Do you write "$a$" or "$-a$" for a variable with a negative value?

Question

If you're writing a variable and the variable's value is negative, can you write it as "$a$", or do you have to write it as "$-a$"?

Answer 1

The notation $a$ means the value of the variable $a$, which can be either positive or negative (or zero). The notation $-a$ means the negation of the value of $a$, which can also be either positive (if $a$ has a negative value) or negative (if $a$ has a positive value).

Write whichever of these you want to express.

---

Historical note: It was a significant breakthrough in the history of algebra when it was realized that the same variable letter can stand for either positive or negative values, and the formulas involving this letter would then usually be valid both for positive or negative values.

For example, in the common form of the quadratic equation $ax^2+bx+c=0$, each of the constants $a$, $b$ or $c$ can have a value of whichever sign you want, and the quadratic formula for the roots $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ will work no matter what the signs are (as long as $a$ is nonzero).

Before this invention, algebra texts would handle each combination of different signs as its own case with a separate solution procedure and correctness proof. For example, $2x^2+3x=5$ and $2x^2=3x+5$ would have counted as different types of problems; allowing $b$ to stand for either $3$ or $-3$ let them be combined into a single case.

Answer 2

You can write it as it will be of more help to you. Sometimes it is of use to write it as $-a $ because you are more interested in its absolute value and will be referring to it more than to the number as a negative quantity. Nonetheless, there is no problem in just having $a $ as a negative variable