Rework number concept

concepts/numbers/about.md

@@ -1,70 +1,183 @@
-# Haskell Built-in Numeric Types
+# Numbers
 
-## Numbers
-
-Haskell's standard library, known as ***Prelude***, defines most basic numeric types: fixed sized integers (`Int`), arbitrary precision integers (`Integer`), single precision floating (`Float`), and double precision floating (`Double`).
-Other numeric types such as rationals and complex numbers are defined in additional libraries.
+Haskell has a type, `Num`, which is the base type for all numeric types.
+There are four main sub-types inside the Number type: [`Int`][int], [`Integer`][integer], [`Double`][double]  and [`Float`][float].
 
+There are two types of integers types in Haskell, which represents whole numbers.
+`Int` is a fixed-size integer, while `Integer` is an arbitrary precision integer.
 `Integer`s hold any number no matter how big, up to the limit of your machine's memory, while `Int` corresponds to the set of positive and negative integers that can be expressed in 32 or 64 bits (guaranteed at least -2^29 to 2^29).
 
+Note that when writting negative numbers, you should use parentheses to avoid ambiguity.
+
+```haskell
+1
+
+(-1)
+```
+
 `Float` corresponds to the set of real numbers, limited by the precision of the computer.
 Operations defined on numbers usually work on one type or the other, but not both.
+`Double` is a floating-point number with double precision.
+
+```haskell
+10.5
+```
+
+## Converting between integers and floats
+
+Using the `toFloat` function, allows you to convert an `Int` or `Integer` to a `Float`.
+
+```haskell
+toFloat 1
+-- -> 1.0
+```
+
+There is also `fromIntegral` which converts any number types to the inferred type number type needed.
+
+```haskell
+fromIntegral 1
+-- -> 1.0
+```
+
+To convert a `Float` to an `Int`, you can use the `round`, `floor`, or `ceiling` functions.
+The `round` function rounds to the nearest integer, `floor` rounds down, and `ceiling` rounds up.
+
+```haskell
+round 1.5
+-- -> 2
+
+floor 1.5
+-- -> 1
 
-Functions to convert between `Float` and `Integer` include, among others, `toFloat` which converts `Int`/`Integer` to `Float` and `ceiling` which converts a `Float` to an `Int`.
+ceiling 1.5
+-- -> 2
+```
 
 ## Arithmetic operators
 
-Haskell has three "raise to the power" operators which work differently and take different argument types.
+You can use the [basic arithmetic operators][math] on the numeric types.
+The operators are `+`, `-`, `*`, `/`, and `%`.
+When using these operators, the types of the numbers must match.
+The `fromIntegral` function will convert so that the types match.
 
-- `**` Takes two **floating point numbers** and uses logarithms to compute the power.
-- `^^` Takes a **floating point** and raises it to a positive or negative **integer** power.
-- `^` Takes **any numerical type** and raises it to a **positive integer** power.
+### Addition & Subtraction & Multiplication
+
+The `+` operator is used for addition, the `-` operator is used for subtraction, and the `*` operator is used for multiplication.
 
-Conversion from an integer type (`Int` or `Integer`) to any other numeric type is done using the function `fromIntegral`.
-The target type is inferred automatically.
-For example:
+| Operator | Example        |
+| -------- | -------------- |
+| `+`      | `4 + 6 => 10`  |
+| `-`      | `15 - 10 => 5` |
+| `*`      | `2 * 3 => 6`   |
+
+### Division
+
+Division is used to divide numbers.
+The `/` operator is used for division.
+The result will always be a float.
 
 ```haskell
-n :: Integer
-n = 6
-x :: Float
-x = fromIntegral n      --> x = 6.0
-
-m :: Int
-m = 7
-y :: Double
-y = fromIntegral m      --> y = 7.0
+4.0 / 2.5
+-- -> 1.6
+
+4 / 2
+-- -> 2.0
 ```
 
-Division of integers is a little complicated.
-If you use the ordinary `/` operator on integers, then you will get an error message (although the expression `4/3` does work because Haskell helpfully promotes literal integers to floats where necessary).
-Instead, integer division is done using a collection of named operators.
+~~~~exercism/caution
+In some programming languages, when dividing by zero, the result will be an error.
 
