Arbitrary-precision Integer Calculators
Important Note: The software posted on this page is for
demonstration purposes only. Although the author is unaware of any
bugs, the software has undergone very little testing. In particular, the
algorithms performing aribitrary-precision arithmetic in the
LargeInteger class are quite complicated, and hence
Exhibit 1: An arbitrary-precision integer calculator using
Exhibit 2:An arbitrary-precision integer calculator using
Assuming the applets are running correctly, the two calculators above
will appear to be identical. If you browser won't run them, you may
want to download the program and run it as an application.
The two calculators are, in fact, functionally very
nearly the same. However, the calculator on the left uses an
arbitrary-precision integer arithmetic written by Rod Howell, the
author of this page, whereas the one on the right uses the
java.math.BigInteger library written by Josh Bloch and
Michael McCloskey of Sun Microsystems, Inc. They are posted here so
that you may compare their performance (see below for a user's
guide). I have found that computing 2200,000 is sufficient
to demonstrate the difference in performance. You may need to adjust
the value of the exponent depending on how fast your machine is.
So is this a fair comparison of the two arbitrary-precision integer
libraries? In a word, no. To see why, take a look at my comparison
of the two libraries.
We describe here the operation of the above applets. We then describe
how to run the program as an application.
Both calculators use RPN (Reverse Polish Notation) data entry. The
computational model is stack-based, with operations taking their
arguments from the top of the stack and replacing them with their
results. The value in the display is always the value at the top of
the stack. The size of the stack is bounded only by the available
Data may be entered using the ten buttons
labeled with digits, and/or the keyboard. Data may be copied from the
system clipboard using the "Paste" operation as defined by the system
look-and-feel (Ctrl-V on Windows). The text caret may be moved with
the mouse, the cursor arrow keys, and/or the "Home" and "End" keys.
"Backspace", "Delete", and/or the "Cut" operation (Ctrl-X on Windows)
may be used to correct typing errors. Valid characters depend upon
the value of the Base: if the base has value b, the first
b of the following characters are valid digits:
In addition, "-" may be inserted as the first character. Valid digits
may be inserted anywhere except to the left of a "-". Invalid
characters are ignored, no matter how they are entered. Upper-case
letters are automatically translated to their lower-case equivalents.
Operations Operations may be entered only by pressing a
button other than a digit or by selecting a different value in the
"Base" menu. Entering an operation terminates the entry of any data
to be used for that (or a subsequent) operation. Thus, whatever is in
the display at the time the operation is entered is at the top of the
stack if and when the operation attempts to acquire data from the
stack. If a single "-" has been entered, or if digits have been
entered but subsequently deleted, the value at the top of the stack is
taken to be 0. If the operation requires more operands than are
present on the stack, a "Stack Underflow" error occurs. Note that an
empty display may indicate one of two possibilities: either the stack
is empty (as occurs initially or after a "CS" operation has been
performed) or data has been entered and deleted, resulting in a 0.
Individual operations are as follows:
- Base menu - Selecting a new value causes any data in the display
to be converted to the new base. If the stack is empty, no error
is reported. In either case, subsequent data are interpreted
in this new base.
- CS - Empties the stack, including the display.
- Up - Navigates up the stack by removing the bottom
element and placing it on top, so that it appears in the display.
- Down - Navigates down the stack by removing the top
element and placing it on the bottom.
- Rem - Divides the second element by the top element and
replaces them by the remainder of the division.
- x^y - Raises the second element to the power of the top
element and replaces them by the result.
- 2^x - Raises 2 to the power of the top element and replaces
that element by the result.
- 10^x - Raises 10 to the power of the top element and replaces
that element by the result.
- / - Divides the second element by the top element and
replaces them by the result of the division.
- * - Multiplies the top two elements and replaces them by
- - - Subtracts the top element from the second element
and replaces them by the result.
- + - Adds the top two elements and replaces them by
- +/- - Replaces the top element by its negation.
- Pop - Removes the top element.
- Enter - If nothing has been input since the completion
of the last operation, a copy of the top element is placed on the
top of the stack; otherwise, the input is simply terminated (to
facilitate entry of a new data item).
