes zero, and one doesn’t. For various reasons, the definition including zero has grown in popularity in certain circles, but both are still used routinely. (If you’re interested in reading more, start with the Wikipedia article about natural numbers.) And while I’m personally fond of the definition that includes zero, it’s been pointed out to me that many—if not most—textbooks used in schools do not include zero in the set of natural numbers.
Now, I don’t want to lead anybody astray, so I thought it was important to address the question: Which definition should you use? And if there’s not a “right” answer, why does any of this even matter? Well, my response is simple—and it’s especially important if you’re a student. Here’s the quick and dirty tip: Always solve problems using the definitions preferred by your instructor. If they include zero, you should too. If not, then you shouldn’t either. Otherwise, your answers to their questions might be wrong—and I really don’t want that to happen! The bottom line is pay attention to definitions in math. Otherwise, this mathematical ambiguity could bite!
Wrap Up
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