## The Emacs Calculator

Did you know that Emacs comes with a calculator? Woop-dee-doo! Call the presses! Wow, a whole calculator! Sounds a bit lame, right?

Actually, it's much more than just a simple calculator. It's a computer algebra system! It is officially called a calculator, which isn't fair. It's an understatement, and I am sure has caused many people to overlook it. I finally ran into it during a thorough (re)reading of the Emacs manuals and almost skipped over it myself.

Ever see that demonstration by Will Wright for the game Spore several years ago? The player starts as a single-cell organism and evolves into a civilization with interstellar presence. When he started the demo he showed a cell through what looked like a microscope. No one had any idea yet what the game was about, so every time he increased the scope, from bacteria to animal, animal to civilization, civilization to space travel, interplanetary travel to interstellar travel, there was a huge reaction from the audience. It was like those infomercials: "But that's not all!!!"

As I made my way through the Emacs calc manual I was continually amazed by its power, with a similar constant increase in scope. Each new page was almost saying, "But that's not all!!!"

Like an infomercial I'm going to run through some of its features. See the calc manual for a real thorough introduction. It has practice exercises that shows some gotchas and interesting feature interactions.

Fire it up with `C-x * c` or `M-x calc`. There will be two new windows (Emacs windows, that is), one with the calculator and the other with usage history (the "trail").

First of all, the calculator operates on a stack and so its basic use is done with RPN. The stack builds vertically, downwards. Type in numbers and hit enter to push them onto the stack. Operators can be typed right after the number, so no need to hit enter all the time. Because negative (`-`) is reserved for subtraction an underscore `_` is used to type a negative number. An example stack with 3, 4, and 10,

```3:  3
2:  4
1:  10
.
```

10 is at the "top" of the stack (indicated by the "1:"), so if we type a `*` the top two elements are multiplied. Like so,

```2:  3
1:  40
.
```

The calculator has no limitations on the size of integers, so you work with large numbers without losing precision. For example, we'll take `2^200`.

```2:  2
1:  200
.
```

Apply the `^` operator,

```1:  1606938044258990275541962092341162602522202993782792835301376
.
```

But that's not all!!! It has a complex number type, which is entered in pairs (real, imaginary) with parenthesis. They can be operated on like any other number. Take `-1 + 2i` minus ```4 + 2i```,

```2:  (-1, 2)
1:  (4, 2)
.
```

Subtract with `-`,

```1:  -5
.
```

Then take the square root of that using `Q`, the square root function.

```1:  (0., 2.2360679775)
.
```

We can set the calculator's precision with `p`. The default is 12 places, showing here `1 / 7`.

```1:  0.142857142857
.
```

If we adjust the precision to 50 and do it again,

```2:  0.142857142857
1:  0.14285714285714285714285714285714285714285714285714
.
```

Numbers can be displayed in various notations, too, like fixed-point, scientific notation, and engineering notation. It will switch between these without losing any information (the stored form is separate from the displayed form).

But that's not all!!! We can represent rational numbers precisely with ratios. These are entered with a `:`. Push on `1/7`, `3/14`, and `17/29`,

```3:  1:7
2:  3:13
1:  17:29
.
```

And multiply them all together, which displays in the lowest form,

```1:  51:2842
.
```

There is a mode for working in these automatically.

But that's not all!!! We can change the radix. To enter a number with a different radix, which prefix it with the radix and a `#`. Here is how we enter 29 in base-2,

```2#11101
```

We can change the display radix with `d r`. With 29 on the stack, here's base-4,

```1:  4#131
.
```

Base-16,

```1:  16#1D
.
```

Base-36,

```1:  36#T
.
```

But that's not all!!! We can enter algebraic expressions onto the stack with apostrophe, `'`. Symbols can be entered as part of the expression. Note: these expressions are not entered in RPN.

```1:  a^3 + a^2 b / c d - a / b
.
```

There is a "big" mode (`d B`) for easier reading,

```          2
3   a  b   a
1:  a  + ---- - -
c d    b

.
```

We can assign values to variables to have the expression evaluated. If we assign `a` to 10 and use the "evaluates-to" operator,

```          2
3   a  b   a             100 b   10
1:  a  + ---- - -  =>  1000 + ----- - --
c d    b              c d    b

.
```

But that's not all!!! There is a vector type for working with vectors and matrices and doing linear algebra. They are entered with brackets, `[]`.

```2:  [4, 1, 5]
1:  [ [ 1, 2, 3 ]
[ 4, 5, 6 ]
[ 6, 7, 8 ] ]
.
```

And take the dot product, then take cross product of this vector and matrix,

```2:  [38, 48, 58]
1:  [ [ -14, -18, -22 ]
[ -19, -18, -17 ]
[ 15,  18,  21  ] ]
.
```

Any matrix and vector operator you could probably think of is available, including map and reduce (and you can define your own expression to apply).

