calc: Integer Truncation
8.2 Integer Truncation
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There are four commands for truncating a real number to an integer,
differing mainly in their treatment of negative numbers. All of these
commands have the property that if the argument is an integer, the
result is the same integer. An integer-valued floating-point argument
is converted to integer form.
If you press ‘H’ (‘calc-hyperbolic’) first, the result will be
expressed as an integer-valued floating-point number.
The ‘F’ (‘calc-floor’) [‘floor’ or ‘ffloor’] command truncates a real
number to the next lower integer, i.e., toward minus infinity. Thus
‘3.6 F’ produces 3, but ‘_3.6 F’ produces -4.
The ‘I F’ (‘calc-ceiling’) [‘ceil’ or ‘fceil’] command truncates
toward positive infinity. Thus ‘3.6 I F’ produces 4, and ‘_3.6 I F’
produces -3.
The ‘R’ (‘calc-round’) [‘round’ or ‘fround’] command rounds to the
nearest integer. When the fractional part is .5 exactly, this command
rounds away from zero. (All other rounding in the Calculator uses this
convention as well.) Thus ‘3.5 R’ produces 4 but ‘3.4 R’ produces 3;
‘_3.5 R’ produces -4.
The ‘I R’ (‘calc-trunc’) [‘trunc’ or ‘ftrunc’] command truncates
toward zero. In other words, it “chops off” everything after the
decimal point. Thus ‘3.6 I R’ produces 3 and ‘_3.6 I R’ produces -3.
These functions may not be applied meaningfully to error forms, but
they do work for intervals. As a convenience, applying ‘floor’ to a
modulo form floors the value part of the form. Applied to a vector,
these functions operate on all elements of the vector one by one.
Applied to a date form, they operate on the internal numerical
representation of dates, converting a date/time form into a pure date.
There are two more rounding functions which can only be entered in
algebraic notation. The ‘roundu’ function is like ‘round’ except that
it rounds up, toward plus infinity, when the fractional part is .5.
This distinction matters only for negative arguments. Also, ‘rounde’
rounds to an even number in the case of a tie, rounding up or down as
necessary. For example, ‘rounde(3.5)’ and ‘rounde(4.5)’ both return 4,
but ‘rounde(5.5)’ returns 6. The advantage of round-to-even is that the
net error due to rounding after a long calculation tends to cancel out
to zero. An important subtle point here is that the number being fed to
‘rounde’ will already have been rounded to the current precision before
‘rounde’ begins. For example, ‘rounde(2.500001)’ with a current
precision of 6 will incorrectly, or at least surprisingly, yield 2
because the argument will first have been rounded down to ‘2.5’ (which
‘rounde’ sees as an exact tie between 2 and 3).
Each of these functions, when written in algebraic formulas, allows a
second argument which specifies the number of digits after the decimal
point to keep. For example, ‘round(123.4567, 2)’ will produce the
answer 123.46, and ‘round(123.4567, -1)’ will produce 120 (i.e., the
cutoff is one digit to the _left_ of the decimal point). A second
argument of zero is equivalent to no second argument at all.
To compute the fractional part of a number (i.e., the amount which,
when added to ‘floor(N)’, will produce N) just take N modulo 1 using the
‘%’ command.
Note also the ‘\’ (integer quotient), ‘f I’ (integer logarithm), and
‘f Q’ (integer square root) commands, which are analogous to ‘/’, ‘B’,
and ‘Q’, respectively, except that they take integer arguments and
return the result rounded down to an integer.
Info Catalog
calc: Basic Arithmetic
calc: Arithmetic
calc: Complex Number Functions