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Physics of Media Arts: Conceptual Physics for Students of the Media Arts

Aaron Wong

Section 2.1 Calculators

“Calculators can only calculate - they cannot do mathematics.”
―John A. Van de Walle
One of the most basic tools of physics is the calculator. While it is certainly possible to do most of the calculations by hand, there’s not a lot of value to that in the context of this course. But with that said, it puts an additional layer of importance for using your calculator correctly.
One of the challenges of this section is that there are many different types of calculators out there, and they don’t all have the same set of features. In fact, not all calculators behave the same way, and it’s possible for two calculators to give different results for the same button presses. Rather than going into the details of different brands, we’re going to take a high level overview of calculators and develop some basic strategies for how to avoid errors, regardless of how your specific calculator works.

Order of Operations.

One of the main errors (aside from pressing the wrong button) that can arise is when the desired calculation and the calculation that the calculator understands are not in alignment with each other. These are some of the most difficult errors to identify on the fly because your only indication that something went wrong is that the number is wrong. In some situations, the number you get is "obviously" wrong (such as getting a negative number when you know you should be getting a positive number). But the most common result is that a student dutifully copies the number the calculator gives them onto their paper, then moves forward from there without the slightest bit of hesitation.
Underneath this problem is an understanding of the order of operations. When used in the mathematical context, this is interpreted as PEMDAS (Please Excuse My Dear Aunt Sally) or BOMDAS, which is understood to be a set of rules for how to interpret a mathematical expression. More broadly, however, it just means the order in which you do things. For example, socks-before-shoes is an order of operations. The reason we need to think about this broader sense is because not all calculators understand the mathematical order of operations the way that it’s set up in mathematics.
Consider the calculation \(1 + 2 \times 3\text{.}\) You would probably immediately recognize that you need to do the multiplication before the addition, and get the mathematically correct result of 7. But in order for you to know that, you had to see the whole problem before you strated. You had to know that the multiplication was at the end, and you used that information to determine how you were going to do the computation. Some calculators have this ability, but others do not.
Some calculators use the immediate execution method of computation. This means that it tries to do the calculation as you type it in. And even within this category, there are calculators that can hold values in memory and those that can’t. Let’s analyze how one of these calculators might see this calculation.
  • \(\boxed{1}\text{:}\) The calculator understands that you’re starting to type in a number. As long as you keep typing in digits, it will keep thinking that you’re still typing in the number.
  • \(\boxed{+}\text{:}\) You’ve now pressed a mathematical operation, so it knows that you’re done with the first number. The next thing it is expecting is another number.
  • \(\boxed{2}\text{:}\) Just as with the first step, as long as you’re typing in numbers, it’s waiting for you to finish telling it the number.
  • \(\boxed{\times}\text{:}\) At this point, things can go in one of two ways. Some calculators will understand that you’re done with the second number of the sum and do that calculation right away. Then it will use that result with the multiplication that you’re about to give it. On the other hand, some calculators know about the mathematical order of operations, and it will know that it needs to do this multiplication first, then go back and add the other result. But in order to do that, it will also need to hold that original 1 in memory.
While there is more to the calculation, this is the critical step where things can fall apart. And this is why it’s critical for you to know how your calculator works. Lots of students get problems wrong because of something as simple as this.
It’s worth noting that there are some calculators that take the whole expression in before doing the calculation. These are sometimes called infix notation calculators. These behave more like how you would behave when looking at a calculation. You would use a logical system to determine which calculation needs to be performed first based on the order of operations, and then you would simplify from there. But infix notation also has order of operations pitfalls.
Consider the calculation \(\frac{5}{2+3}\text{.}\) You can easily see that this is equal to 1. But if you don’t type it into the calculator correctly, it won’t give you the desired result. Here is the button sequences that students would typically press: \(\boxed{\strut 5}\boxed{\div}\boxed{2}\boxed{+}\boxed{3}\text{.}\) And some students might even see \(5/2+3\) in the display and feel as though everything is correct. But how would the calculator see it?
The calculator is programmed to try to follow the order of operations, and so it will scan through the different operations to help it determine the order in which it will do the calculations. Based on this, it will see the division symbol and prioritize that one. This would lead to the first calculation being \(5 \div 2 = 2.5\text{.}\) This will be added to 3, giving 5.5 as the result.
The problem is now clear. You need to tell the calculator that the \(2+3\) is all in the denominator for the division calculation. The way to do this is to use parentheses (which most infix calculators have). Once you type it as \(\boxed{\strut 5}\boxed{\div}\boxed{(}\boxed{2}\boxed{+}\boxed{3}\boxed{)}\text{,}\) it will give you the correct result.

A Basic Calculator Strategy.

One of the best calculator strategies to develop is to get in the habit of only doing a few calculations at a time. As the expressions being evaluated become increasingly complex, the chances of making a mistake increase significantly. These errors can be as simple as pressing the wrong button (or failing to press a button) while typing in the calculation to errors related to the discussion above. As mentioned before, these are the types of errors that can be difficult to identify because you can hit the equal key and never realize that there was an error.
The technique is to use the order of operations to scan the expression for groups of calculations that can be executed together. In physics courses, this often reduces to one of two strategies: (1) Calculating the numerator and denominator of a fraction separately before taking the quotient; (2) Calculating all of the products before adding them together.
When you do this, you will need to write down the intermediate results. This is because you will need to reference the results you get in future calculations. This also provides you an opportunity to check your work by checking that you get a consistent result on the shorter calculations. It is generally good to keep at least 4 decimals in the middle of the calculation in order to avoid rounding errors. This is especially true for division calculations, where errors can quickly become magnified (typically when dividing by a small value).

Activity 2.1. Calculator Practice.

Intro Text

Instructions.

Instructions
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