We can use this to solve a linear system. Find `x` and `y` in terms of `a` and `b`,

```x + a y = 6
x + b y = 10
```

Enter it (note we are using symbols),

```2:  [6, 10]
1:  [ [ 1, a ]
[ 1, b ] ]
.
```

And divide,

```          4 a     4
1:  [6 + -----, -----]
a - b  b - a

.
```

But that's not all!!! We can create graphs if gnuplot is installed. We can give it two vectors, or an algebraic expression. This plot of `sin(x)` and `x cos(x)` was made with just a few keystrokes, But that's not all!!! There is an HMS type for handling times and angles. For 2 hours, 30 minutes, and 4 seconds, and some others,

```3:  2@ 30' 4"
2:  4@ 22' 13"
1:  1@ 2' 56"
.
```

Of course, the normal operators work as expected. We can add them all up,

```1:  7@ 55' 13"
.
```

We can convert between this and radians, and degrees, and so on.

But that's not all!!! The calculator also has a date type, entered inside angled brackets, `<>` (in algebra entry mode). It is really flexible on input dates. We can insert the current date with `t N`.

```1:  <6:59:34pm Tue Jun 23, 2009>
.
```

```1:  <6:59:34pm Sat Jun 27, 2009>
.
```

It works with the HMS format from before too. Subtract ```2@ 3' 15"```.

```1:  <4:56:32pm Sat Jun 27, 2009>
.
```

But that's not all!!! There is a modulo form for performing modulo arithmetic. For example, 17 mod 24,

```1:  17 mod 24
.
```

```1:  3 mod 24
.
```

This is most useful for forms such as `n^p mod M`, which this will handle efficiently. For example, ```3^100000 mod 24```. The naive way would be to find `3^100000` first, then take the modulus. This involves a computationally expensive middle step of calculating `3^100000`, a huge number. The modulo form does it smarter.

But that's not all!!! The calculator can do unit conversions. The version of Emacs (22.3.1) I am typing in right now knows about 159 different units. For example, I push 65 mph onto the stack,

```1:  65 mph
.
```

Convert to meters per second with `u c`,

```1:  29.0576 m / s
.
```

It is flexible about mixing type of units. For example, I enter 3 cubic meters,

```       3
1:  3 m

.
```

I can convert to gallons,

```1:  792.516157074 gal
.
```

I work in a lab without Internet access during the day, so when I need to do various conversions Emacs is indispensable.

The speed of light is also a unit. I can enter `1 c` and convert to meters per second,

```1:  299792458 m / s
.
```

But that's not all!!! As I said, it's a computer algebra system so it understands symbolic math. Remember those algebraic expressions from before? I can operate on those. Let's push some expressions onto the stack,

```3:  ln(x)

2   a x
2:  a x  + --- + c
b

1:  y + c

.
```

Multiply the top two, then add the third,

```                2   a x
1:  ln(x) + (a x  + --- + c) (y + c)
b

.
```

Expand with `a x`, then simplify with `a s`,

```                 2   a x y              2   a c x    2
1:  ln(x) + a y x  + ----- + c y + a c x  + ----- + c
b                      b

.
```

Now, one of the coolest features: calculus. Differentiate with respect to x, with `a d`,

```    1             a y             a c
1:  - + 2 a y x + --- + 2 a c x + ---
x              b               b

.
```

Or undo that and integrate it,

```                       3      2                  3        2
a y x    a x  y           a c x    a c x       2
1:  x ln(x) - x + ------ + ------ + c x y + ------ + ------ + x c
3       2 b               3       2 b

.
```

That's just awesome! That's a text editor ... doing calculus!

So, that was most of the main features. It was kind of exhausting going through all of that, and I am only scratching the surface of what the calculator can do.

Naturally, it can be extended with some elisp. It provides a `defmath` macro specifically for this.

I bet (hope?) someday it will have a functions for doing Laplace and Fourier transforms.