calc: Types Tutorial
3.4 Types Tutorial
==================
Calc understands a variety of data types as well as simple numbers. In
this section, we’ll experiment with each of these types in turn.
The numbers we’ve been using so far have mainly been either
“integers” or “floats”. We saw that floats are usually a good
approximation to the mathematical concept of real numbers, but they are
only approximations and are susceptible to roundoff error. Calc also
supports “fractions”, which can exactly represent any rational number.
1: 3628800 2: 3628800 1: 518400:7 1: 518414:7 1: 7:518414
. 1: 49 . . .
.
10 ! 49 <RET> : 2 + &
The ‘:’ command divides two integers to get a fraction; ‘/’ would
normally divide integers to get a floating-point result. Notice we had
to type <RET> between the ‘49’ and the ‘:’ since the ‘:’ would otherwise
be interpreted as part of a fraction beginning with 49.
You can convert between floating-point and fractional format using ‘c
f’ and ‘c F’:
1: 1.35027217629e-5 1: 7:518414
. .
c f c F
The ‘c F’ command replaces a floating-point number with the
“simplest” fraction whose floating-point representation is the same, to
within the current precision.
1: 3.14159265359 1: 1146408:364913 1: 3.1416 1: 355:113
. . . .
P c F <DEL> p 5 <RET> P c F
(•) *Exercise 1.* A calculation has produced the result
1.26508260337. You suspect it is the square root of the product of ‘pi’
and some rational number. Is it? (Be sure to allow for roundoff
error!) 
1 Types Answer 1. (•)
“Complex numbers” can be stored in both rectangular and polar form.
1: -9 1: (0, 3) 1: (3; 90.) 1: (6; 90.) 1: (2.4495; 45.)
. . . . .
9 n Q c p 2 * Q
The square root of -9 is by default rendered in rectangular form
(‘0 + 3i’), but we can convert it to polar form (3 with a phase angle of
90 degrees). All the usual arithmetic and scientific operations are
defined on both types of complex numbers.
Another generalized kind of number is “infinity”. Infinity isn’t
really a number, but it can sometimes be treated like one. Calc uses
the symbol ‘inf’ to represent positive infinity, i.e., a value greater
than any real number. Naturally, you can also write ‘-inf’ for minus
infinity, a value less than any real number. The word ‘inf’ can only be
input using algebraic entry.
2: inf 2: -inf 2: -inf 2: -inf 1: nan
1: -17 1: -inf 1: -inf 1: inf .
. . . .
' inf <RET> 17 n * <RET> 72 + A +
Since infinity is infinitely large, multiplying it by any finite number
(like -17) has no effect, except that since -17 is negative, it changes
a plus infinity to a minus infinity. (“A huge positive number,
multiplied by -17, yields a huge negative number.”) Adding any finite
number to infinity also leaves it unchanged. Taking an absolute value
gives us plus infinity again. Finally, we add this plus infinity to the
minus infinity we had earlier. If you work it out, you might expect the
answer to be -72 for this. But the 72 has been completely lost next to
the infinities; by the time we compute ‘inf - inf’ the finite difference
between them, if any, is undetectable. So we say the result is
“indeterminate”, which Calc writes with the symbol ‘nan’ (for Not A
Number).
Dividing by zero is normally treated as an error, but you can get
Calc to write an answer in terms of infinity by pressing ‘m i’ to turn
on Infinite mode.
3: nan 2: nan 2: nan 2: nan 1: nan
2: 1 1: 1 / 0 1: uinf 1: uinf .
1: 0 . . .
.
1 <RET> 0 / m i U / 17 n * +
Dividing by zero normally is left unevaluated, but after ‘m i’ it
instead gives an infinite result. The answer is actually ‘uinf’,
“undirected infinity.” If you look at a graph of ‘1 / x’ around ‘x = 0’,
you’ll see that it goes toward plus infinity as you approach zero from
above, but toward minus infinity as you approach from below. Since we
said only ‘1 / 0’, Calc knows that the answer is infinite but not in
which direction. That’s what ‘uinf’ means. Notice that multiplying
‘uinf’ by a negative number still leaves plain ‘uinf’; there’s no point
in saying ‘-uinf’ because the sign of ‘uinf’ is unknown anyway.
