Expert: Josh - 9/17/2009

Question
I'm helping my son with Order of Operations problems.  I can do them just fine, and now he can as well.  But what I don't understand is why the need for the Order of Operations.  Where in "real life" would you come into an equation that needed this use?

Answer
Hello JanaLee,

One of the primary reasons is to avoid ambiguity. Without proper rules on the order of arithmetic operation, a mathematical expression that involves several operations can be computed in a variety of ways. If it is open to interpretation, then there can be no universal agreement on what it stands. One would then have to wonder what exactly it actually means, and guess the original intent of the person who wrote the expressions, then deduce the order in which additions and multiplications are done -- without any certainty.

Although it is not a perfect analogy, you can imagine the chaos on the road when several vehicles approach a traffic junction all at the same time, without knowing who gives way to who.

Regarding the second part of your question, the answer has to be "in all walks of life". The challenge on my part is finding a "real-life" example that resonates with your experience. I am not sure if I can do just that, but I will try citing some examples that I am more familiar with.

In science and engineering, the numbers (or letters) that you see, for instance in " A + B / C" take on extra meaning. I think this is the first thing to appreciate. Real life problems are complex and very few can be adequately modeled by a few numbers. Instead of representing numbers, the "A", "B" and "C" might each represent a process or computational recipe that do not commute.

For example, we may need three chemical agents to synthesize a new compound. The chemical substances may behave in such a way that "B" needs to be mixed with "C" first before it is added to "A" for successful synthesis of a polymer. If this strict order is not followed, e.g., if "A" and "B" are allowed to come into contact with each other before being added to "C", an adverse reaction may happen. So, the order of operation needs to be respected.

In other areas such as electronics, for instance, if a signal is not properly conditioned (say a predicted signal is not subtracted from the actual signal) before it gets amplified (multiplied), it may end up compromising performance, or even destroy the circuit. In this example, a portion of the system might be represented by A*(X-P), where X is the actual signal, P the predicted signal and A the level of amplification. If X has a range between 0 and 0.001 and P (the average signal) is 0.0005 and A=24000, normal operation (as described by the equation Y=A*(X-P)) would restrict the output to -12 to +12 units. If, instead, A*X-P is carried out, the worst case output could reach dangerous levels (e.g., if X=0.001, then Y=24000*0.001-0.0005 is approximately 24; twice the safe operating range).