Why is the rate of disappearance negative?

Rate of disappearance is given as $-\frac{\Delta [A]}{\Delta t}$ where $\ce{A}$ is a reactant. However, using this formula, the rate of disappearance cannot be negative.

$\Delta [A]$ will be negative, as $[A]$ will be lower at a later time, since it is being used up in the reaction. Then, $[A]_{\text{final}} - [A]_{\text{initial}}$ will be negative. Therefore, the numerator in $-\frac{\Delta [A]}{\Delta t}$ will be negative.

$\Delta t$ will be positive because final time minus initial time will be positive.

This means that $-\frac{\Delta [A]}{\Delta t}$ will evaluate to $(-)\frac{(-)}{(+)} = (-) \cdot (-) =(+)$

However, we still write the rate of disappearance as a negative number. Also, if you think about it, a negative rate of disappearance is essentially a positive rate of appearance. The reactants disappear at a positive rate, so why isn't the rate of disappearance positive?

Reaction rates are generally by convention given based on the formation of the product, and thus reaction rates are positive. So, for the reaction:

$$\ce{A->B}$$

$$\text{Rate} = \frac{\Delta[\ce{B}]}{\Delta t}$$

To ensure that you get a positive reaction rate, the rate of disappearance of reactant has a negative sign:

$$\text{Rate} = -\frac{\Delta[\ce{A}]}{\Delta t}=\frac{\Delta[\ce{B}]}{\Delta t}$$

When you say "rate of disappearance" you're announcing that the concentration is going down. If you wrote a negative number for the rate of disappearance, then, it's a double negative---you'd be saying that the concentration would be going up!

As you've noticed, keeping track of the signs when talking about rates of reaction is inconvenient. It would be much simpler if we defined a single number for the rate of reaction, regardless of whether we were looking at reactants or products.

We can do this by a) flipping the sign on rates for reactants, so that the rate of reaction will always be a positive number, and b) scaling all rates by their stoichiometric coefficients.

For example, if you have a balanced equation for the reaction $$a \mathrm{A} + b \mathrm{B} \rightarrow c \mathrm{C} + d \mathrm{D}$$ the rate of the reaction $r$ is defined $$r = -\frac{1}{a}\frac{\mathrm{d[A]}}{\mathrm{d}t} = -\frac{1}{b}\frac{\mathrm{d[B]}}{\mathrm{d}t} = \frac{1}{c}\frac{\mathrm{d[C]}}{\mathrm{d}t} = \frac{1}{d}\frac{\mathrm{d[D]}}{\mathrm{d}t}$$

This lets us compute the rate of reaction from whatever concentration change is easiest to measure.