## Positives and Negatives

### Two sides of the same coin

#### math_player

$$\textrm{Income} = \textrm{Expenses} + \textrm{Savings}$$

Ideally, you want your income to be at least as much as your expenses so that you’re in a financially sustainable situation. But in reality, there may be times when your expenses exceed your income, e.g., when there’s a sudden unexpected large expense event. What are your savings in such a scenario?

Colloquially speaking, there are no “savings” in this scenario. Instead, you’d need to draw upon whatever money you had saved previously – or if you didn’t have any, go into debt, e.g., credit card debt. However Math, with its penchant for precision, prefers using the same term, viz., “savings”, for the same quantity, i.e., the difference between income and expenses, regardless of the scenario. The way it accomplishes this feat is by using the concept of “negative numbers”.

Negative numbers are quite straightforward. First, let’s start with the ideal scenario where your income is greater than your expenses in a given period of time. Suppose you had an income of 2500 currency units (c.u.’s) in a month, and expenses of 1500 c.u.’s. Then your savings for that month would be:

$$2500\ \textrm{c.u.} - 1500\ \textrm{c.u.} = 1000\ \textrm{c.u.}$$

The amount of savings in this case, 1000 (in c.u.’s), is technically a “positive number”, which is basically the same thing as an ordinary number. Positive numbers such as 1000 can also be written as +1000 to emphasize that they’re positive numbers, e.g.,

$$\textrm{Savings} = +1000\ \textrm{c.u.}$$

But in common practice, the “+” sign is left out, and it’s understood that when there is no sign in front of a number, it’s a positive number.

Now consider the reverse scenario, viz., your income is 1500 c.u.’s, but your expenses are 2500 c.u.’s in a month, i.e., your income is less than your expenses. In this case, your savings are computed as1:

$$\textrm{Savings} = 1500\ \textrm{c.u.} - 2500\ \textrm{c.u.} = -1000\ \textrm{c.u.}$$

Notice that the only difference from the savings in the previous scenario is that there is a “-” sign2 in front of the number 1000. The “-” sign indicates that the amount of savings is not an ordinary positive number, but rather a negative number obtained by subtracting a bigger number from a smaller number (e.g., subtracting 2500 from 1500 in this case). Mathematically, we say that your savings for the month are -1000 c.u.’s. Thus in both cases, we can use the same term – savings – to denote the difference between income and expenses, and let the sign of the savings amount tell us if our income exceeds our expenses or vice versa. This is in contrast to colloquial language, where the term “savings” only really makes sense when income exceeds expenses.

Another way to understand the sign of savings amount is to consider its effect on the total amount of money you’ve saved so far. A positive savings amount, e.g., +1000 c.u.’s, will add to your “stash of cash”, i.e., total savings so far, as is hinted by the “+” sign in front of the number. On the other hand, a negative savings amount, e.g., -1000 c.u.’s, will subtract from your stash of cash, as is hinted by the “-” sign in front of the number. This is consistent with our earlier observation that when your expenses exceed your income, you’ll need to draw upon, i.e., reduce, your stash of cash.

Positive and negative numbers arise in many other contexts too. For example, your net worth can be positive or negative, depending on whether your assets exceed your liabilities or vice versa respectively. Similarly, your ROI on investments can be positive or negative depending on whether your investments have generated a net profit or loss respectively. From all these examples, it’s easy to see that positive and negative numbers are really two sides of the same coin – in the sense that they provide a concise way to express a quantity in two contrasting scenarios3.

Now you might wonder what’s the point of inventing negative numbers if all they do is allow us to use the same term in different situations? It turns out that negative numbers are more useful than that.

We saw that it’s important to invest at least a portion of our savings in diversified financial assets, but what does that mean quantitatively? In simple terms, it means investing in asset classes such that when some of them increase in value, others tend to decrease in value. Mathematically, this translates to the requirement that the asset classes have a negative correlation amongst themselves, i.e., the mathematical quantity known as "correlation" between these asset classes is a negative number. As an example, the definition of inversely proportional quantities implies that such quantities are negatively correlated to each other.

Now that we’ve seen some use-cases of negative numbers, let’s see how they work arithmetically. Negative numbers have a few quirks of their own, and it’s important to understand them in order to use them correctly.

First, let’s start with the ordering relationship, i.e., comparing two numbers and determining which number is greater or lesser than the other number. To help make sense of the rules, we’ll consider the example of your savings and its effect on your stash of cash from before.

For positive numbers, the matter is quite straightforward: +1 is less than +2, which is less than +3, and so on. This ordering is consistent with the fact that savings of +1 c.u. will add less to your stash of cash than savings of +2 c.u.’s, which in turn will add less to your stash of cash than savings of +3 c.u’s. In other words, savings of +1 c.u. will result in a lower stash of cash than savings of +2 c.u.’s, and so on.

