# What do all the Calculator Buttons Mean?

So, what are we discussing today? Ah yes, the meaning of the calculator buttons. I feel like I should have known all of these, I am studying maths after all, but I did have to Google a few. Enough chatter, on to the discussion!

### Introduction

Did you see what I did there? In the display of the calculator? Genius isn’t it?

I apologise for the dust which has swept its way across my machine like an arid wasteland, but she’s been my trusty Casio since year 7 and that takes its toll.

Umm, ok not sure why I spoke like that but anyway, on with the preamble:

I’m assuming that you’ll know most of the buttons, so I won’t insult you by explaining what the fraction button does.

Instead, I’ll talk about some of the harder buttons, but still include a key as to whether a button is easy or hard, so I don’t waste too much of your precious time.

I’ve also included the yellow text above the buttons, which are accessible by pressing shift as I’m sure you know. To make it look nice, I haven’t included which button to press with shift, but you’re smart, you can figure it out.

Also, note that I’ll only be explaining what you can do on the ‘COMP’ mode, which is the normal, default maths setting.

Let me know if you’d like to see what else you can do in the other modes, since I assume most of you haven’t ever used them, here or in the comments.

## Easy

### $\log_{\blacksquare} \square$

This is used for the log function for an arbitrary base. So, what is the log function?

Well, imagine you had an equation, $a^x = b$. Think of $2^3 = 8$ for example.

Now, if you knew $a$ and $b$, how would you alter this equation to find out what $x$ was?

Of course, you could use trial and error, but this could take ages, so instead you can use the log function.

This will make $x$ the subject as $x = \log_a b$, so that $b = a^{\log_a b}$.

To compare the earlier example, we would have that $\log_2 8 = 3$.

We say that $a$ is the base of the log so that $\log_a b$ is $\log b$ base $a$.

### $\log$

This is pretty similar to the other log button, but with $a = 10$, so that if $10^x = b$, then $x = \log b$.

Really it’s just a shorthand notation since $\log$ base 10 is used quite a lot.

### $\ln$

This is another shorthand notation; it means ‘The Natural Logarithm’, and is just $\log$ base $e$, where $e$ is Euler’s number.

This means that if $b = e^x$, then $x = \ln b$ so that $e^{\ln b} = x$. Easy.

### $(-)$

This button is just another way of writing negative numbers, instead of using the subtract button. Note that you can’t use it as subtract, and trying to do $2 (-) 3$ will result in syntax error.

I think this is mainly used in the other calculator modes where you can make tables and graphs.

### DRG

Wait, there are more different units for angles? Why?

I wish I could answer that for you, but alas, all I can give you are the names of some more:

This button gives you access to: degrees, $^\circ$; gradians, $^g$; and radians, $^r$.

I’m assuming you know degrees, so here are the definitions of the others:

1 grad = 0.9 $^\circ$

1 rad = $\frac{180}{\pi} ^\circ$

So.. that’s what this button does, it allows you to easily convert between the units to your current angle setting using the deg, rad and gra options in the setup menu (shift + MODE).

### Ran#

I never knew a calculator could do so much, and here is another really useful function.

This button generates a random 3 digit number less than one.

Not much else to this really.

### RanInt

This one is similar to Ran# – maybe that’s why they put it on the same base button?

RanInt$(a,b)$ generates a random integer between $a$ and $b$ inclusive, where $a$ and $b$ are integers.

Yet another of the many applications of the comma button (shift + ) )

With this and the Ran# button, you’re able to generate any random number to 3 decimal places!

### $x!$

This is the factorial operation. It is the product of all positive integers up to and including the integer $x$.

i.e. $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$.

Non integer numbers will give a math error if using this button.

### $\boldsymbol{\dot{\blacksquare}}$

This lets you add recurring decimals into a number, like $0. \dot{3} = \frac{1}{3}$, or $1.2 \dot{3} 4 \dot{5} = \frac{4111}{3330}$.

Personally, I’d prefer to just use fractions, or only give a couple of numbers in the decimal expansion since it won’t really make any difference to the outcome.

### FACT

This gives the prime factorisation of the number, which is the number expressed as a unique product of prime numbers.

i.e. $28 = 7 \times 2 \times 2$

This can be useful when calculating by hand the lowest common multiple or highest common factor of two integers.

## Hard

### ${^{\circ}} \, {^{\prime}} \, {^{\prime\prime}}$

What do these symbols even mean? The first one looks like a degree symbol, but I’ve no idea about the others.”

I sounded like this for a long time since this button never really needs to be used. However, it does serve a purpose.

First, we need to define a new unit of measuring angle. We all know that the degree is 1/360th of a whole circle, but what if we want to use a smaller angle, maybe in optics or astronomy?

Well, that’s where the other symbols come in. We define an arcminute as 1/60th of a degree, and an arcsecond as 1/60th of an arcminute so that 1 arcsecond = 1/3600th of a degree.

These units come from the Babylonians, who used base 60 (sexagesimal) for counting, and used these units for astronomy.

So, back to the button: the symbols on the button are $^{\circ}$ for degree, $^{\prime}$ for arcminute and $^{\prime\prime}$ for arcsecond, so that an angle could be described as a mixture of all 3.

For example, an angle may be ${0^{\circ}} \, {25.4^{\prime}} \, {33.2^{\prime\prime}}$.

