# Introduction

“You see, Vergon 6 was once filled with the super-dense substance known as dark matter, each pound of which weighs over 10,000 pounds.”— Futurama, S1E4

Numbat is a statically typed programming language for scientific computations with first class support for physical dimensions and units.

You can use it for simple mathematical computations:

```
>>> 1920/16*9
= 1080
>>> 2^32
= 4294967296
>>> sqrt(1.4^2 + 1.5^2) * cos(pi/3)^2
= 0.512957
```

The real strength of Numbat, however, is to perform calculations with physical units:

```
>>> 8 km / (1 h + 25 min)
8 kilometer / (1 hour + 25 minute)
= 5.64706 km/h [Velocity]
>>> 140 € -> GBP
140 euro ➞ british_pound
= 120.768 £ [Money]
>>> atan2(30 cm, 1 m) -> deg
atan2(30 centimeter, 1 meter) ➞ degree
= 16.6992°
>>> let ω = 2π c / 660 nm
let ω: Frequency = 2 π × c / 660 nanometer
>>> ℏ ω -> eV
ℏ × ω ➞ electronvolt
= 1.87855 eV [Energy]
```

Read the tutorial to learn more about the language. Or jump directly to the syntax reference.

# Tutorial

In this tutorial, you will use Numbat to calculate how many bananas you would need to power a
house. This is based on an article in the great *what if?* series
by the author of the xkcd comics.

Bananas contain potassium. In its natural form, potassium contains a tiny fraction (0.0117%)
of the isotope ^{40}K, which is radioactive. The idea is to
use the radioactive decay energy as a power source. Open an interactive Numbat session by
typing `numbat`

in your favorite terminal emulator. We start by entering a few facts about
potassium-40:

```
let halflife = 1.25 billion years
let occurrence = 0.0117%
let molar_mass = 40 g / mol
```

New constants are introduced with the `let`

keyword. We
define these physical quantities with their respective physical units (`years`

,
`percent`

, `g / mol`

) in order to profit from Numbat’s unit-safety and unit-conversion
features later on.

Our first goal is to compute the radioactivity of natural potassium. Instead of dealing with the
half-life, we want to know the decay rate. When entering the following computation, you can try
Numbat’s auto-completion functionality. Instead of typing out `halflife`

, just type `half`

and press
`Tab`

.

```
let decay_rate = ln(2) / halflife
```

As you can see, we can use typical mathematical functions such as the
natural logarithm `ln`

. Next, we are interested how much radioactivity comes from a certain
mass of potassium:

```
let radioactivity =
N_A * occurrence * decay_rate / molar_mass -> Bq / g
```

The `-> Bq / g`

part at the end converts the expression to Becquerel per gram. If you type
in `radioactivity`

, you should see a result of roughly `31 Bq / g`

, i.e. 31 radioactive
decays per second, per gram of potassium.

The unit conversion also serves another purpose. If anything would be wrong with our calculation at the units-level, Numbat would detect that and show an error. Unit safety is a powerful concept not just because you can eliminate an entire category of errors, but also because it makes your computations more readable.

We are interested in the radioactivity of bananas, so we first introduce a new (base) unit:

```
unit banana
```

This lets us write readable code like

```
let potassium_per_banana = 451 mg / banana
let radioactivity_banana = potassium_per_banana * radioactivity -> Bq / banana
```

and should give you a result of roughly `14 Bq / banana`

. Adding unit conversions at the end
of unit definitions is one way to enforce unit safety. An even more powerful way to do this
is to add *type annotations*: For example, to define the decay energy for a single
potassium-40 atom,
you can optionally add a `: Energy`

annotation that will be enforced by Numbat:

```
let energy_per_decay: Energy = 11 percent × 1.5 MeV + 89 percent × 1.3 MeV
```

This also works with custom units since Numbat adds new physical dimensions (types) implicitly:

```
let power_per_banana: Power / Banana = radioactivity_banana * energy_per_decay
```

You’ll also notice that types can be combined via mathematical operators such as `/`

in this example.

How many bananas we need to power a household is going to depend on the average power consumption of that household. So we are defining a simple function

```
fn household_power(annual_consumption: Energy) -> Power = annual_consumption / year
```

This allows us to finally answer the original question (for a typical US household in 2021)

```
household_power(10000 kWh) / power_per_banana
```

This should give you a result of roughly 4×10^{14} bananas^{1}.

## Attribution

The images in this tutorial are from https://what-if.xkcd.com/158/. They are licensed under the Creative Commons Attribution-NonCommercial 2.5 License. Details and usage notes can be found at https://xkcd.com/license.html.

^{1}

Interestingly, the *what if* article comes up with a result of 300 quadrillion bananas,
or 3 × 10^{17}. This is a factor of 1000 higher. This seems like a
mistake in the original source. All of our other intermediate results are consistent with what has
been computed in the original article.

# Examples

This chapter shows some exemplary Numbat programs from various disciplines and sciences.

# Acidity

```
# Compute the pH (acidity) of a solution based
# on the activity of hydrogen ions
#
# https://en.wikipedia.org/wiki/PH
fn pH_acidity(activity_hplus: Molarity) -> Scalar =
- log10(activity_hplus / (mol / L))
print(pH_acidity(5e-6 mol / L))
```

# Barometric formula

```
# This script calculates the air pressure at a specified
# height above sea level using the barometric formula.
let p0: Pressure = 1 atm
let t0: Temperature = 288.15 K
dimension TemperatureGradient = Temperature / Length
let lapse_rate: TemperatureGradient = 0.65 K / 100 m
fn air_pressure(height: Length) -> Pressure =
p0 · (1 - lapse_rate · height / t0)^5.255
print("Air pressure 1500 m above sea level: {air_pressure(1500 m) -> hPa}")
```

# Body mass index

```
# This script calculates the Body Mass Index (BMI) based on
# the provided mass and height values.
unit BMI: Mass / Length^2 = kg / m^2
fn body_mass_index(mass: Mass, height: Length) =
mass / height² -> BMI
print(body_mass_index(70 kg, 1.75 m))
```

# Factorial

```
# Naive factorial implementation to showcase recursive
# functions and conditionals.
fn factorial(n: Scalar) -> Scalar =
if n < 1
then 1
else n × factorial(n - 1)
# Compare result with the builtin factorial operator
assert_eq(factorial(10), 10!)
```

# Flow rate in a pipe

```
# This script calculates and prints the flow rate in a pipe
# using the Hagen-Poiseuille equation. It assumes the dynamic
# viscosity of water and allows for inputs of pipe radius,
# pipe length, and pressure difference.
let μ_water: DynamicViscosity = 1 mPa·s
fn flow_rate(radius: Length, length: Length, Δp: Pressure) -> FlowRate =
π × radius^4 × Δp / (8 μ_water × length)
let pipe_radius = 1 cm
let pipe_length = 10 m
let Δp = 0.1 bar
let Q = flow_rate(pipe_radius, pipe_length, Δp)
print("Flow rate: {Q -> L/s}")
```

# Medication dosage

```
# This script calculates the total daily dose and per intake
# dose of a medication based on a person's body weight.
@aliases(takings)
unit taking
let body_weight = 75 kg
let dosage = (60 mg / kg) / day
let frequency = 3 takings / day
let total_daily_dose = dosage * body_weight -> mg / day
print("Total daily dose: {total_daily_dose}")
let single_dose = total_daily_dose / frequency
print("Single dose: {single_dose}")
```

# Molarity

```
# This script calculates and prints the molarity of a salt
# water solution, given a fixed mass of salt (NaCl) and a
# volume of water.
let molar_mass_NaCl = 58.44 g / mol
fn molarity(mass: Mass, volume: Volume) -> Molarity =
(mass / molar_mass_NaCl) / volume
let salt_mass = 9 g
let water_volume = 1 L
print(molarity(salt_mass, water_volume) -> mmol / l)
```

