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Géry Ogam
Géry Ogam

Summary of Edsger Dijkstra’s notes Why Numbering Should Start at Zero.

Range convention

There are 4 conventions to denote a range of natural numbers a, a + 1, …, b:

  1. a ≤ i < b + 1.
  2. a − 1 < i ≤ b.
  3. a ≤ i ≤ b.
  4. a − 1 < i < b + 1.

Excluding the lower bound like in range conventions 2 and 4 gives a lower bound of −1, which is not a natural number, to denote a range starting at 0 (convention 2: −1 < i ≤ b; convention 4: −1 < i < b + 1). Including the upper bound like in range conventions 2 and 3 gives a negative upper bound, which is not a natural number, to denote a range starting at 0 that has shrunk to the empty range (convention 2: −1 < i ≤ b, where b < 0; convention 3: 0 ≤ i ≤ b, where b < 0). Including the lower bound and excluding the upper bound like in range convention 1 gives a lower bound of 0, which is a natural number, to denote a range starting at 0 (0 ≤ i < b + 1), and an upper bound of 0, which is a natural number, to denote a range starting at 0 that has shrunk to the empty range (0 ≤ i < 0). So range convention 1 should be preferred.

Index convention

There are 2 conventions to index n elements:

  1. Starting at 0.
  2. Starting at 1.

Starting at 1 like in index convention 2 gives the index range 1 ≤ i < n + 1, following the preferred range convention. Starting at 0 like in index convention 1 gives the index range 0 ≤ i < n, following the preferred range convention, which is nicer. So index convention 1 should be preferred.

Summary of Edsger Dijkstra’s notes Why Numbering Should Start at Zero.

Range convention

There are 4 conventions to denote a range of natural numbers a, a + 1, …, b:

  1. a ≤ i < b + 1.
  2. a − 1 < i ≤ b.
  3. a ≤ i ≤ b.
  4. a − 1 < i < b + 1.

Excluding the lower bound like in range conventions 2 and 4 gives a lower bound of −1, which is not a natural number, to denote a range starting at 0. Including the upper bound like in range conventions 2 and 3 gives a negative upper bound, which is not a natural number, to denote a range starting at 0 that has shrunk to the empty range. Including the lower bound and excluding the upper bound like in range convention 1 gives a lower bound of 0, which is a natural number, to denote a range starting at 0, and an upper bound of 0, which is a natural number, to denote a range starting at 0 that has shrunk to the empty range. So range convention 1 should be preferred.

Index convention

There are 2 conventions to index n elements:

  1. Starting at 0.
  2. Starting at 1.

Starting at 1 like in index convention 2 gives the index range 1 ≤ i < n + 1, following the preferred range convention. Starting at 0 like in index convention 1 gives the index range 0 ≤ i < n, following the preferred range convention, which is nicer. So index convention 1 should be preferred.

Summary of Edsger Dijkstra’s notes Why Numbering Should Start at Zero.

Range convention

There are 4 conventions to denote a range of natural numbers a, a + 1, …, b:

  1. a ≤ i < b + 1.
  2. a − 1 < i ≤ b.
  3. a ≤ i ≤ b.
  4. a − 1 < i < b + 1.

Excluding the lower bound like in range conventions 2 and 4 gives a lower bound of −1, which is not a natural number, to denote a range starting at 0 (convention 2: −1 < i ≤ b; convention 4: −1 < i < b + 1). Including the upper bound like in range conventions 2 and 3 gives a negative upper bound, which is not a natural number, to denote a range starting at 0 that has shrunk to the empty range (convention 2: −1 < i ��� b, where b < 0; convention 3: 0 ≤ i ≤ b, where b < 0). Including the lower bound and excluding the upper bound like in range convention 1 gives a lower bound of 0, which is a natural number, to denote a range starting at 0 (0 ≤ i < b + 1), and an upper bound of 0, which is a natural number, to denote a range starting at 0 that has shrunk to the empty range (0 ≤ i < 0). So range convention 1 should be preferred.

