calc: Infinities
5.5 Infinities
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The word ‘inf’ represents the mathematical concept of “infinity”. Calc
actually has three slightly different infinity-like values: ‘inf’,
Variables::); you should avoid using these names for your own variables
because Calc gives them special treatment. Infinities, like all
variable names, are normally entered using algebraic entry.
Mathematically speaking, it is not rigorously correct to treat
“infinity” as if it were a number, but mathematicians often do so
informally. When they say that ‘1 / inf = 0’, what they really mean is
that ‘1 / x’, as ‘x’ becomes larger and larger, becomes arbitrarily
close to zero. So you can imagine that if ‘x’ got “all the way to
infinity,” then ‘1 / x’ would go all the way to zero. Similarly, when
they say that ‘exp(inf) = inf’, they mean that ‘exp(x)’ grows without
bound as ‘x’ grows. The symbol ‘-inf’ likewise stands for an infinitely
negative real value; for example, we say that ‘exp(-inf) = 0’. You can
have an infinity pointing in any direction on the complex plane:
‘sqrt(-inf) = i inf’.
The same concept of limits can be used to define ‘1 / 0’. We really
want the value that ‘1 / x’ approaches as ‘x’ approaches zero. But if
all we have is ‘1 / 0’, we can’t tell which direction ‘x’ was coming
from. If ‘x’ was positive and decreasing toward zero, then we should
say that ‘1 / 0 = inf’. But if ‘x’ was negative and increasing toward
zero, the answer is ‘1 / 0 = -inf’. In fact, ‘x’ could be an imaginary
number, giving the answer ‘i inf’ or ‘-i inf’. Calc uses the special
symbol ‘uinf’ to mean “undirected infinity”, i.e., a value which is
infinitely large but with an unknown sign (or direction on the complex
plane).
Calc actually has three modes that say how infinities are handled.
Normally, infinities never arise from calculations that didn’t already
have them. Thus, ‘1 / 0’ is treated simply as an error and left
unevaluated. The ‘m i’ (‘calc-infinite-mode’) command (
Infinite
Mode) enables a mode in which ‘1 / 0’ evaluates to ‘uinf’ instead.
There is also an alternative type of infinite mode which says to treat
zeros as if they were positive, so that ‘1 / 0 = inf’. While this is
less mathematically correct, it may be the answer you want in some
cases.
Since all infinities are “as large” as all others, Calc simplifies,
e.g., ‘5 inf’ to ‘inf’. Another example is ‘5 - inf = -inf’, where the
‘-inf’ is so large that adding a finite number like five to it does not
affect it. Note that ‘a - inf’ also results in ‘-inf’; Calc assumes
that variables like ‘a’ always stand for finite quantities. Just to
show that infinities really are all the same size, note that ‘sqrt(inf)
= inf^2 = exp(inf) = inf’ in Calc’s notation.
It’s not so easy to define certain formulas like ‘0 * inf’ and ‘inf /
inf’. Depending on where these zeros and infinities came from, the
answer could be literally anything. The latter formula could be the
limit of ‘x / x’ (giving a result of one), or ‘2 x / x’ (giving two), or
‘x^2 / x’ (giving ‘inf’), or ‘x / x^2’ (giving zero). Calc uses the
symbol ‘nan’ to represent such an “indeterminate” value. (The name
“nan” comes from analogy with the “NAN” concept of IEEE standard
arithmetic; it stands for “Not A Number.” This is somewhat of a
misnomer, since ‘nan’ _does_ stand for some number or infinity, it’s
just that _which_ number it stands for cannot be determined.) In Calc’s
notation, ‘0 * inf = nan’ and ‘inf / inf = nan’. A few other common
indeterminate expressions are ‘inf - inf’ and ‘inf ^ 0’. Also, ‘0 / 0 =
nan’ if you have turned on Infinite mode (as described above).
Infinities are especially useful as parts of “intervals”.
Interval Forms.
Info Catalog
calc: Complex Numbers
calc: Data Types
calc: Vectors and Matrices