Finally, we add ‘uinf’ to our ‘nan’, yielding ‘nan’ again. It’s easy to
see that, because ‘nan’ means “totally unknown” while ‘uinf’ means
“unknown sign but known to be infinite,” the more mysterious ‘nan’ wins
out when it is combined with ‘uinf’, or, for that matter, with anything
else.
(•) *Exercise 2.* Predict what Calc will answer for each of these
formulas: ‘inf / inf’, ‘exp(inf)’, ‘exp(-inf)’, ‘sqrt(-inf)’,
‘sqrt(uinf)’, ‘abs(uinf)’, ‘ln(0)’. 2 Types Answer 2. (•)
(•) *Exercise 3.* We saw that ‘inf - inf = nan’, which stands for an
unknown value. Can ‘nan’ stand for a complex number? Can it stand for
infinity? 3 Types Answer 3. (•)
“HMS forms” represent a value in terms of hours, minutes, and
seconds.
1: 2@ 30' 0" 1: 3@ 30' 0" 2: 3@ 30' 0" 1: 2.
. . 1: 1@ 45' 0." .
.
2@ 30' <RET> 1 + <RET> 2 / /
HMS forms can also be used to hold angles in degrees, minutes, and
seconds.
1: 0.5 1: 26.56505 1: 26@ 33' 54.18" 1: 0.44721
. . . .
0.5 I T c h S
First we convert the inverse tangent of 0.5 to degrees-minutes-seconds
form, then we take the sine of that angle. Note that the trigonometric
functions will accept HMS forms directly as input.
(•) *Exercise 4.* The Beatles’ _Abbey Road_ is 47 minutes and 26
seconds long, and contains 17 songs. What is the average length of a
song on _Abbey Road_? If the Extended Disco Version of _Abbey Road_
added 20 seconds to the length of each song, how long would the album
be? 4 Types Answer 4. (•)
A “date form” represents a date, or a date and time. Dates must be
entered using algebraic entry. Date forms are surrounded by ‘< >’
symbols; most standard formats for dates are recognized.
2: <Sun Jan 13, 1991> 1: 2.25
1: <6:00pm Thu Jan 10, 1991> .
.
' <13 Jan 1991>, <1/10/91, 6pm> <RET> -
In this example, we enter two dates, then subtract to find the number of
days between them. It is also possible to add an HMS form or a number
(of days) to a date form to get another date form.
1: <4:45:59pm Mon Jan 14, 1991> 1: <2:50:59am Thu Jan 17, 1991>
. .
t N 2 + 10@ 5' +
The ‘t N’ (“now”) command pushes the current date and time on the stack;
then we add two days, ten hours and five minutes to the date and time.
Other date-and-time related commands include ‘t J’, which does Julian
day conversions, ‘t W’, which finds the beginning of the week in which a
date form lies, and ‘t I’, which increments a date by one or several
months. Date Arithmetic, for more.
(•) *Exercise 5.* How many days until the next Friday the 13th?
5 Types Answer 5. (•)
(•) *Exercise 6.* How many leap years will there be between now and
the year 10001 AD? 6 Types Answer 6. (•)
An “error form” represents a mean value with an attached standard
deviation, or error estimate. Suppose our measurements indicate that a
certain telephone pole is about 30 meters away, with an estimated error
of 1 meter, and 8 meters tall, with an estimated error of 0.2 meters.
What is the slope of a line from here to the top of the pole, and what
is the equivalent angle in degrees?
1: 8 +/- 0.2 2: 8 +/- 0.2 1: 0.266 +/- 0.011 1: 14.93 +/- 0.594
. 1: 30 +/- 1 . .
.
8 p .2 <RET> 30 p 1 / I T
This means that the angle is about 15 degrees, and, assuming our
original error estimates were valid standard deviations, there is about
a 60% chance that the result is correct within 0.59 degrees.