Index convention

There are 2 conventions to index n elements:

  1. Starting at 0.
  2. Starting at 1.

Starting at 1 like in index convention 2 gives the index range 1 ≤ i < n + 1, following the preferred range convention. Starting at 0 like in index convention 1 gives the index range 0 ≤ i < n, following the preferred range convention, which is nicer. So index convention 1 should be preferred.

added 28 characters in body
Source Link
Géry Ogam
Géry Ogam

Summary of Edsger Dijkstra’s notes Why Numbering Should Start at Zero.

Range convention

There are 4 conventions to denote a range of natural numbers a, a + 1, …, b:

  1. a ≤ i < b + 1.
  2. a − 1 < i ≤ b.
  3. a ≤ i ≤ b.
  4. a − 1 < i < b + 1.

Excluding the lower bound like in range conventions 2 and 4 gives a lower bound of −1, which is not a natural number, to denote a range starting at 0. Including the upper bound like in range conventions 2 and 3 gives a negative upper bound, which is not a natural number, to denote a range starting at 0 that has shrunk to the empty range. Including the lower bound and excluding the upper bound like in range convention 1 gives a lower bound of 0, which is a natural number, to denote a range starting at 0, and an upper bound of 0, which is a natural number, to denote a range starting at 0 that has shrunk to the empty range. So range convention 1 should be preferred.

Index convention

There are 2 conventions to index n elements:

  1. Starting at 0.
  2. Starting at 1.

Starting at 1 like in index convention 2 gives the index range 1 ≤ i < n + 1, following the preferred range convention. Starting at 0 like in index convention 1 gives the index range 0 ≤ i < n, following the preferred range convention, which is nicer. So index convention 1 should be preferred.

Summary of Edsger Dijkstra’s notes Why Numbering Should Start at Zero.

Range convention

There are 4 conventions to denote a range of natural numbers a, a + 1, …, b:

  1. a ≤ i < b + 1.
  2. a − 1 < i ≤ b.
  3. a ≤ i ≤ b.
  4. a − 1 < i < b + 1.

Excluding the lower bound like in range conventions 2 and 4 gives a lower bound of −1, which is not a natural number, to denote a range starting at 0. Including the upper bound like in range conventions 2 and 3 gives a negative upper bound, which is not a natural number, to denote a range starting at 0 that has shrunk to the empty range. Including the lower bound and excluding the upper bound like in range convention 1 gives a lower bound of 0, which is a natural number, to denote a range starting at 0, and an upper bound of 0 to denote a range starting at 0 that has shrunk to the empty range. So range convention 1 should be preferred.

Index convention

There are 2 conventions to index n elements:

  1. Starting at 0.
  2. Starting at 1.

Starting at 1 like in index convention 2 gives the index range 1 ≤ i < n + 1, following the preferred range convention. Starting at 0 like in index convention 1 gives the index range 0 ≤ i < n, following the preferred range convention, which is nicer. So index convention 1 should be preferred.

Summary of Edsger Dijkstra’s notes Why Numbering Should Start at Zero.

Range convention

There are 4 conventions to denote a range of natural numbers a, a + 1, …, b:

  1. a ≤ i < b + 1.
  2. a − 1 < i ≤ b.
  3. a ≤ i ≤ b.
  4. a − 1 < i < b + 1.

Excluding the lower bound like in range conventions 2 and 4 gives a lower bound of −1, which is not a natural number, to denote a range starting at 0. Including the upper bound like in range conventions 2 and 3 gives a negative upper bound, which is not a natural number, to denote a range starting at 0 that has shrunk to the empty range. Including the lower bound and excluding the upper bound like in range convention 1 gives a lower bound of 0, which is a natural number, to denote a range starting at 0, and an upper bound of 0, which is a natural number, to denote a range starting at 0 that has shrunk to the empty range. So range convention 1 should be preferred.

