# Negative Ratio----Math

I have always studied the ratios of the type $a:b$, where a and b are natural numbers and I can also understand the ratio where both a and b are negative.

But what I don't get is that when a is positive and b is negative and vice versa.

Because ratios are used to compare quantities having the same unit, how can this last type of the ratio be used to compare quantities even if they have the same unit? Please help me.

• Think of a positively sloped line, every movement of a to the right the line moves up b, the negatives of both a and b refer to moving to the left a and down b which is identical to the positive ratio Commented May 12, 2016 at 20:20

The ratio of the type $a:b$ where $a$ and $b$ are natural numbers is the easiest to understand since as you state it has a clear basis in physical reality.

Let me see if I can argue why a generalization of this concept makes sense.

Let's say we have 8 oranges and two apples; then the ratio of oranges to apples is 4:1 because $$\frac{\text{number of oranges}}{\text{number of applies}}=\frac{8}{2}=\frac{4}{1}$$

Likewise the ratio of apples to oranges is 1:4 because $$\frac{\text{number of applies}}{\text{number of oranges}}=\frac{2}{8}=\frac{1}{4}$$

From this perspective, the ratio $a:b$ then really is just

$$\text{numerator}:\text{denominator}$$

with respect to the corresponding fraction.

So if we take quantities that are not necessarily positive and which don't have to be integers, say amps of electric current (a negative sign indicating current flowing in the opposite direction), we can define more generalized ratios. For example, let's say we have $\pi$ amps of current flowing in one direction and $\sqrt{2}$ amps of current flowing in the other direction; the corresponding fraction is:

$$\frac{\pi}{-\sqrt{2}}$$

so we can write the ratio $\pi : -\sqrt{2}$, where both are measured in amps.

Note: This might be a potential cause of confusion. Looking at the above fraction, we can also see that it equals

$$\frac{\pi}{-\sqrt{2}}=\frac{\pi}{(-1)\sqrt{2}}=\frac{1}{(-1)}\frac{\pi}{\sqrt{2}}=(-1)\frac{\pi}{\sqrt{2}}=\frac{-\pi}{\sqrt{2}}$$

hence the above ratio is also equivalent to $-\pi: \sqrt{2}$, since the corresponding fractions are equal.

If you think of a minus sign as indicating the "direction" we are measuring units in, the fact that these two ratios are equivalent is the same as noting that we can arbitrarily set

• current to flowing to the right as positive and current flowing to the left as negative

OR

• current flowing to the left as positive and current flowing to the right as negative

Neither direction has an "inherent" sign -- the only thing is true is that both directions can not have the SAME sign. Thus the fact that $\pi:-\sqrt{2}$ and $-\pi:\sqrt{2}$ are the same ratios actually makes more sense than the opposite being true.

I hope this helps.

• Thanks. I also thought that it could mean what you have written i.e the difference in direction like one object is moving away from the origin along the positive x-axis of the coordinate system while the other is moving away along the negative x-axis then the distances covered by both the objects can be put in the form of a positive and negative ratio. but then I thought I might be wrong. Commented May 12, 2016 at 20:48
• I believe what you are saying is exactly the same as what I wrote above. Moving away along the negative x-axis is just moving away from the origin going left; moving away along the positive x-axis is moving away from the origin going right. The difference is the direction. Commented May 12, 2016 at 20:55

"The ratio $a:b$" and "$a/b$" are equivalent expressions as long as $b \neq 0$. The value of $a$ can be anything, and $b$ can be anything except zero.

Here's an example. Let's say that you are a poor businessperson and sell items at $\$1$while each item costs$\$5$ to make. Your net income to cost ratio is $-4$ to $5$. You get $-\$4$in income (a loss of$\$4$) for each $\$5\$ you incur in cost.

• I know that but I am asking about comparing two quantities represented by a positive number and the other being negative. Commented May 12, 2016 at 20:34