# Musical note frequency

```
# Musical note frequencies in the 12 equal temperament system
let frequency_A4: Frequency = 440 Hz # the A above middle C, A4
fn note_frequency(n: Scalar) -> Frequency = frequency_A4 * 2^(n / 12)
print("A5: {note_frequency(12)}") # one octave higher up, 880 Hz
print("E4: {note_frequency(7)}")
print("C4: {note_frequency(-3)}")
```

# Population growth

```
# Exponential model for population growth
let initial_population = 50_000 people
let growth_rate = 2% per year
fn predict_population(t: Time) =
initial_population × e^(growth_rate·t) // round
print("Population in 20 years: {predict_population(20 years)}")
print("Population in 100 years: {predict_population(1 century)}")
```

# Recipe

```
# Scale ingredient quantities based on desired servings.
@aliases(servings)
unit serving
let original_recipe_servings = 2 servings
let desired_servings = 3 servings
fn scale<D>(quantity: D) -> D =
quantity × desired_servings / original_recipe_servings
print("Milk: {scale(500 ml)}")
print("Flour: {scale(250 g)}")
print("Sugar: {scale(2 cups)}")
print("Baking powder: {scale(4 tablespoons)}")
```

# XKCD 687

```
# Dimensional analysis
#
# https://xkcd.com/687/
let core_pressure = 3.5 million atmospheres
let prius_milage = 50 miles per gallon
let min_width_channel = 21 miles
# Make sure that the result is dimensionless:
let r: Scalar =
planck_energy / core_pressure × prius_milage / min_width_channel
print("{r} ≈ π ?")
```

# XKCD 2585

```
# Rounding
#
# https://xkcd.com/2585/
17 mph
ans -> meters/sec // round
ans -> knots // round
ans -> fathoms/sec // round
ans -> furlongs/min // round
ans -> fathoms/sec // round
ans -> kph // round
ans -> knots // round
ans -> kph // round
ans -> furlongs/hour // round
ans -> mph // round
ans -> m/s // round
ans -> furlongs/min // round
ans -> yards/sec // round
ans -> fathoms/sec // round
ans -> m/s // round
ans -> mph // round
ans -> furlongs/min // round
ans -> knots // round
ans -> yards/sec // round
ans -> fathoms/sec // round
ans -> knots // round
ans -> furlongs/min // round
ans -> mph // round
print("I can ride my bike at {ans}.")
print("If you round.")
```

# XKCD 2812

```
# Solar panel placement
#
# Solar energy tip: To maximize sun exposure, always
# orient your panels downward and install them on the
# surface of the sun.
#
# https://xkcd.com/2812/
#
# [1] https://en.wikipedia.org/wiki/Solar_luminosity
# [2] https://en.wikipedia.org/wiki/Sun
unit $: Money
let net_metering_rate = $ 0.20 / kWh
let panel_area = 1 m²
let panel_efficiency = 20 %
fn savings(i: Irradiance) -> Money / Time =
net_metering_rate × i × panel_area × panel_efficiency -> $/year
print("Option A: On the roof, south facing")
let savings_a = savings(4 kWh/m²/day)
print(savings_a // round)
print()
print("Option B: On the sun, downward facing")
dimension Luminosity = Power
let sun_luminosity: Luminosity = 3.828e26 W # [1]
let sun_area: Area = 6.09e12 km^2 # [2]
let savings_b = savings(sun_luminosity / sun_area)
print(savings_b // round)
```

# IDE / editor integration

There is syntax highlighting support for the following IDEs / text editors:

# Basics

This chapter introduces language features that are required to perform basic computations in Numbat or to write small programs.

# Number notation

Numbers in Numbat can be written in the following forms:

- Integer notation
`12345`

`12_345`

— with decimal separators

- Floating point notation
`0.234`

`.234`

— without the leading zero

- Scientific notation
`1.234e15`

`1.234e+15`

`1e-9`

`1.0e-9`

- Non-decimal bases notation
`0x2A`

— Hexadecimal`0o52`

— Octal`0b101010`

— Binary

# Unit notation

Most units can be entered in the same way that they would appear in textbook calculations. They
usually have a long form (`meter`

, `degrees`

, `byte`

, …), a plural form (`meters`

, `degrees`

, `bytes`

),
and a short alias (`m`

, `°`

, `B`

). For a full list of supported units, see
this page.

All SI-accepted units support metric prefixes (`mm`

, `cm`

, `km`

, … or `millimeter`

, `centimeter`

, `kilometer`

, …)
and — where sensible — units allow for binary prefixes (`MiB`

, `GiB`

, … or `mebibyte`

, `gibibyte`

, …). Note
that the short-form prefixes can only be used with the short version of the unit, and vice versa (that is: `kmeter`

and `kilom`

are *not* allowed, only `km`

and `kilometer`

).

Units can be combined using mathematical operations such as multiplication, division and exponentiation: `kg * m/s^2`

, `km/h`

, `m²`

, `meter per second`

.

The following snippet shows various styles of entering units:

```
2 min + 1 s
150 cm
sin(30°)
50 mph
6 MiB
2 minutes + 1 second
150 centimeters
sin(30 degrees)
50 miles per hour
6 mebibyte
```

Note that Numbat also allows you to define new units.

# Operations and precedence

Numbat operators and other language constructs, ordered by precedence form *high* to *low*:

Operation / operator | Syntax |
---|---|

square, cube, … | `x²` , `x³` , `x⁻¹` , … |

factorial | `x!` |

exponentiation | `x^y` , `x**y` |

multiplication (implicit) | `x y` (whitespace) |

unary negation | `-x` |

division | `x per y` |

division | `x / y` , `x ÷ y` |

multiplication (explicit) | `x * y` , `x · y` , `x × y` |

subtraction | `x - y` |

addition | `x + y` |

comparisons | `x < y` , `x <= y` , `x ≤ y` , … `x == y` , `x != y` |

unit conversion | `x -> y` , `x → y` , `x ➞ y` , `x to y` |

conditionals | `if x then y else z` |

reverse function call | `x // f` |

Note that *implicit* multiplication has a higher precedence than division, i.e. `50 cm / 2 m`

will be parsed as `50 cm / (2 m)`

.

Also, note that `per`

-division has a higher precedence than `/`

-division. This means `1 / meter per second`

will be parsed as `1 / (meter per second)`

.

If in doubt, you can always look at the pretty-printing output (second line in the snippet below) to make sure that your input was parsed correctly:

```
>>> 1 / meter per second
1 / (meter / second)
= 1 s/m
```

# Constants

New constants can be introduced with the `let`

keyword:

```
let pipe_radius = 1 cm
let pipe_length = 10 m
let Δp = 0.1 bar
```

Definitions may contain a type annotation after the identifier (`let Δp: Pressure = 0.1 bar`

). This annotation will be verified by the type checker. For more complex definitions
it can be desirable to add type annotations, as it often improves readability and allows
you to catch potential errors early:

```
let μ_water: DynamicViscosity = 1 mPa·s
let Q: FlowRate = π × pipe_radius^4 × Δp / (8 μ_water × pipe_length)
```

# Unit conversions

The conversion operator `->`

attempts to convert the physical quantity on its left hand side to
the *unit of the expression* on its right hand side. This means that you can write an arbitrary
expression on the right hand side — but only the unit part will be extracted. For example:

```
# simple unit conversion:
> 120 km/h -> mph
= 74.5645 mi/h
# expression on the right hand side:
> 120 m^3 -> km * m^2
= 0.12 m²·km
# convert x1 to the same unit as x2:
> let x1 = 50 km / h
> let x2 = 3 m/s -> x1
x2 = 10.8 km/h
```

# Function definitions

Numbat comes with a large number of predefined functions, but
it is also possible to add new functions. A function definition is introduced with
the `fn`

keyword:

```
fn max_distance(v: Velocity, θ: Angle) -> Length = v² · sin(2 θ) / g0
```

This exemplary function computes the maximum distance of a projectile under the
influence of Earths gravity. It takes two parameters (The initial velocity `v`

and
the launch angle `θ`

), which are both annotated with their corresponding physical
dimension (their type). The function returns a distance, and so the return type
is specified as `Length`

.