Index convention

There are 2 conventions to index n elements:

  1. Starting at 0.
  2. Starting at 1.

Starting at 1 like in index convention 2 gives the index range 1 ≤ i < n + 1, following the preferred range convention. Starting at 0 like in index convention 1 gives the index range 0 ≤ i < n, following the preferred range convention, which is nicer. So index convention 1 should be preferred.

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Géry Ogam
Géry Ogam

Summary of Edsger Dijkstra’s notes Why Numbering Should Start at Zero.

Range convention

There are 4 conventions to denote a range of natural numbers a, a + 1, …, b:

  1. a ≤ i < b + 1.
  2. a − 1 < i ≤ b.
  3. a ≤ i ≤ b.
  4. a − 1 < i < b + 1.

Excluding the lower bound like in range conventions 2 and 4 gives a lower bound of −1, which is not a natural number, to denote a range starting at 0. Including the upper bound like in range conventions 2 and 3 gives a negative upper bound, which is not a natural number, to denote a range starting at 0 that has shrunk to the empty range. Including the lower bound and excluding the upper bound like in range convention 1 gives a lower bound of 0, which is a natural number, to denote a range starting at 0, and any integeran upper bound less than or equal to the lower boundof 0 to denote a range starting at 0 that has shrunk to the empty range. So range convention 1 should be preferred.

Index convention

There are 2 conventions to index n elements:

  1. Starting at 0.
  2. Starting at 1.

Starting at 1 like in index convention 2 gives the index range 1 ≤ i < n + 1, following the preferred range convention. Starting at 0 like in index convention 1 gives the index range 0 ≤ i < n, following the preferred range convention, which is nicer. So index convention 1 should be preferred.

Summary of Edsger Dijkstra’s notes Why Numbering Should Start at Zero.

Range convention

There are 4 conventions to denote a range of natural numbers a, a + 1, …, b:

  1. a ≤ i < b + 1.
  2. a − 1 < i ≤ b.
  3. a ≤ i ≤ b.
  4. a − 1 < i < b + 1.

Excluding the lower bound like in range conventions 2 and 4 gives a lower bound of −1, which is not a natural number, to denote a range starting at 0. Including the upper bound like in range conventions 2 and 3 gives a negative upper bound, which is not a natural number, to denote the empty range. Including the lower bound and excluding the upper bound like in range convention 1 gives a lower bound of 0, which is a natural number, to denote a range starting at 0, and any integer upper bound less than or equal to the lower bound to denote the empty range. So range convention 1 should be preferred.

Index convention

There are 2 conventions to index n elements:

  1. Starting at 0.
  2. Starting at 1.

Starting at 1 like in index convention 2 gives the index range 1 ≤ i < n + 1, following the preferred range convention. Starting at 0 like in index convention 1 gives the index range 0 ≤ i < n, following the preferred range convention, which is nicer. So index convention 1 should be preferred.

Summary of Edsger Dijkstra’s notes Why Numbering Should Start at Zero.

Range convention

There are 4 conventions to denote a range of natural numbers a, a + 1, …, b:

  1. a ≤ i < b + 1.
  2. a − 1 < i ≤ b.
  3. a ≤ i ≤ b.
  4. a − 1 < i < b + 1.

Excluding the lower bound like in range conventions 2 and 4 gives a lower bound of −1, which is not a natural number, to denote a range starting at 0. Including the upper bound like in range conventions 2 and 3 gives a negative upper bound, which is not a natural number, to denote a range starting at 0 that has shrunk to the empty range. Including the lower bound and excluding the upper bound like in range convention 1 gives a lower bound of 0, which is a natural number, to denote a range starting at 0, and an upper bound of 0 to denote a range starting at 0 that has shrunk to the empty range. So range convention 1 should be preferred.

Index convention

There are 2 conventions to index n elements:

  1. Starting at 0.
  2. Starting at 1.

Starting at 1 like in index convention 2 gives the index range 1 ≤ i < n + 1, following the preferred range convention. Starting at 0 like in index convention 1 gives the index range 0 ≤ i < n, following the preferred range convention, which is nicer. So index convention 1 should be preferred.

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Géry Ogam
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