To input a value like this into the calculator, you would simply press 0 and ${^{\circ}} \, {^{\prime}} \, {^{\prime\prime}}$, then 25.4 and ${^{\circ}} \, {^{\prime}} \, {^{\prime\prime}}$, and finally 33.2 and ${^{\circ}} \, {^{\prime}} \, {^{\prime\prime}}$.

Note that when typing these into a calculator, the screen will initially only show as three degree symbols, as shown here:

In that example, the calculator transformed the values slightly just to make the arcminute value an integer since $0.4^{\prime} = 24^{\prime\prime}$

You may have seen this when talking about co-ordinates in a map.

You can then use these angles as you would with any other unit system on the calculator; adding, multiplying, or applying trigonometric functions.

### hyp

I used to press this at school and had no idea what they meant, other than they looked like the familiar sin, cos and tan. This is because they are similar.

This button gives you access to the hyperbolic functions, denoted by sinh ( pronounced shine/sinch), cosh, and tanh(tansh), and their inverses.

These are just other functions which are sometimes used in physics or maths, and are defined as follows:

### ENG

I haven’t ever used this button, but it is actually quite useful, especially in an exam where you’re liable to easily make silly mistakes.

An initial press of ENG rewrites the result in engineering notation.

For example: $544 \rightarrow 544 \times 10^0$ or $1234 \rightarrow 1.234 \times 10^3$.

If you don’t know what engineering notation is, it’s just like scientific notation, but the power of ten must be a multiple of three. This is useful to convert between SI prefixes like Mega or Kilo.

Each subsequent press of ENG moves the decimal place to the right three spaces and reduces the power of 10 by three.

For example, $544 \rightarrow 544 \times 10^0 \rightarrow 544000 \times 10^-3 \rightarrow$ …

Similarly, the $\leftarrow$ button, shift + ENG, performs the same operation, but moves the decimal place to the left, and increases the power of 10 by three.

i.e. $544 \rightarrow 544 \times 10^0 \rightarrow 0.544 \times 10^3 \rightarrow$ …

### nCr

This button looks completely random, but it’s actually incredibly useful.

So, what does it do? Well, imagine you have a set of $n$ elements, and you want to work out how many different ways there are to uniquely arrange $r$ elements of the set. That’s what this button will do.

The number of ways will be equal to pressing $n$, then “nCr”, then $r$ on the calculator.

What does this actually mean? Let’s see an example:

Imagine you have the numbers 1 to 5. How many distinct subsets of three numbers are there?

123, 124, 125, 134, 135, 145, 234, 235, 245, 345… So 10.

This means that $\binom{5}{3} = 10$.

The above is just the normal way of writing out what the calculator would display as $5 \, \mathbb{C} \, 3$.

If you want to say it aloud, you say “5 choose 3”.

How does the calculator know how to do this? I’m glad you asked:

There is a (relatively) simple formula to calulate this:

$$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$

### nPr

This is similar to nCr, but this time, the combinations do not have to be distinct, and the order matters, although you still can’t repeat elements.

We say this is the number of $r$ sized permutations of the set of $n$ elements.

Using the previous example, there would now be 5 “pick” 3 = 60 ways, since combinations like 123, 132, 213, 231, 321, 312 would all be allowed.

You can easily calculate this as:

$$\binom{n}{r} r!$$

### Pol

This one would be really helpful just to double check in an exam, but unfortunately, we can’t take calculators into our maths exams.

Anyway, suppose, you have a set of Cartesian coordinates, $(x,y)$, and you want to convert them into polar coordinates, $(r, \theta)$.

If you don’t know what polar coordinates are, they just describe a point in space by its distance from the origin, $r$, and the angle it makes with the $x$ axis, $\theta$.

That’s exactly what this button does. Pol$(x,y)$ transforms $(x,y)$ to $(r, \theta)$!

To do this, you’ll also need the comma, which is just shift + ), to separate $x$ and $y$.

### Rec

This one’s another useful one, it’s just the opposite of Pol. It turns polar coordinates into Cartesian, or Rectangular, coordinates– I wonder if that’s where the button gets its name…

$(x,y) =$ Rec$(r, \theta)$.

### :

I have never used this in my life, and quite frankly it seems a bit useless to me, but nevertheless here it is:

It allows you to perform two calculations, one after the other, with the press of =.

For example, writing $54 + 2: 12 -3$ would display 56 after a single press of =, and then 9 after a second press.

I suppose this could be useful, since you can use the Ans button in the second expression to manipulate the answer obtained in the first.

For example, writing $10 + 2$ : Ans $+ 5$ would display 12 after one press of =, and, you guessed it, 17 after a second.

### Storing Numbers

You may not have realised why there are so many red letters. This is part of the functionality to store numbers/constants. So, how do you do it?

Say you have a really long calculation, and you’re going to need the answer from it again. Rather than writing down the full accuracy and retyping every time, you can save the answer as follows.

Your answer is on the screen and you want to save it. You then press shift + RCL (STO), and then press the key of the letter you’d like to save the number as, without pressing alpha first.

For example, I want to save the result of 2+3 as the variable D. I type 2+3 and press = to display the result. I then press shift + RCL, and then press sin. Note after pressing STO, you will see a STO option appear on the screen.

The value of 5 has then been saved to the letter D.

M is a special value of memory which has an interesting option.

You can add or subtract a number to M very easily by pressing M+ or M- (shift + M+).

For example, if M contains the value of 7, and you want to add or subtract 5 to it, then you just press 5 and = to get it on the screen, and then press M+ or M-.

M will now contain either 12 or 2 depending on whether M+ or M- was pressed.