## Type inference

The return type annotation may be omitted, but it is often desirable to add it for better readability of the code and in order to catch potential errors.

The parameter types can also (sometimes) be omitted, in which case Numbat tries to infer their type. However, this often leads to overly generic function signatures. For example, consider the following function to compute the kinetic energy of a massive object in motion:

```
fn kinetic_energy(mass, speed) = 1/2 * mass * speed^2
```

Without any type annotations, this function has an overly generic type where
`mass`

and `speed`

can have arbitrary dimensions (and the return type is
`type(mass) * type(speed)^2`

). So for this case, it is probably better to add
parameter and return types.

## Generic functions

Sometimes however, it *is* useful to write generic functions. For example, consider
`max(a, b)`

— a function that returns the larger of the two arguments. We might
want to use that function with *dimensionful* arguments such as `max(1 m, 1 yd)`

.
To define such a generic function, you can introduce *type parameters* in angle
brackets:

```
fn max<T>(a: T, b: T) -> T = if a > b then a else b
```

This function signature tells us that `max`

takes two arguments of *arbitrary*
type `T`

(but they need to match!), and returns a quantity of the same type `T`

.

Note that you can perform the usual operations with type parameters, such as multiplying/dividing them with other types, or raising to rational powers. For example, consider this cube-root function

```
fn cube_root<T>(x: T^3) -> T = x^(1/3)
```

that can be called with a scalar (`cube_root(8) == 2`

) or a dimensionful
argument (`cube_root(1 liter) == 10 cm`

).

Note: `cube_root`

can also be defined as `fn cube_root<T>(x: T) -> T^(1/3)`

,
which is equivalent to the definition above.

## Recursive functions

It is also possible to define recursive functions. In order to do so, you currently need to specify the return type — as the type signature can not (yet) be inferred otherwise.

For example, a naive recursive implementation to compute Fibonacci numbers in Numbat looks like this:

```
fn fib(n: Scalar) -> Scalar =
if n ≤ 2
then 1
else fib(n - 2) + fib(n - 1)
```

# Conditionals

Numbat has `if-then-else`

conditional expressions with the following
syntax

```
if <cond> then <expr1> else <expr2>
```

where `<cond>`

is a condition that evaluates to a Boolean value, like
`3 ft < 3 m`

. The types of `<expr1>`

and `<expr2>`

need to match.

For example, you can defined a simple step function using

```
fn step(x: Scalar) -> Scalar = if x < 0 then 0 else 1
```

# Printing, testing, debugging

## Printing

Numbat has a builtin `print`

procedure that can be used to print the value of an expression:

```
print(2 km/h)
print(3 ft < 1 m)
```

You can also print out simple messages as strings. This is particularly useful when combined with string interpolation to print results of a computation:

```
let radius: Length = sqrt(footballfield / 4 pi) -> meter
print("A football field would fit on a sphere of radius {radius}")
```

You can use almost every expression inside a string interpolation field. For example:

```
print("3² + 4² = {hypot2(3, 4)}²")
let speed = 25 km/h
print("Speed of the bicycle: {speed} ({speed -> mph})")
```

## Testing

The `assert_eq`

procedure can be used to test for (approximate) equality of two quantities.
This is often useful to make sure that (intermediate) results in longer calculations have
a certain value, e.g. when restructuring the code. The general syntax is

```
assert_eq(q1, q2)
assert_eq(q1, q2, ε)
```

where the first version tests for exact equality while the second version tests for approximate equality \( |q_1-q_2| < \epsilon \) with a specified accuracy of \( \epsilon \). For example:

```
assert_eq(2 + 3, 5)
assert_eq(1 ft × 77 in², 4 gal)
assert_eq(alpha, 1 / 137, 1e-4)
assert_eq(3.3 ft, 1 m, 1 cm)
```

There is also a plain `assert`

procedure that can test any boolean condition. For example:

```
assert(1 yard < 1 meter)
assert(π != 3)
```

A runtime error is thrown if an assertion fails. Otherwise, nothing happens.

## Debugging

You can use the builtin `type`

procedure to see the type (or physical dimension) of a quantity:

```
>>> type(g0)
Length / Time²
>>> type(2 < 3)
Bool
```

# Advanced

This chapter covers more advanced topics, like defining custom physical units or new physical dimensions.

# Unit definitions

New units of measurement can be introduced with the `unit`

keyword. There are two types of units: base units and derived units.

A new **base unit** can be defined by specifying the *physical dimension* it represents. For example, in the
International System of Units (SI), the *second* is the base unit for measuring times:

```
unit second: Time
```

Here, `Time`

denotes the physical dimension. To learn more, you can read the corresponding chapter. But for now, we can just assume that they are already given.

**Derived units** are also introduced with the `unit`

keyword. But unlike base units, they are defined through their relation to
other units. For example, a *minute* can be defined as

```
unit minute: Time = 60 second
```

Here, the `: Time`

annotation is optional. If a dimension is specified, it will be used to verify that the right hand side expression (`60 second`

) is indeed of physical dimension `Time`

. This is apparent in this simple example, but can be useful for more complicated unit definitions like

```
unit farad: Capacitance = ampere^2 second^4 / (kilogram meter^2)
```

## Prefixes

If a unit may be used with metric prefixes such as `milli`

/`m`

, `kilo`

/`k`

or `mega`

/`M`

, we can prepend the unit definition with the `@metric_prefixes`

decorator:

```
@metric_prefixes
unit second: Time
```

This allows identifiers such as `millisecond`

to be used in calculations. See the section below how prefixes interact with aliases.

Similarly, if a unit should be prependable with *binary* (IEC) prefixes such as `kibi`

/`Ki`

, `mebi`

/`Mi`

or `gibi`

/`Gi`

, you can
add the `@binary_prefixes`

decorator. A unit might also allow for both metric and binary prefixes, for example:

```
@binary_prefixes
@metric_prefixes
unit byte = 8 bit
```

This allows the usage of both `mebibyte`

(1024² byte) as well as `megabyte`

(1000² byte).

## Aliases

It is often useful to define alternative names for a unit. For example, we might want to use the plural form `seconds`

or the commonly
used short version `s`

. We can use the `@aliases`

decorator to specify them:

```
@metric_prefixes
@aliases(meters, metre, metres, m: short)
unit meter: Length
```

In addition to the name, we can also specify how aliases interact with prefixes using `: long`

(the default), `: short`

, `: both`

or
`: none`

. The actual unit name (`meter`

) and all `long`

aliases will accept the long version of prefixes (…, `milli`

, `kilo`

, `mega`

, `giga`

, …).
All `short`

aliases (`m`

in the example above) will only accept the respective short versions of the prefixes (…, `m`

, `k`

, `M`

, `G`

, …).
Aliases annotated with `: both`

or `: none`

accept either both long *and* short prefixes, or none of them.
The unit definition above allows all of following expressions:

```
millimeter
kilometer
millimeters
kilometers
millimetre
kilometre
millimetres
kilometres
mm
km
...
```

# Dimension definitions

New (physical) dimensions can be introduced with the `dimension`

keyword. Similar to units, there are base dimensions (like *length*, *time* and *mass*) and dimensions that are derived from those base dimensions (like *momentum*, which is *mass* · *length* / *time*). Base dimensions are simply introduced by declaring their name:

```
dimension Length
dimension Time
dimension Mass
```

Derived dimensions need to specify their relation to base dimensions (or other derived dimensions). For example:

```
dimension Velocity = Length / Time
dimension Momentum = Mass * Velocity
dimension Force = Mass * Acceleration = Momentum / Time
dimension Energy = Momentum^2 / Mass = Mass * Velocity^2 = Force * Length
```

In the definition of `Force`

and `Energy`

, we can see that *alternative definitions* can be given. This is entirely optional. If specified, the compiler will make sure that all definitions are equivalent.

