[All Your Base] draft approaches doc

exercises/practice/all-your-base/.approaches/config.json

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+{
+  "introduction": {
+    "authors": ["colinleach",
+                "BethanyG"],
+    "contributors": []
+  }
+}

exercises/practice/all-your-base/.approaches/introduction.md

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+# Introduction
+
+The main aim of this exercise is to understand how non-negative integers work in different bases.
+
+Given that mathematical understanding, the code to implement it can be relatively simple.
+
+For this exercise, no attempt was made to benchmark performance, as this would distract from the main focus of writing clear, correct code.
+
+## General guidance
+
+Essentially all succesful solutions involve three steps:
+
+1. Check that inputs are valid.
+2. Convert the input list to a Python `int`.
+3. Convert that `int` to an output list in the new base.
+
+Some programmers prefer to separate the two conversions into separate functions, others put everything in a single function.
+
+This is largely a matter of taste, and either structure can be made reasonably concise and readable.
+
+## 1. Check the inputs
+
+```python
+    if input_base < 2:
+        raise ValueError("input base must be >= 2")
+
+    if not all( 0 <= digit < input_base for digit in digits) :
+        raise ValueError("all digits must satisfy 0 <= d < input base")
+
+    if not output_base >= 2:
+        raise ValueError("output base must be >= 2")
+
+```
+
+A valid number base must be `>=2` and all digits must be at least zero and strictly less than the number base.
+
+For the familiar base-10 system, this means 0 to 9.
+
+As implemented, the tests require that invalid input raise a `ValueError` with a suitable error message.
+
+## 2. Convert the input digits to an `int`
+
+These four code fragments all do essentially the same thing:
+
+```python
+# Simplest loop
+    val = 0
+    for digit in digits:
+        val = input_base * val + digit
+
+# Loop, separating the arithmetic steps
+    val = 0
+    for digit in digits:
+        val *= input_base
+        val += digit
+
+# Sum a comprehension over reversed digits
+    val = sum(digit * input_base ** pos for pos, digit in enumerate(reversed(digits)))
+
+# Sum a comprehension with alternative reversing
+    val = sum((digit * (input_base ** (len(digits) - 1 - i)) for i, digit in enumerate(digits)))
+```
+
+In the first two, the `val *= input_base` step essentially left-shifts all the previous digits, and `val += digit` adds a new digit on the right.
+
+In the two comprehensions, an exponentation like `input_base ** pos` left-shifts the current digit to the appropriate position in the output.
+
+*Please think about this until it makes sense:* these short code fragments are the main point of the exercise.
+
+In each code fragment, the Python `int` is called `val`, a deliberately neutral identifier.
+
+Surprisingly many students use names like `decimal` or `base10` for the intermediate value, which is misleading.
+
+A Python `int` is an object with a complicated (but largely hidden) implementation.
+
+There are methods to convert an `int` to string representations such as decimal, binary or hexadecimal, but the internal representation of `int` is certainly not decimal.
+
+## 3. Convert the intermediate `int` to output digits
+
+Now we have to reverse step 2, with a different base.
+
+```python
+    out = []
+
+# Step forward, insert new digits at beginning
+    while val > 0:
+        out.insert(0, val % output_base)
+        val = val // output_base
+
+# Insert at end, then reverse
+    while val:
+        out.append(val % output_base)
+        val //= output_base
+    out.reverse()
+
+# Use divmod()
+    while val:
+        div, mod = divmod(val, output_base)
+        out.append(mod)
+        val = div
+    out.reverse()
+```
+
+Again, there are multiple code snippets shown above, which all do the same thing.
+
+In each case, we essentially need the value and remainder of an integer division.
+
+The first snippet above adds new digits at the start of the list, while the next two add at the end.
+
+This is a choice of where to take the performance hit: appending to the end is a faster way to grow the list, but needs an extra reverse step.
+
+The choice of append-reverse would be obvious in Lisp or SML, but the difference is less important in Python.
+
+```python
+# return, with guard for empty list
+    return out or [0]
+```
+
+Finally, we return the digits just calculated.
+
+A minor complcation is that a zero value should be `[0]`, not `[]`.
+
+Here, we cover this case in the `return` statement, but it could also have been trapped at the beginning of the program, with an early `return`.
+
+## Recursion option
+
+```python
+def base2dec(input_base: int, digits: list[int]) -> int:
+    if not digits:
+        return 0
+    return input_base * base2dec(input_base, digits[:-1]) + digits[-1]
+
+
+def dec2base(number: int, output_base: int) -> list[int]:
+    if not number:
+        return []
+    return [number % output_base] + dec2base(number // output_base, output_base)
+```
+
+An unusual solution to the two conversions is shown above.
+
+It works, and the problem is small enough to avoid stack overflow (Python has no tail recursion).
+
+In practice, few Python programmers would take this approach in a language without the appropriate performance optimizations.
+
+To simplify: Python only *allows* recursion, it does nothing to *encourage* it: in contrast to Scala, Elixir, and similar languages.
+