Secrets of the Calculator, Part 1

In my tutoring I am continually surprised by how little most students know about their calculators.

It is true that most students know the basics:  the four operations, the exponent key, the square root key, the Pi key, and maybe some trig fundamentals, like sin, cos 60 and tan. But aside from these basic keys and keystrokes, many students have little to no idea what the other keys do.

The funny thing is that there are so many keys and keystrokes that students would just “love” if they only knew about them.

So to help students out a bit here, I’m starting an occasional series whose name is just below.

Yes, we will get into the deep, dark calculator secrets that all math students are burning to know. Along the way we might just learn a thing or two about technology, too. And we’ll help students improve their overall performance and grades, make math easier in general, and help students complete their work in less time, a big crowd pleaser among kids, I’ve found.

Calculator “secret” for today:  the “1/x” key.

Basically every scientific or graphing calculator will have a 1/x key, also known as the x^(–1) key, because some calculators display it this way.

So what in the world does this mysterious key do?

Basically, it takes whatever value you feed it, and then it shoves that value into the denominator of a fraction, putting an invisible a 1 on top in the numerator. And then, when you follow up by pressing the “Enter” or “=” key, it gives you the value of the whole fraction.

Easy example:  input a 2. Then hit the 1/x key, followed by “Enter” or “=.”  What do you get?  .5. Why? Because the calculator just took the 2, put it in the denominator, and stuck an invisible 1 on top in the numerator, giving you the fraction:  1/2 whose decimal value is .5

Slightly more complex example: suppose that you need to get the decimal value for the fraction:  1/(12 – 2.5).

Using the 1/x key, you would first just input the denominator, like this:   12 – 2.5, then hit “Enter” or “=.”

At this point your screen should read:  9.5

Now the big moment:  press the 1/x key, and your screen will read:  0.10526…

This is the value of the fraction:  1/9.5, which is the same as 10/95, or 2/19

Again that is because the calculator took the 9.5, shoved it down into the denominator, placed an invisible 1 on top in the numerator, and gave you the value. All that with just a couple of keystrokes:  1/x   and  =

Starting to see how it works?

Here’s an example that shows still more clearly how this key can save you time and trouble.

Suppose the expression you need to simplify is this:  4/[5sin45° – 3(.5cos75° – 8.6)]   Yuck, right?

Right, because if you don’t know how to use the 1/x key, you’d probably enter this with this long and cumbersome series of keystrokes:

4   ÷   [  5   x   sin  (  45  )   –  3  (  .5  x  (   cos  )  7 5   –   8.6    )    ]

Note that it is easy to forget the brackets on the far left and far right of the denominator, which you need to enter in order to ensure that the calculator treats the denominator as one unified quantity. And in general, whenever you have a long expression like this, which needs to go into the denominator, students often get confused as to how many start and end parentheses or start and end bracket marks they need to put in.

But if you understand and use the 1/x key, you can save yourself trouble, entering the expression just like this:

First you enter the denominator, without brackets:  5  sin  (  45  ) –  3  (  .5  cos  (  75  )  –  8.6  )  =

You should get:  28.9473…

Next press the 1/x key, followed by “Enter” or “=,” and this whole quantity goes to the denominator. Now you get:   .0345 …

IMPORTANT:  Note that in this case you did not need starting and ending brackets. That is because you are just finding the value of this expression; you are not cordoning it off as a unified denominator term.

After you use the 1/x key, the calculator has put the denominator IN the denominator. Since it is already down there with an invisible 1 standing in the numerator, now you just need to multiply by the numerator, which is 4.

So just take the value you have on screen (.0345 …) and multiply by 4.

The result you get, .1382 … is the simplified form of the original expression.

And that is that.

My advice is to just play around with the 1/x key until you feel comfortable with how it works. Once you’ve got the hang of it, you will find it VERY useful!

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