For negative numbers, the same logic applies. Savings of -1 c.u. will reduce your stash of cash less than savings of -2 c.u.’s. In other words, savings of -1 c.u. will result in a higher stash of cash than savings of -2 c.u.’s. Therefore, -1 is considered greater than -2. By the same token, -2 is greater than -3, and so on. Thus, the ordering relation for negative numbers follows the rule:

$$-1 > -2 > -3 > \ldots$$

Finally, since any positive savings add to your stash of cash, while any negative savings reduce your stash of cash, any positive savings will result in a greater stash of cash than any negative savings. Therefore, all positive numbers are greater than all negative numbers. Similarly, since savings of 0 c.u. doesn’t change your stash of cash, but any negative savings will reduce it, 0 is considered greater than any negative number4.

Thus, the ordering relation between positive and negative numbers follows the rule:

$$\ldots < -3 < -2 < -1 < 0 < 1 < 2 < 3 < \ldots$$

In the “number line” shown in the cover image, the right direction is traditionally taken to be the direction of increasing numbers, so negative numbers are marked to the left of positive numbers and 0.

Next, let’s consider the basic arithmetic operations with positive and negative numbers. To understand how these operations works, again consider the example of savings and stash of cash from before.

Adding two positive numbers, say $2+3$ is pretty straightforward, and doesn’t require much comment. In terms of savings and stash of cash, this is the same as having savings of 2 c.u.’s in one month followed by savings of 3 c.u.’s in the next month. The net effect of these two savings is an increase in your stash of cash by 5 c.u.’s, so $2+3=5$.

What about adding a negative number to a positive number, say $2 + (-3)$?

As before, $2 + (-3)$ can be interpreted in terms of savings as having savings of 2 c.u.’s in one month followed by savings of -3 c.u.’s in the next month. Since savings of 2 c.u.’s will increase your stash of cash by 2 c.u.’s, and savings of 3 c.u.’s will reduce it by 3 c.u.’s, the net effect on your stash of cash would be a reduction by 1 c.u. This is equivalent to savings of -1 c.u. as that would also reduce your stash of cash by 1 c.u. Therefore, $2 + (-3) = -1$.

Notice that $2 - 3$ also results in -1. In other words:

$$2 - 3 = 2 + (-3)$$

This shows that the operation of subtraction is equivalent to adding the negative of the second number, a.k.a., its “additive inverse” (a.i.)5.

What about adding a positive number to a negative number, say $(-3) + 2$? Since addition is a "commutative operator", i.e., the order of the numbers in the operation doesn’t matter, $(-3) + 2 = 2 + (-3) = -1$.

Finally, what about adding two negative numbers, say $(-2) + (-3)$? In terms of savings and stash of cash, this is the same as having savings of -2 c.u.’s in one month followed by savings of -3 c.u.’s in the next month. The net effect of these two savings is a decrease in your stash of cash by 5 c.u.’s, so $(-2)+(-3) = -5$.

To summarize, when adding two numbers of the same sign, you simply add them as ordinary numbers, and the sign of the result is the same as the signs of the two numbers. On the other hand, for two numbers of the opposite sign, you subtract them as ordinary numbers, and the sign of the result is the same as the sign of the greater of the two ordinary numbers you subtracted6.

The rules for subtraction with negative numbers follow from the general rule that subtraction is equivalent to addition of the a.i. For example, when subtracting a negative number from a positive number, say:

$$2 - (-3) = 2 + 3 = 5$$

where we added the a.i. of (-3), viz. 3, to 2 instead of subtracting (-3) from 2.

Finally, for multiplication and division, you multiply or divide as you would with ordinary numbers, and assign a sign to the result following this general rule: if both numbers have the same sign, the result is positive, but if they have different signs, the result is negative7.

For example:

$$(-2) \times (-3) = +6$$

Familiarizing yourself with negative numbers, and becoming comfortable working with them is a key step in developing fluency in the Math language, as well as developing a deeper understanding of your personal Finances.

1. N.B. How we got the result below will become clear when the rules of arithmetic with negative numbers are discussed later in the post. ↩︎

2. Unlike the “+” sign, the “-” sign is never left out, otherwise there’d be confusion when there’s no sign in front of a number whether it’s a positive or negative number! ↩︎

3. instead of having to come up with two different terms for the same quantity ↩︎

4. N.B. 0 itself is considered to be neither positive nor negative ↩︎

5. N.B. Just as the a.i. of 3 is -3, the a.i. of -3 is 3. In other words, a.i. is a "symmetric relation"↩︎

6. e.g., when computing $1500-2500$, first subtract the smaller number from the bigger number, i.e, $2500-1500=1000$, and then attach the sign of the bigger number, which in this case is 2500, to the final result, i.e., final result is $-1000$. ↩︎

7. This rule is a consequence of the "distributive law" ↩︎

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