(•) *Exercise 7.* The volume of a torus (a donut shape) is
‘2 pi^2 R r^2’ where ‘R’ is the radius of the circle that defines the
center of the tube and ‘r’ is the radius of the tube itself. Suppose
‘R’ is 20 cm and ‘r’ is 4 cm, each known to within 5 percent. What is
the volume and the relative uncertainty of the volume? 7 Types
Answer 7. (•)
An “interval form” represents a range of values. While an error form
is best for making statistical estimates, intervals give you exact
bounds on an answer. Suppose we additionally know that our telephone
pole is definitely between 28 and 31 meters away, and that it is between
7.7 and 8.1 meters tall.
1: [7.7 .. 8.1] 2: [7.7 .. 8.1] 1: [0.24 .. 0.28] 1: [13.9 .. 16.1]
. 1: [28 .. 31] . .
.
[ 7.7 .. 8.1 ] [ 28 .. 31 ] / I T
If our bounds were correct, then the angle to the top of the pole is
sure to lie in the range shown.
The square brackets around these intervals indicate that the
endpoints themselves are allowable values. In other words, the distance
to the telephone pole is between 28 and 31, _inclusive_. You can also
make an interval that is exclusive of its endpoints by writing
parentheses instead of square brackets. You can even make an interval
which is inclusive (“closed”) on one end and exclusive (“open”) on the
other.
1: [1 .. 10) 1: (0.1 .. 1] 2: (0.1 .. 1] 1: (0.2 .. 3)
. . 1: [2 .. 3) .
.
[ 1 .. 10 ) & [ 2 .. 3 ) *
The Calculator automatically keeps track of which end values should be
open and which should be closed. You can also make infinite or
semi-infinite intervals by using ‘-inf’ or ‘inf’ for one or both
endpoints.
(•) *Exercise 8.* What answer would you expect from ‘1 / (0 .. 10)’?
What about ‘1 / (-10 .. 0)’? What about ‘1 / [0 .. 10]’ (where the
interval actually includes zero)? What about ‘1 / (-10 .. 10)’?
8 Types Answer 8. (•)
(•) *Exercise 9.* Two easy ways of squaring a number are ‘<RET> *’
and ‘2 ^’. Normally these produce the same answer. Would you expect
this still to hold true for interval forms? If not, which of these will
result in a larger interval? 9 Types Answer 9. (•)
A “modulo form” is used for performing arithmetic modulo M. For
example, arithmetic involving time is generally done modulo 12 or 24
hours.
1: 17 mod 24 1: 3 mod 24 1: 21 mod 24 1: 9 mod 24
. . . .
17 M 24 <RET> 10 + n 5 /
In this last step, Calc has divided by 5 modulo 24; i.e., it has found a
new number which, when multiplied by 5 modulo 24, produces the original
number, 21. If M is prime and the divisor is not a multiple of M, it is
always possible to find such a number. For non-prime M like 24, it is
only sometimes possible.
1: 10 mod 24 1: 16 mod 24 1: 1000000... 1: 16
. . . .
10 M 24 <RET> 100 ^ 10 <RET> 100 ^ 24 %
These two calculations get the same answer, but the first one is much
more efficient because it avoids the huge intermediate value that arises
in the second one.
(•) *Exercise 10.* A theorem of Pierre de Fermat says that ‘x^(n-1)
mod n = 1’ if ‘n’ is a prime number and ‘x’ is an integer less than ‘n’.
If ‘n’ is _not_ a prime number, this will _not_ be true for most values
of ‘x’. Thus we can test informally if a number is prime by trying this
formula for several values of ‘x’. Use this test to tell whether the
following numbers are prime: 811749613, 15485863. 10 Types
Answer 10. (•)
It is possible to use HMS forms as parts of error forms, intervals,
modulo forms, or as the phase part of a polar complex number. For
example, the ‘calc-time’ command pushes the current time of day on the
stack as an HMS/modulo form.
1: 17@ 34' 45" mod 24@ 0' 0" 1: 6@ 22' 15" mod 24@ 0' 0"
. .
x time <RET> n
This calculation tells me it is six hours and 22 minutes until midnight.