## Custom dimensions

It is often useful to introduce ‘fictional’ physical dimensions. For example, we might want to do calculations with
screen resolutions and ‘dot densities’. Introducing a new dimension for *dots* then allows us to define units like `dpi`

without sacrificing unit safety:

```
dimension Dot
@aliases(dots)
unit dot: Dot
unit dpi = dots / inch
fn inter_dot_spacing(resolution: Dot / Length) -> Length = 1 dot / resolution
inter_dot_spacing(72 dpi) -> µm # 353 µm
```

There is also a shorthand notation for creating a new dimension and a corresponding unit:

```
unit book
@aliases(pages)
unit page
@aliases(words)
unit word
let words_per_book = 500 words/page × 300 pages/book
```

Here, the base unit definitions will implicitly create new dimensions which are capitalized
versions of the unit names (`Book`

, `Page`

, `Word`

). This allows you to count books, pages
and words independently without any risk of mixing them. The `words_per_book`

constant in this
examples has a type of `Word / Book`

.

# Syntax overview

```
# This is a line comment. It can span over
# multiple lines
# 1. Imports
use prelude # This is not necessary. The 'prelude'
# module will always be loaded upon startup
use units::stoney # Load a specific module
# 2. Numbers
12345 # integer notation
12_345 # optional decimal separators
0.234 # floating point notation
.234 # without the leading zero
1.234e15 # scientific notation
1.234e+15
1e-9
1.0e-9
0x2A # hexadecimal
0o52 # octal
0b101010 # binary
# 3. Simple expressions
3 + (4 - 3) # Addition and subtraction
1920 / 16 * 9 # Multiplication, division
1920 ÷ 16 × 9 # Unicode-style, '·' is also multiplication
2 pi # Whitespace is implicit multiplication
meter per second # 'per' keyword can be used for division
2^3 # Exponentiation
2**3 # Python-style
2³ # Unicode exponents
2^-3 # Negative exponents
mod(17, 4) # Modulo
3 in -> cm # Unit conversion, can also be → or ➞
3 in to cm # Unit conversion with the 'to' keyword
cos(pi/3 + pi) # Call mathematical functions
pi/3 + pi // cos # Same, 'arg // f' is equivalent to 'f(arg)'
# The '//' operator has the lowest precedence
# which makes it very useful for interactive
# terminals (press up-arrow, and add '// f')
# 4. Constants
let n = 4 # Simple numerical constant
let q1 = 2 m/s # Right hand side can be any expression
let q2: Velocity = 2 m/s # With optional type annotation
let q3: Length / Time = 2 m/s # more complex type annotation
# 5. Function definitions
fn foo(z: Scalar) -> Scalar = 2 * z + 3 # A simple function
fn speed(len: Length, dur: Time) -> Velocity = len / dur # Two parameters
fn my_sqrt<T>(q: T^2) -> T = q^(1/2) # A generic function
fn is_non_negative(x: Scalar) -> Bool = x ≥ 0 # Returns a bool
# 6. Dimension definitions
dimension Fame # A new base dimension
dimension Deceleration = Length / Time^2 # A new derived dimension
# 7. Unit definitions
@aliases(quorks) # Optional aliases-decorator
unit quork = 0.35 meter # A new derived unit
@metric_prefixes # Optional decorator to allow 'milliclonk', etc.
@aliases(ck: short) # short aliases can be used with short prefixes (mck)
unit clonk: Time = 0.2 seconds # Optional type annotation
@metric_prefixes
@aliases(wh: short)
unit warhol: Fame # New base unit for the "Fame" dimension
unit thing # New base unit with automatically generated
# base dimension "Thing"
# 8. Conditionals
fn step(x: Scalar) -> Scalar = # The construct 'if <cond> then <expr> else <expr>'
if x < 0 # is an expression, not a statement. It can span
then 0 # multiple lines.
else 1
# 9. Procedures
print(2 kilowarhol) # Print the value of an expression
print("hello world") # Print a message
print("value of pi = {pi}") # String interpolation
print("sqrt(10) = {sqrt(10)}") # Expressions in string interpolation
assert(1 yard < 1 meter) # Assertion
assert_eq(1 ft, 12 in) # Assert that two quantities are equal
assert_eq(1 yd, 1 m, 10 cm) # Assert that two quantities are equal, up to
# the given precision
type(2 m/s) # Print the type of an expression
```

# The prelude

Numbat comes with a special module called `prelude`

that is always loaded on
startup (unless `--no-prelude`

is specified on the command line). This module
is split into multiple submodules and sets up a useful default environment with
mathematical functions, constants but also dimension definitions, unit
definitions and physical constants.

You can find the full source code of the standard library on GitHub.

This chapter is a reference to the prelude module.

# Predefined functions

## Utility

```
fn unit_of<T>(x: T) -> T
fn value_of<T>(x: T) -> Scalar
```

## Math

### Basics

```
fn abs<T>(x: T) -> T
fn round<T>(x: T) -> T
fn floor<T>(x: T) -> T
fn ceil<T>(x: T) -> T
fn mod<T>(a: T, b: T) -> T
fn sqrt<D>(x: D^2) -> D
fn sqr<D>(x: D) -> D^2
```

### Exponential and logarithm

```
fn exp(x: Scalar) -> Scalar
fn ln(x: Scalar) -> Scalar
fn log(x: Scalar) -> Scalar
fn log10(x: Scalar) -> Scalar
fn log2(x: Scalar) -> Scalar
```

### Trigonometry

Basic:

```
fn cos(x: Scalar) -> Scalar
fn sin(x: Scalar) -> Scalar
fn tan(x: Scalar) -> Scalar
fn asin(x: Scalar) -> Scalar
fn acos(x: Scalar) -> Scalar
fn atan(x: Scalar) -> Scalar
fn atan2<T>(y: T, x: T) -> Scalar
```

Hyperbolic:

```
fn sinh(x: Scalar) -> Scalar
fn cosh(x: Scalar) -> Scalar
fn tanh(x: Scalar) -> Scalar
fn asinh(x: Scalar) -> Scalar
fn acosh(x: Scalar) -> Scalar
fn atanh(x: Scalar) -> Scalar
```

Extra:

```
fn cot(x: Scalar) -> Scalar
fn acot(x: Scalar) -> Scalar
fn coth(x: Scalar) -> Scalar
fn acoth(x: Scalar) -> Scalar
fn secant(x: Scalar) -> Scalar
fn arcsecant(x: Scalar) -> Scalar
fn cosecant(x: Scalar) -> Scalar
fn csc(x: Scalar) -> Scalar
fn acsc(x: Scalar) -> Scalar
fn sech(x: Scalar) -> Scalar
fn asech(x: Scalar) -> Scalar
fn csch(x: Scalar) -> Scalar
fn acsch(x: Scalar) -> Scalar
```

### Others

```
fn gamma(x: Scalar) -> Scalar
```

### Statistics

```
fn mean<D>(xs: D…) -> D
fn maximum<D>(xs: D…) -> D
fn minimum<D>(xs: D…) -> D
```

### Geometry

```
fn hypot2<T>(x: T, y: T) -> T
fn hypot3<T>(x: T, y: T, z: T) -> T
fn circle_area<L>(radius: L) -> L^2
fn circle_circumference<L>(radius: L) -> L
fn sphere_area<L>(radius: L) -> L^2
fn sphere_volume<L>(radius: L) -> L^3
```

## Physics

### Temperature conversion

```
fn from_celsius(t_celsius: Scalar) -> Temperature
fn to_celsius(t_kelvin: Temperature) -> Scalar
fn from_fahrenheit(t_fahrenheit: Scalar) -> Temperature
fn to_fahrenheit(t_kelvin: Temperature) -> Scalar
```

## Strings

```
fn str_length(s: String) -> Scalar
fn str_slice(s: String, start: Scalar, end: Scalar) -> String
fn str_append(a: String, b: String) -> String
fn str_contains(haystack: String, needle: String) -> Bool
fn str_replace(s: String, pattern: String, replacement: String) -> String
fn str_repeat(a: String, n: Scalar) -> String
```

# Constants

## Mathematical

`pi`

,`π`

`τ`

`e`

`golden_ratio`

,`φ`

## Named numbers

Large numbers

`hundred`

`thousand`

`million`

`billion`

`trillion`

`quadrillion`

`quintillion`

`googol`

Unicode fractions:

`½`

,`⅓`

,`⅔`

,`¼`

,`¾`

, …

Colloquial:

`quarter`

`half`

`semi`

`double`

`triple`

`dozen`

## Physics

Description | Identifier | Dimension |
---|---|---|

The speed of light in vacuum | `speed_of_light` , `c` | `Velocity` |

The Newtonian constant of gravitation | `gravitational_constant` , `G` | `Force × Length^2 / Mass^2` |

Standard acceleration of gravity on earth | `gravity` , `g0` | `Acceleration` |

The Planck constant | `planck_constant` , `ℎ` | `Mass × Length^2 / Time` |

The reduced Planck constant | `h_bar` , `ℏ` | `Mass × Length^2 / Time` |

Mass of the electron | `electron_mass` | `Mass` |

Elementary charge (charge of the electron) | `elementary_charge` , `electron_charge` | `ElectricCharge` |

Magnetic constant (vacuum magnetic permeability) | `magnetic_constant` , `µ0` , `mu0` | `Force / Current^2` |

Electric constant (vacuum electric permittivity) | `electric_constant` , `ε0` , `eps0` | `Capacitance / Length` |

Bohr magneton | `bohr_magneton` , `µ_B` | `Energy / MagneticFluxDensity` |

Fine structure constant | `fine_structure_constant` , `alpha` , `α` | `Scalar` |

Proton mass | `proton_mass` | `Mass` |

Neutron mass | `neutron_mass` | `Mass` |

Avogadro constant | `avogadro_constant` , `N_A` | `1 / AmountOfSubstance` |

Boltzmann constant | `boltzmann_constant` , `k_B` | `Energy / Temperature` |

Stefan-Boltzmann constant | `stefan_boltzmann_constant` | `Power / (Area × Temperature^4)` |

Ideal gas constant | `gas_constant` , `R` | `Energy / (AmountOfSubstance × Temperature)` |

Planck length | `planck_length` | `Length` |

Planck mass | `planck_mass` | `Mass` |

Planck time | `planck_time` | `Time` |

Planck temperature | `planck_temperature` | `Temperature` |

Planck energy | `planck_energy` | `Energy` |

Bohr radius | `bohr_radius` , `a0` | `Length` |

Rydberg constant | `rydberg_constant` | `Wavenumber` |

# List of supported units

See also: Unit notation.

All SI-accepted units support metric prefixes (`mm`

, `cm`

, `km`

, … or `millimeter`

, `centimeter`

, `kilometer`

, …)
and — where sensible — units allow for binary prefixes (`MiB`

, `GiB`

, … or `mebibyte`

, `gibibyte`

, …).

Dimension | Unit name | Identifier(s) |
---|---|---|

`AbsorbedDose` | Gray | `gray` , `grays` , `Gy` |

`Activity` | Becquerel | `becquerel` , `becquerels` , `Bq` |

`AmountOfSubstance` | Mole | `mol` , `mole` , `moles` |

`Angle` | Minute of arc | `arcmin` , `arcminute` , `arcminutes` |

`Angle` | Second of arc | `arcsec` , `arcsecond` , `arcseconds` |

`Angle` | Degree | `deg` , `degree` , `degrees` , `°` |

`Angle` | Gradian | `gon` |

`Angle` | Radian | `rad` , `radian` , `radians` |

`Angle` | Revolution | `rev` , `revolution` , `revolutions` |

`Angle` | Turn | `turn` , `turns` |

`Area` | Acre | `acre` , `acres` |

`Area` | Are | `are` |

`Area` | Barn | `barn` , `barns` |

`Area` | Football field | `footballfield` |

`Area` | Hectare | `ha` , `hectare` , `hectares` |

`Capacitance` | Farad | `F` , `farad` , `farads` |

`CatalyticActivity` | Katal | `kat` , `katal` , `katals` |

`Current` | Ampere | `A` , `ampere` , `amperes` |

`DigitalInformation` | Bit | `bit` , `bits` |

`DigitalInformation` | Byte | `B` , `byte` , `Byte` , `bytes` , `Bytes` , `octet` , `Octet` , `octets` , `Octets` |

`DigitalInformation / Time` | Bits per second | `bps` |

`Dot` | Dot | `dot` , `dots` |

`Dot / Length` | Dots per inch | `dpi` |

`DynamicViscosity` | Poise | `poise` |

`ElectricCharge` | Coulomb | `C` , `coulomb` , `coulombs` |

`ElectricConductance` | Siemens | `S` , `siemens` |

`ElectricResistance` | Ohm | `ohm` , `ohms` , `Ω` |

`Energy` | British thermal unit | `BTU` , `Btu` |

`Energy` | Calorie | `cal` , `calorie` , `calories` |

`Energy` | Electronvolt | `electronvolt` , `electronvolts` , `eV` |

`Energy` | Erg | `erg` , `ergs` |

`Energy` | Hartree | `hartree` , `hartrees` |

`Energy` | Joule | `J` , `joule` , `joules` |

`Energy` | Planck energy | `planck_energy` |

`Energy` | Rydberg unit of energy | `Ry` |

`Energy` | Watt-hour | `watthour` , `Wh` |

`EquivalentDose` | Sievert | `sievert` , `sieverts` , `Sv` |

`Force` | Dyne | `dyn` , `dyne` |

`Force` | Kilogram-force | `kgf` , `kilogram_force` |

`Force` | Newton | `N` , `newton` , `newtons` |

`Force` | Ounce-force | `ounce_force` , `ozf` |

`Force` | Pound-force | `lbf` , `pound_force` |

`Force / Volume` | Mercury | `Hg` |

`Frame` | Frame | `frame` , `frames` |

`Frame / Time` | Frames per second | `fps` |

`Frequency` | Hertz | `hertz` , `Hz` |

`Frequency` | Revolutions per minute | `rpm` , `RPM` |

`Illuminance` | Foot-candle | `fc` , `footcandle` , `footcandles` |

`Illuminance` | Lux | `lux` , `lx` |

`Inductance` | Henry | `H` , `henries` , `henry` , `henrys` |

`KinematicViscosity` | Stokes | `St` , `stokes` |

`Length` | Ångström | `angstrom` , `angstroms` , `Å` |

`Length` | Astronomical unit | `astronomicalunit` , `astronomicalunits` , `au` , `AU` |

`Length` | Bohr | `bohr` |

`Length` | Fathom | `fathom` , `fathoms` |

`Length` | Fermi | `fermi` |

`Length` | Foot | `feet` , `foot` , `ft` |

`Length` | Furlong | `furlong` , `furlongs` |

`Length` | Inch | `in` , `inch` , `inches` |

`Length` | League | `league` , `leagues` |

`Length` | Light-year | `lightyear` , `lightyears` , `ly` |

`Length` | Metre | `m` , `meter` , `meters` , `metre` , `metres` |

`Length` | Micron | `micron` |

`Length` | Mile | `mi` , `mile` , `miles` |

`Length` | Nautical Mile | `nautical_mile` , `nautical_miles` , `NM` , `nmi` |

`Length` | Parsec | `parsec` , `parsecs` , `pc` |

`Length` | Planck length | `planck_length` |

`Length` | US rod | `perch` , `rod` , `rods` |

`Length` | Smoot | `smoot` |

`Length` | Stoney length | `stoney_length` |

`Length` | Thousandth of an inch | `mil` , `mils` , `thou` |

`Length` | Yard | `yard` , `yards` , `yd` |

`Length / Volume` | Miles per gallon | `mpg` |

`LuminousFlux` | Lumen | `lm` , `lumen` , `lumens` |

`LuminousIntensity` | Candela | `candela` , `candelas` , `cd` |

`MagneticFieldStrength` | Oersted | `Oe` , `oersted` |

`MagneticFlux` | Maxwell | `maxwell` , `Mx` |

`MagneticFlux` | Weber | `Wb` , `weber` , `webers` |

`MagneticFluxDensity` | Gauss | `gauss` |

`MagneticFluxDensity` | Tesla | `T` , `tesla` , `teslas` |

`Mass` | Dalton | `Da` , `dalton` , `daltons` |

`Mass` | Firkin | `firkin` , `firkins` |

`Mass` | Grain | `grain` , `grains` |

`Mass` | Gram | `g` , `gram` , `gramme` , `grammes` , `grams` |

`Mass` | Hundredweight | `cwt` , `long_hundredweight` |

`Mass` | Long ton | `long_ton` , `long_tons` |

`Mass` | Ounce | `ounce` , `ounces` , `oz` |

`Mass` | Planck mass | `planck_mass` |

`Mass` | Pound | `lb` , `lbs` , `pound` , `pounds` |

`Mass` | Stone | `stone` |

`Mass` | Stoney mass | `stoney_mass` |

`Mass` | Tonne | `metricton` , `ton` , `tonne` , `tonnes` , `tons` |

`Molality` | Molal | `molal` |

`Molarity` | Molar | `molar` |

`Money` | Australian dollar | `A$` , `AUD` , `australian_dollar` , `australian_dollars` |

