chriswarbo-net: ffd89477db028dabc509d70719d29ef18343145a

     1: ---
     2: title: Joining Parentheses
     3: ---
     4: 
     5: <small>Part of [my units pages](/projects/units)</small>
     6: 
     7: TODO: Maybe just draw boxes?
     8: 
     9: Parentheses are used to 'group together' parts of an expression; for example
    10: $8 + ((3 + 2) \times (1 + 4))$. When many parentheses are nested, a common
    11: problem is having to physically *count them* to figure out which ones match up;
    12: or, in other words, to determine the extent of what's in them.
    13: 
    14: Prior to the use of parentheses, it was common to use an over-bar (called a
    15: 'vinculum') to express grouping. This would render the above like:
    16: $8 + \overbar{\overbar{3 + 2} \times \overbar{1 + 4}}$
    17: 
    18: <div style="display: none;">
    19: 
    20: \newcommand{\overparens}[1]{\ensuremath{\bigl(\text{$\overline{\mathstrut\smash{\text{#1}}}$}\bigr)}}
    21: \newcommand{\underparens}[1]{\ensuremath{\bigl(\text{$\underline{\mathstrut\smash{\text{#1}}}$}\bigr)}}
    22: 
    23: </div>
    24: 
    25: The vinculum makes its extent more clear, since it is physically present above
    26: its entire contents. However, I still appreciate the way parentheses 'split up'
    27: their interior from their exterior: we can combine both by using the vinculum to
    28: 'join the parentheses', like so: $\overparens{3 + 2} \times \overparens{1 + 4}$
    29: 
    30: Note that this may cause clutter when using
    31: [overbars for negatives](negative_bar_notation.html), in which case it may be
    32: preferable to join parentheses *below* their enclosed expression:
    33: $\underparens{3 + 2} \times \underparens{1 + 4}$
    34: 
    35: When there's no overbar ambiguity, and we have heavily-nested parentheses, I
    36: find it helpful to alternate between over- and under-bars at each level of
    37: nesting. This original example above would thus become:
    38: 
    39: $$
    40: 8 + \overparens{\underparens{3 + 2} \times \underparens{1 + 4}}
    41: $$
    42: 
    43: I use this extensively when working in combinatory logic (which makes extensive
    44: use of nested tree structures!). For example, here's function composition:
    45: 
    46: $$
    47: \overparens{\underparens{\overparens{C f} g} x}
    48:   &= \overparens{\underparens{\overparens{\underparens{\overparens{S \underparens{K S}} K} f} g} x} \\
    49:   &= \overparens{\underparens{\overparens{\underparens{\overparens{K S} f} \underparens{K f}} g} x} \\
    50:   &= \overparens{\underparens{\overparens{S \underparens{K f}} g} x} \\
    51:   &= \overparens{\underparens{\overparens{K f} x} \underparens{g x}} \\
    52:   &= \overparens{f \underparens{g x}} \\
    53: $$