(•) *Exercise 11.* A rule of thumb is that one year is about
‘pi * 10^7’ seconds. What time will it be that many seconds from right
now? 11 Types Answer 11. (•)
(•) *Exercise 12.* You are preparing to order packaging for the CD
release of the Extended Disco Version of _Abbey Road_. You are told
that the songs will actually be anywhere from 20 to 60 seconds longer
than the originals. One CD can hold about 75 minutes of music. Should
you order single or double packages? 12 Types Answer 12. (•)
Another kind of data the Calculator can manipulate is numbers with
“units”. This isn’t strictly a new data type; it’s simply an
application of algebraic expressions, where we use variables with
suggestive names like ‘cm’ and ‘in’ to represent units like centimeters
and inches.
1: 2 in 1: 5.08 cm 1: 0.027778 fath 1: 0.0508 m
. . . .
' 2in <RET> u c cm <RET> u c fath <RET> u b
We enter the quantity “2 inches” (actually an algebraic expression which
means two times the variable ‘in’), then we convert it first to
centimeters, then to fathoms, then finally to “base” units, which in
this case means meters.
1: 9 acre 1: 3 sqrt(acre) 1: 190.84 m 1: 190.84 m + 30 cm
. . . .
' 9 acre <RET> Q u s ' $+30 cm <RET>
1: 191.14 m 1: 36536.3046 m^2 1: 365363046 cm^2
. . .
u s 2 ^ u c cgs
Since units expressions are really just formulas, taking the square root
of ‘acre’ is undefined. After all, ‘acre’ might be an algebraic
variable that you will someday assign a value. We use the
“units-simplify” command to simplify the expression with variables being
interpreted as unit names.
In the final step, we have converted not to a particular unit, but to
a units system. The “cgs” system uses centimeters instead of meters as
its standard unit of length.
There is a wide variety of units defined in the Calculator.
1: 55 mph 1: 88.5139 kph 1: 88.5139 km / hr 1: 8.201407e-8 c
. . . .
' 55 mph <RET> u c kph <RET> u c km/hr <RET> u c c <RET>
We express a speed first in miles per hour, then in kilometers per hour,
then again using a slightly more explicit notation, then finally in
terms of fractions of the speed of light.
Temperature conversions are a bit more tricky. There are two ways to
interpret “20 degrees Fahrenheit”—it could mean an actual temperature,
or it could mean a change in temperature. For normal units there is no
difference, but temperature units have an offset as well as a scale
factor and so there must be two explicit commands for them.
1: 20 degF 1: 11.1111 degC 1: -6.666 degC
. . . .
' 20 degF <RET> u c degC <RET> U u t degC <RET>
First we convert a change of 20 degrees Fahrenheit into an equivalent
change in degrees Celsius (or Centigrade). Then, we convert the
absolute temperature 20 degrees Fahrenheit into Celsius.
For simple unit conversions, you can put a plain number on the stack.
Then ‘u c’ and ‘u t’ will prompt for both old and new units. When you
use this method, you’re responsible for remembering which numbers are in
which units:
1: 55 1: 88.5139 1: 8.201407e-8
. . .
55 u c mph <RET> kph <RET> u c km/hr <RET> c <RET>
To see a complete list of built-in units, type ‘u v’. Press
‘C-x * c’ again to re-enter the Calculator when you’re done looking at
the units table.
(•) *Exercise 13.* How many seconds are there really in a year?
13 Types Answer 13. (•)
(•) *Exercise 14.* Supercomputer designs are limited by the speed of
light (and of electricity, which is nearly as fast). Suppose a computer
has a 4.1 ns (nanosecond) clock cycle, and its cabinet is one meter
across. Is speed of light going to be a significant factor in its
design? 14 Types Answer 14. (•)
(•) *Exercise 15.* Sam the Slug normally travels about five yards in
an hour. He has obtained a supply of Power Pills; each Power Pill he
eats doubles his speed. How many Power Pills can he swallow and still
travel legally on most US highways? 15 Types Answer 15. (•)
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