`Money` | Brazilian real | `brazilian_real` , `brazilian_reals` , `BRL` , `R$` |

`Money` | Pound sterling | `british_pound` , `GBP` , `pound_sterling` , `£` |

`Money` | Bulgarian lev | `BGN` , `bulgarian_lev` , `bulgarian_leva` |

`Money` | Canadian dollar | `C$` , `CAD` , `canadian_dollar` , `canadian_dollars` |

`Money` | Czech koruna | `czech_koruna` , `czech_korunas` , `CZK` , `Kč` |

`Money` | Danish krone | `danish_krone` , `danish_kroner` , `DKK` |

`Money` | US dollar | `$` , `dollar` , `dollars` , `USD` |

`Money` | Euro | `EUR` , `euro` , `euros` , `€` |

`Money` | Hong Kong dollar | `HK$` , `HKD` , `hong_kong_dollar` , `hong_kong_dollars` |

`Money` | Hungarian forint | `Ft` , `HUF` , `hungarian_forint` , `hungarian_forints` |

`Money` | Icelandic króna | `icelandic_krona` , `icelandic_kronur` , `icelandic_króna` , `icelandic_krónur` , `ISK` |

`Money` | Indian rupee | `indian_rupee` , `indian_rupees` , `INR` , `₹` |

`Money` | Indonesian rupiah | `IDR` , `indonesian_rupiah` , `indonesian_rupiahs` , `Rp` |

`Money` | Israeli new shekel | `ILS` , `israeli_new_shekel` , `israeli_new_shekels` , `NIS` , `₪` |

`Money` | Malaysian ringgit | `malaysian_ringgit` , `malaysian_ringgits` , `MYR` , `RM` |

`Money` | New Zealand dollar | `new_zealand_dollar` , `new_zealand_dollars` , `NZ$` , `NZD` |

`Money` | Norwegian krone | `NOK` , `norwegian_krone` , `norwegian_kroner` |

`Money` | Philippine peso | `philippine_peso` , `philippine_pesos` , `PHP` , `₱` |

`Money` | Polish złoty | `PLN` , `polish_zloty` , `polish_zlotys` , `zł` |

`Money` | Chinese yuan | `CNY` , `renminbi` , `元` |

`Money` | Romanian leu | `lei` , `romanian_leu` , `romanian_leus` , `RON` |

`Money` | Singapore dollar | `S$` , `SGD` , `singapore_dollar` , `singapore_dollars` |

`Money` | South African rand | `south_african_rand` , `ZAR` |

`Money` | South Korean won | `KRW` , `south_korean_won` , `south_korean_wons` , `₩` |

`Money` | Swedish krona | `SEK` , `swedish_krona` , `swedish_kronor` |

`Money` | Swiss franc | `CHF` , `swiss_franc` , `swiss_francs` |

`Money` | Thai baht | `thai_baht` , `thai_bahts` , `THB` , `฿` |

`Money` | Turkish lira | `TRY` , `turkish_lira` , `turkish_liras` , `₺` |

`Money` | Japanese yen | `JPY` , `yen` , `yens` , `¥` , `円` |

`Person` | Person | `capita` , `people` , `person` , `persons` |

`Piece` | Piece | `piece` , `pieces` |

`Pixel` | Pixel | `pixel` , `pixels` , `px` |

`Pixel / Length` | Pixels per inch | `ppi` |

`Power` | Metric horsepower | `horsepower` , `hp` |

`Power` | Watt | `W` , `watt` , `watts` |

`Pressure` | Standard atmosphere | `atm` , `atmosphere` , `atmospheres` |

`Pressure` | Bar | `bar` , `bars` |

`Pressure` | Inch of mercury | `inHg` |

`Pressure` | Millimeter of mercury | `mmHg` |

`Pressure` | Pascal | `Pa` , `pascal` , `pascals` |

`Pressure` | Pound-force per square inch | `psi` , `PSI` |

`Pressure` | Torr | `torr` |

`Scalar` | Billion | `billion` |

`Scalar` | dozen | `dozen` |

`Scalar` | Hundred | `hundred` |

`Scalar` | Million | `million` |

`Scalar` | Parts per billion | `partsperbillion` , `ppb` |

`Scalar` | Parts per million | `partspermillion` , `ppm` |

`Scalar` | Parts per quadrillion | `partsperquadrillion` , `ppq` |

`Scalar` | Parts per trillion | `partspertrillion` , `ppt` |

`Scalar` | Percent | `%` , `pct` , `percent` |

`Scalar` | Quadrillion | `quadrillion` |

`Scalar` | Quintillion | `quintillion` |

`Scalar` | Thousand | `thousand` |

`Scalar` | Trillion | `trillion` |

`SolidAngle` | Steradian | `sr` , `steradian` , `steradians` |

`Temperature` | Kelvin | `K` , `kelvin` , `kelvins` |

`Temperature` | Planck temperature | `planck_temperature` |

`Time` | Century | `centuries` , `century` |

`Time` | Day | `d` , `day` , `days` |

`Time` | Decade | `decade` , `decades` |

`Time` | Fortnight | `fortnight` , `fortnights` |

`Time` | Hour | `h` , `hour` , `hours` , `hr` |

`Time` | Julian year | `julian_year` , `julian_years` |

`Time` | Millennium | `millennia` , `millennium` |

`Time` | Minute | `min` , `minute` , `minutes` |

`Time` | Month | `month` , `months` |

`Time` | Planck time | `planck_time` |

`Time` | Second | `s` , `sec` , `second` , `seconds` |

`Time` | Sidereal day | `sidereal_day` , `sidereal_days` |

`Time` | Stoney time | `stoney_time` |

`Time` | Week | `week` , `weeks` |

`Time` | Gregorian year | `year` , `years` , `yr` |

`Velocity` | Knot | `kn` , `knot` , `knots` , `kt` |

`Velocity` | Kilometres per hour | `kph` |

`Velocity` | Miles per hour | `mph` |

`Voltage` | Volt | `V` , `volt` , `volts` |

`Volume` | Cubic centimetre | `cc` , `ccm` |

`Volume` | US cup | `cup` , `cups` |

`Volume` | US fluid ounce | `floz` , `fluidounce` , `fluidounces` |

`Volume` | US liquid gallon | `gal` , `gallon` , `gallons` |

`Volume` | US hogshead | `hogshead` , `hogsheads` |

`Volume` | Litre | `l` , `L` , `liter` , `liters` , `litre` , `litres` |

`Volume` | US liquid pint | `pint` , `pints` |

`Volume` | Swimming pool | `swimmingpool` |

`Volume` | US tablespoon | `tablespoon` , `tablespoons` , `tbsp` |

`Volume` | US teaspoon | `teaspoon` , `teaspoons` , `tsp` |

# Installation

## Linux

### Ubuntu

*… and other Debian-based Linux distributions.*

Download the latest `.deb`

package from the release page
and install it via `dpkg`

. For example:

```
curl -LO https://github.com/sharkdp/numbat/releases/download/v1.8.0/numbat_1.8.0_amd64.deb
sudo dpkg -i numbat_1.8.0_amd64.deb
```

### Arch Linux

In Arch Linux and Arch based distributions, you can install the prebuilt package of Numbat from the AUR:

```
yay -S numbat-bin
```

You can also install the numbat AUR package, which will download the source and compile it.

```
yay -S numbat
```

## Void Linux

You can install the `numbat`

package using

```
sudo xbps-install -S numbat
```

## NixOs

*… or any distribution where nix is installed.*

Install numbat to your profile:

```
nix-env -iA nixpkgs.numbat
```

Or add it to your NixOs Configuration:

```
environment.systemPackages = [
pkgs.numbat
];
```

## macOS

### Homebrew

You can install Numbat with Homebrew:

```
brew install numbat
```

## Windows

### Scoop

You can install the numbat package using scoop:

```
scoop install main/numbat
```

## From pre-built binaries

Download the latest release for your system from this page. Unpack
the archive and place the `numbat`

/`numbat.exe`

binary in a folder that is on your `PATH`

.