-Haskell has a neat trick with operators: you can take any function that takes two arguments and use it like an operator by enclosing the name in back-ticks.
-So the following two lines mean exactly the same thing:
+In Haskell, when dividing by zero, the result is `Infinity` or `-Infinity`.
+The only exception is dividing zero by zero, resulting in `NaN` (Not a Number).
 
 ```haskell
- d = 7 `div` 3
- d = div 7 3
+1 / 0
+-- -> Infinity
+
+(-1) / 0
+-- -> -Infinity
+
+0 / 0
+-- -> NaN
+```
+~~~~
+
+## Integer division
+
+Integer division is used to divide numbers and get the whole part of the result.
+The result will always be rounded down to an Int.
+
+```haskell
+2 `div` 2
+-- -> 2
+
+5 `div` 2
+-- -> 2
 ```
 
-With that in mind, here are the integer division operators:
+~~~~exercism/caution
+Dividing by zero when using integer division results in a Exception.
+This is different from normal division.
+~~~~
+
+### Modulus
 
-- `quot`: Returns the quotient of the two numbers.
-  This is the result of division which is then truncated towards zero.
-- `rem`: Returns the remainder from the quotient.
-- `div`: Similar to `quot`, but is rounded down towards minus infinity.
-- `mod`: Returns the modulus of the two numbers.
-  This is similar to `rem`, but has different rules when `div` returns a negative number.
+Modulus is used to get the remainder of a division.
+The `mod` operator is used for modulus.
 
-Just as you can convert a function with two arguments into an operator, you can convert an operator into a function with two arguments by putting it in parentheses.
-So the following two lines mean the same thing:
+```haskell
+5 `mod` 2
+-- -> 1
+
+5 `mod` 3
+-- -> 2
+```
+
+~~~~exercism/caution
+Dividing by zero when using modulo results in an Exception.
+This is different from normal division.
 
 ```haskell
-(+) 3 4
-3 + 4
+1 `mod` 0
+# Exception: divide by zero
+```
+~~~~
+
+## Exponentiation
+
+Haskell has three "raise to the power" operators which work differently and take different argument types.
+
+- `**` Takes two **floating point numbers** and uses logarithms to compute the power.
+- `^^` Takes a **floating point** and raises it to a positive or negative **integer** power.
+- `^` Takes **any numerical type** and raises it to a **positive integer** power.
+
+```haskell
+3.0 ** 2.1
+-- -> 10.04510856630514
+
+2.5 ^^ 2
+-- -> 6.25
+
+2 ^ 3
+-- -> 8
+```
+
+## Priority and parentheses
+
+Haskell allows parentheses(`()`), which can be used to group expressions.
+This is useful when you want to change the order of operations.
+
+When using multiple arithmetic operators, the order of operations is the same as in mathematics, also known as [PEMDAS][pemdas].
+It follows the order of parentheses(`()`), exponents(`**`), multiplication(`*`) and division(`/`), and addition(`+`) and subtraction(`-`).
+
+```haskell
+2 + 3 - 4 * 4
+-- -> -11
+
+(2 + 3 - 4) * 4
+-- -> 4
 ```
 
----
-### Credits:
-The above text is modified from __The Haskell 98 Report__ by Simon Peyton Jones, used with permission to copy, distribute and modify for any purpose with this note included.
+[pemdas]: https://en.wikipedia.org/wiki/Order_of_operations
+[floor]: https://hackage.haskell.org/package/base/docs/Prelude.html#v:floor
+[ceiling]: https://hackage.haskell.org/package/base/docs/Prelude.html#v:ceiling