Sign conventions, Or, "when to add positives and subtract negatives"

As a student of math, physics or engineering, you have probably already learned a few useful equations and formulae for solving problems.

Here are some examples:

To solve a quadratic, ax²+bx+c=0, the two answers are x = (-b±sqrt(b²-4ac))/2a.

To work out the current (I) flowing in a resistor, I=V/R, where V is the potential difference across the resistor and the resistor has a resistance of R.

To get the distance travelled (s) given the initial speed (u), the acceleration (g) and the time interval (t), s=ut+0.5(gt²).

Each of these equations can be thought of as a simple mathematical model of some aspect of the world around us.

The coefficients a, b, c, and the variables V, R, s, u, t, g, can take positive or negative values. (Not sure what a resistor of negative value would look like in real life, but the math makes it possible).

To make our model useful, we have to make sure that the numerical values of these data are correct, and that they have the appropriate sign. What is important is to get all the signs consistent. The aim of this piece is to try to explain that elusive concept of consistency.

In the first example, if we want to solve 2x²+10x+12 = 0, it is clear that

a=2 b=10 c=12

So, substituting these values in the main equation, the solutions are given by (–10±sqrt(100-96))/4 or (-10±2)/4. -2 and -3

This makes sense, because the equation can be written as 2(x+2)(x+3)=0

If we want to solve2x²-10x-12 = 0, then

a= 2 b= - 10 c= - 12

Again, by substitution, the roots are given by (10±sqrt(100+96))/4 or (10±14)/4. 6 and –1

This also makes sense: 2(x-6)(x+1)=0

For coefficients, then, the rules are straightforward. When we have a negative coefficient, we substitute the negative number into the equation, and then solve. Sometimes it’s difficult to remember to subtract a negative (which means add), but thinking through the sum carefully, taking each operation one step at a time and concentrating should get you through it OK.

The tricky bit is knowing when to make one of the variables negative. Try a simple example.

Ohm's Law: V=IR (the voltage drop across a resistor is equal to the current times the resistance)

If the voltage at one end (end ‘A’) of a 1-ohm resistor is 10V and at the other end (end ‘B’) is 5V, what is the current? Simple question, but should the voltage difference be 5 (10-5) or -5 (5-10)?

      A ________________ B
       |                |
  -----| 1-Ohm Resistor |------
 V=10V |________________|  V=5 Volts

       Current flow ---»

Your teacher will tell you that it’s 5V. She's right, but that is not the whole story. You can choose whichever sign convention you like, so long as you do it consistently.

Let’s ignore teacher for a moment and say the voltage difference is -5 Volts, instead of 5V.

      A ________________ B
       |                |
  -----| 1-Ohm Resistor |------
 V=10V |________________|  V=5 Volts

    «------ Current flow (negative value)

We have chosen to subtract the voltage at end A from the voltage at end B. This means we have implicitly assumed the current is flowing from B toward A. We’ll do our sum, and find out that the current is –5 amps. Sounds a bit silly? No, because we assumed the current flow is going from B to A.

Twist it round, and a negative current in one direction corresponds to a positive current in the other. So Ohm’s law still works, and it tells us which way the current is flowing. It flows from A to B. From a higher voltage to the lower.

Ready for something a bit more complicated? Try to find how high a ball goes as it is thrown up in the air.

s=ut+0.5(gt²).

Do we make s, u, and g positive or negative? Again, it doesn’t matter, so long as we are consistent. The natural way of thinking about it is to make all things going upwards positive, (like in an x-y graph). If we do it that way round, u, the initial upward velocity is positive, because we have chosen upwards as positive. The acceleration, g, on the other hand is pulling things downwards, so we have to give it a negative sign.

When we have worked out our sums, a positive s will be a distance above the start point, and a negative s will be below our start point.

Try it the other way around. The same equation works. It’s a universal equation, so we don’t change that in any way.

This time, take downwards to be positive. For a ball thrown upwards, the initial velocity, u, is negative, but the acceleration is positive, and a positive s will be below the start point, whereas a negative answer is a distance above the start point.

There is another hidden sign convention here as well: time. If we choose to put in a negative time (t=-2) then we can project backwards in time. We have automatically assumed that time runs forwards, but the equation has no preference for time running forwards or backwards, so long as we are consistent about it.

Now to extend the idea further. You can choose any frame of reference you like, so long as all the signs you give to values are consistent with that frame of reference.

In the first example, teacher advised us to look at current flows from A to B. We ignored her advice, and looked at current flowing from B to A, and found that a negative voltage produces a negative current. This is perfectly good and acceptable, if a little confusing.

In the second example, we tried it both ways round. First, we put a frame around the experiment with positive values going upwards. That gave us some answers. Then we turned the frame upside down, and made the positive values go downwards. It was a little counter-intuitive, but it gave exactly the same answers.

If it helps, draw a little diagram at the side of your notes, showing your frame of reference, indicating which direction is positive, and which negative. Or list all the main variables and write down their sign conventions. As you work through the problem, refer back to these notes whenever you need to assign a sign to one of the variables

And if you feel silly doing that, I'll let you into a secret. Even the best engineers and physicists tend to draw those little diagrams, partly to remind themselves which way is up, and partly to explain to others what sign convention they have used.

Good luck!