Note that the `modules`

folder that is included in the archives is not strictly required to run Numbat. It serves more
as a reference for interested users. However, if you want to get the best possible experience or if you are
a package maintainer, please follow these guidelines.

## From source

Clone the Git repository, and build Numbat with `cargo`

:

```
git clone https://github.com/sharkdp/numbat
cd numbat/
cargo install -f --path numbat-cli
```

Or install the latest release using

```
cargo install numbat-cli
```

## Guidelines for package maintainers

Thank you for packaging Numbat! This section contains instructions that are not strictly necessary to create a Numbat package, but provide users with the best-possible experience on your target platform.

Numbat has a standard library that is written in Numbat itself. The sources for this
so called “prelude” are available in the `numbat/modules`

folder.
We also include this `modules`

folder in the pre-built GitHub releases.
Installing this folder as part of the package installation is not necessary for Numbat to work, as the prelude is also
stored inside the `numbat`

binary. But ideally, this folder should be made available for users. There are three reasons for this:

- Users might want to look at the code in the standard library to get a better understanding of the language itself.
- For some error messages, Numbat refers to locations in the source code. For example, if you type
`let meter = 2`

, the compiler will let you know that this identifier is already in use, and has been previously defined at a certain location inside the standard library. If the corresponding module is available as a file on the users system, they will see the proper path and can read the corresponding file. - Users might want to make changes to the prelude. Ideally, this should be done via a user module folder, but the system-wide folder can serve as a template.

In order for this to work, the `modules`

folder should ideally be placed in the standard location for the
target operating system. If this is not possible, package maintainers can customize
numbat during compilation by setting the environment variable `NUMBAT_SYSTEM_MODULE_PATH`

to the final locatiom.
If this variable is set during compilation, the specified path will be compiled into the `numbat`

binary.

In order to test that everything is working as intended, you can open `numbat`

and type `let meter = 2`

. The
path in the error message should point to the specified location (and *not* to `<builtin>/…`

).

# Usage

## Modes

You can run the Numbat command-line application in three different modes:

Mode | Command to run |
---|---|

Start an interactive session (REPL) | `numbat` |

Run a Numbat program | `numbat script.nbt` |

Evaluate a single expression | `numbat -e '30 km/h -> mi/h'` |

## Command-line options

See `numbat --help`

for more information.

## Interactive sessions

Interactive sessions allow you to perform a sequence of calculations. You can use the special identifiers
`ans`

or `_`

to refer to the result of the last calculation. For example:

```
>>> 60 kW h / 150 kW
= 0.4 h
>>> ans -> minutes
= 24 min
```

### Commands

There is a set of special commands that only work in interactive mode:

Command | Action |
---|---|

`list` , `ls` | List all functions, dimensions, variables and units |

`list <what>` | Where `<what>` can be `functions` , `dimensions` , `variables` , `units` |

`info <identifier>` | Get more information about units and variables |

`clear` | Clear screen |

`help` , `?` | View short help text |

`quit` , `exit` | Quit the session |

### Key bindings

In interactive command-line mode, you can use the following key bindings. Most importantly,
`Tab`

for auto-completion, arrow keys and `Ctrl-R`

for browsing the command history, and
`Ctrl-D`

for exiting the interactive session.

Key sequence | Action |
---|---|

`Tab` , `Ctrl` -`I` | Auto-completion |

`Ctrl` -`D` | Quit |

`Ctrl` -`L` | Clear screen |

`Up` , `Down` | Browse command history |

`Ctrl` -`R` | Search command history |

`Ctrl` -`C` | Clear the current line |

`Alt` -`Enter` | Insert newline |

`Home` , `Ctrl` -`A` | Move cursor to the beginning of the line |

`End` , `Ctrl` -`E` | Move cursor to the end of the line |

`Ctrl` -`W` | Delete word leading up to cursor |

# Customization

## Startup

By default, Numbat will load the following modules/files during startup, in order:

- Numbat Prelude (a module called
`prelude`

, either from`<module-path>/prelude.nbt`

if available, or the builtin version) - The user initialization file, if available (a file called
`init.nbt`

from`<config-path>/init.nbt`

)

### Config path

Numbat’s configuration folder (`<config-path>`

above) can be found under:

Platform | Path |
---|---|

Linux | `$HOME/.config/numbat` or `$XDG_CONFIG_HOME/numbat` |

macOS | `$HOME/Library/Application Support/numbat` |

Windows | `C:\Users\Alice\AppData\Roaming\numbat` |

## Module paths

Numbat will load modules from the following sources. Entries higher up in the list take precedence.

Location | Description |
---|---|

`$NUMBAT_MODULES_PATH` | This environment variable can point to a single directory or contain a `:` -separatedlist of paths |

`<config-path>/modules` | User-customized module folder |

`/usr/share/numbat/modules` | System-wide module folder (Linux and macOS) |

`C:\Program Files\numbat\modules` | System-wide module folder (Windows) |

`<builtin>` | Builtin modules inside the `numbat` binary |

Note that the System-location might be different for some installation methods. Refer to your package manager for details.

## Customization

### Configuration

Numbat’s configuration file is called `config.toml`

, and it needs to be placed in
`<config-path>`

described above (`~/.config/numbat/config.toml`

on Linux). You
can generate a default configuration by calling

```
numbat --generate-config
```

The most important fields are:

```
# Controls the welcome message. Can be "long", "short", or "off".
intro-banner = "long"
# Controls the prompt character(s) in the interactive terminal.
prompt = ">>> "
# Whether or not to pretty-print expressions before showing the result.
# Can be "always", "never" or "auto". The latter uses pretty-printing
# only in interactive mode.
pretty-print = "auto"
[exchange-rates]
# When and if to load exchange rates from the European Central Bank for
# currency conversions. Can be "on-startup" to always fetch exchange rates
# in the background when the application is started. With "on-first-use",
# Numbat only fetches exchange rates when they are needed. Exchange rate
# fetching can also be disabled using "never". The latter will lead to
# "unknown identifier" errors when a currency unit is being used.
fetching-policy = "on-startup"
```

### Custom functions, constants, units

If you want to add custom functions, constants, or units to your default environment,
create a `init.nbt`

file in your config folder (`~/.config/numbat/init.nbt`

on Linux).

### Custom modules

You can also create your own modules that can be loaded on demand. To this end,
create a new file, say `<module-path>/user/finance.nbt`

in one of the module folders
(e.g. `~/.config/numbat/modules/custom/finance.nbt`

on Linux). This module can then be
loaded using

```
use custom::finance
```

in your Numbat scripts or in the REPL. You can also load custom modules from `init.nbt`

if you want to have them available all the time.

You can also organize modules into subfolders (e.g. `<module-path>/custom/finance/functions.nbt`

).
In that case, you can load them using

```
use custom::finance::functions
```

In fact, the `custom`

folder is just a convention to avoid name clashes with the
standard library.

# Usage

The browser-based version of Numbat is available at https://numbat.dev/.

## Interactive terminal

The terminal allows you to perform a sequence of calculations.
You can use the arrow keys to browse through the command history.
The special identifiers `ans`

and `_`

refer to the result of the last calculation. For example:

```
>>> 60 kW h / 150 kW
= 0.4 h
>>> ans -> minutes
= 24 min
```

## Commands

There is a set of special commands that only work in the web version:

Command | Action |
---|---|

`list` , `ls` | List all constants, units, and dimensions |

`list <what>` | Where `<what>` can be `functions` , `dimensions` , `variables` , `units` |

`info <identifier>` | Get more information about units and variables |

`help` , `?` | View short help text |

`reset` | Reset state (clear constants, functions, units, …) |

`clear` | Clear screen |

## Key bindings

In interactive command-line mode, you can use the following key bindings. Most importantly,
`Tab`

for auto-completion, arrow keys and `Ctrl-R`

for browsing the command history, and
`Ctrl-D`

for exiting the interactive session.

Key sequence | Action |
---|---|

`Tab` | Auto-completion |

`Ctrl` -`L` | Clear screen |

`Up` , `Down` | Browse command history |

`Ctrl` -`R` | Search command history |

`Ctrl` -`C` | Clear the current line |

`Shift` -`Enter` | Insert newline |

`Home` , `Ctrl` -`A` | Move cursor to the beginning of the line |

`End` , `Ctrl` -`E` | Move cursor to the end of the line |

`Ctrl` -`Left` , `Ctrl` -`Right` | Move cursor one word left/right |

`Ctrl` -`K` | Remove text to the right of the cursor |

`Ctrl` -`U` | Remove text to the left of the cursor |

## Sharing calculations

To share the result of a calculation with someone else, you can just copy the URL from
your browers address bar. As you enter new lines in the terminal, your input will be
appended to the URL to build up something like
`https://numbat.dev/?q=let+P0+%3D+50_000+people%0A…`

that you can just copy and share. To reset the state and clear the URL, use the `reset`

command (see above).

# Type system

Numbat is a language with a special type system that treats *physical dimensions* as types.
A type checker infers types for every expression in the program and ensures that everything is correct in terms of physical dimensions, which implies correctness in terms of physical *units*.
For example, the expression `2 meter`

has a *type* of `Length`

.
The expression `3 inch`

*also* has a type of `Length`

.
The combined expression `2 meter + 3 inch`

is therefore well-typed.
On the other hand, `2 meter + 3 second`

is ill-typed, as `3 second`

is of type `Time`

.

The type system is *static* which means that the correctness of a Numbat program is verified before the program starts executing.
Note that certain *runtime* errors (like division-by-zero) can still occur.

## Algebra of types

Types in Numbat can be combined in various ways to produce new types.
In its most general form, a type can be thought of as a product of physical (base) dimensions \( D_k \) with exponents \( \alpha_k \in \mathbb{Q} \):
\[ \prod_k D_k^{\alpha_k} \]
For example, the type *Energy* can be represented as *Mass¹ × Length² × Time⁻²*.

### Multiplication

This naturally allows us to *multiply* types (by combining the factors of both products into a single product).
We can use the `*`

operator to construct types for physical dimensions that are products of two or more (base) dimensions. For example:

```
dimension Time
dimension Current
dimension ElectricCharge = Current * Time
```

### Exponentiation

We can also raise units to arbitrary powers \( n \in \mathbb{Q} \), by simply multiplying each \( \alpha_k \) with \( n \). The syntax uses the `^`

exponentiation operator:

```
dimension Length
dimension Volume = Length^3
dimension Time
dimension Frequency = Time^(-1)
```

### Division

Once we have multiplication and exponentiation, we can define the *division* of two types as

```
TypeA / TypeB ≡ TypeA * TypeB^(-1)
```

This is mostly for convenience. It allows us to write definitions like

```
dimension Power = Energy / Time
```

Note:When we talk about products of types in this section, we mean actual, literal products. Type theory also has the notion ofproduct typeswhich denote something else: compound types — like tuples or structs — that are built by combining two or more types. If we think of types in terms of the sets of all possible values that they represent, then product types represent the Cartesian product of those.

## Type inference and type annotations

The type checker can infer the types of (most) expressions without explicitly declaring them. For example, the following definition does not mention any types:

```
let E_pot = 80 kg × 9.8 m/s² × 5 m
```

However, it is often helpful to specify the type anyway. This way, we can make sure that no mistakes were made:

```
let E_pot: Energy = 80 kg × 9.8 m/s² × 5 m
```

The type checker will compare the inferred type with the specified type and raise an error in case of inconsistency.

Function definitions also allow for type annotations, both for the parameters as well as the return type. The following example shows a function that takes a quantity of type `Length`

and returns a `Pressure`

:

```
let p0: Pressure = 101325 Pa
let t0: Temperature = 288.15 K
let gradient = 0.65 K / 100 m
fn air_pressure(height: Length) -> Pressure = p0 · (1 - gradient · height / t0)^5.255
```

## Generic types

Numbat’s type system also supports generic types (type polymorphism). These can be used for functions that work regardless of the physical dimension of the argument(s). For example, the type signature of the absolute value function is given by

```
fn abs<D>(x: D) -> D
```

where the angle brackets after the function name introduce new type parameters (`D`

).
This can be read as: `abs`

takes an arbitrary physical quantity of dimension `D`

and returns a quantity of the *same* physical dimension `D`

.

As a more interesting example, we can look at the `sqrt`

function. Its type signature can be written as

```
fn sqrt<D>(x: D^2) -> D
```

Alternatively, it could also be specified as `fn sqrt<D>(x: D) -> D^(1/2)`

.

## Limitations

The static type system also has some limitations. Let’s look at an exponentiation expression like

```
expr1 ^ expr2
```

where `expr1`

and `expr2`

are arbitrary expressions. In order for that expression
to properly type check, the *type* of `expr2`

must be `Scalar`

— something like
`2^meter`

does not make any sense. *If* the type of `expr1`

is also `Scalar`

,
everything is well and the type of the total expression is also `Scalar`

. An example
for this trivial case is an expression like `e^(-x²/σ²)`

. As long as the type
of `x`

is the same as the type of `σ`

, this is fine.

A more interesting case arises if `expr1`

is dimensionfull, as in `meter^3`

. Here,
things become difficult: in order to compute the *type* of the total expression
`expr1 ^ expr2`

, we need to know the *value* of `expr2`

. For the `meter^3`

example,
the answer is `Length^3`

. This seems straightforward. However, the syntax of the
language allows arbitrary expressions in the exponent. This is important to support
use cases like the above `e^(-x²/σ²)`

. But it poses a problem for the type checker.
In order to compute the type of `expr1 ^ expr2`

, we need to fully *evaluate*
`expr2`

at compile time. This is not going to work in general. Just think of a
hypothetical expression like `meter^f()`

where `f()`

could do *anything*. Maybe even
get some input from the user at runtime.

Numbat’s solution to this problem looks like this: If `expr1`

is *not* dimensionless,
we restrict `expr2`

to a small subset of allowed operations that can be fully
evaluated at compile time (similar to `constexpr`

expressions in C++, `const`

expressions in Rust, etc). Expressions like `meter^(2 * (2 + 1) / 3)`

are completely
fine and can be typechecked (`Length^2`

), but things like function calls are not
allowed and will lead to a compile time error.

To summarize: Given an exponentiation expression like `expr1 ^ expr2`

, the type checker
requires that:

`expr2`

is of type`Scalar`

- One of the following:
`expr1`

is also of type`Scalar`

`expr2`

can be*evaluated at compile time*and yields a rational number.

#### Remark

We would probably need to enter the world of *dependent types* if we wanted to fully
support exponentiation expressions without the limitations above. For example, consider
the function `f(x, n) = x^n`

. The return type of that function *depends on the value*
of the parameter `n`

.

# Contact us

To contact us, either open a GitHub issue
or discussion, or pop into
`#numbat`

on Libera.Chat (link
to webchat).