After a result has been displayed, it
may not be edited directly. Any attempt to delete characters from it
is ignored. Any valid characters inserted will replace the entire
result in the display; the value will be retained as the second
element on the stack. Editing of computed results may be accomplished
by first selecting the entire result (Ctrl-A on Windows), then copying
and pasting it (Ctrl-C Ctrl-V on Windows). This has the effect of
inserting an editable copy into the display, above the original on the
Below the display, to the right of the Base
Menu, is a checkbox labeled, "Sticky". If this option is checked (the
default), values in the display will always be shown in the most
recently selected base. Otherwise, each value will be displayed in
the base in which it was originally entered or computed, and the Base
menu will automatically be updated to reflect the current base.
Note: In the calculator using the
java.math.BigInteger class, bases are always sticky;
unchecking this option has no effect.
When an error occurs, a message will be displayed
in a dialog box and the stack will be restored to its contents prior
to the offending operation. The following messages may occur:
- Stack Underflow - An operation required more data items
than were present on the stack.
- Invalid Operand - An attempt was made to divide by 0 or
use a negative exponent.
- Insufficient Memory for Operation - The Java Virtual
Machine does not have enough memory available to perform the
Any other messages probably indicate bugs in the program.
Example Given a Mersenne prime p
(i.e., a prime number of the form 2n-1), a
perfect number can be computed by p(p+1)/2. We can express
this in terms of n as
(2n-1)(2n-1). For example,
231-1 is a Mersenne prime. We may compute the
perfect number (231-1)(231-1) with
the following series of keystrokes:
31 [Enter] 1 [-] [2^x] [Enter] [Enter] [+] 1 [-] [*]
Thus, n = 31 is first pushed onto the stack, followed by
1. The difference is then computed (n-1). Then
2n-1 is computed, and two more copies are pushed
onto the stack; thus, the stack then contains three copies of
2n-1. The top two of these are added and replaced
by their sum, 2n. 1 is then pushed, then
2n is replaced by 2n-1.
Finally, 2n-1 and 2n-1 are
replaced by their product,
If you are patient and/or have a reasonably fast machine, you might
try computing the 37th even perfect number by using n =
Downloading and Running the Program
As long as you
have the JavaTM SE Runtime Environment (which can be
downloaded from the
Oracle® Software Downloads page) installed, there are two
ways to install the program:
- The simplest way to install the program on your machine is to
click one of the following links:
When running the program in this way, the program has restricted
access to the
system clipboard. In particular, it can no longer be accessed via the
standard keyboard shortcuts. In order to provide access to the system
clipboard, a popup
menu has been attached to the calculator's display. This popup menu
is accessed via the mouse in a way defined by the particular system
look-and-feel (right-click on most system). This
menu contains "Cut", "Copy", and "Paste" entries, which provide access
to the system clipboard in a controlled way. Specifically, when you
try to access the clipboard via one of these menu items, a warning
will be displayed asking you to confirm. You then have the
option to disable this warning, but it will appear on any program
execution the first time you try to write to the clipboard and the
first time you try to read the clipboard.
- If you wish to install the program manually, first download calculator.jar. If this file is downloaded,
it may be run simply by double-clicking on its icon in most operating
systems, or by
issuing the command
java -jar calculator.jar
on any operating system on which the JavaTM runtime is
When the program is run in this way, the user interface is slightly
different. An "Edit" menu with "Select All", "Cut", "Copy", and
"Paste" entries is present, and the "Sticky" checkbox is in a menu
called "Options". The purpose of the "Edit" menu is to provide a
uniform mechanism for accessing the system clipboard across
platforms. However, these menu items actually result in inconsistent
behavior on the various platforms. The interface described above is
To use the interface described above, or to use the calculator model
based on java.math.BigInteger, one or two command-line
arguments must be supplied to the java command given above.
java -jar calculator.jar ModelLocation
will run the calculator using the calculator model defined by
ModelLocation, which must be a package name terminated
by a period. The currently-implemented possibilities are:
- EDU.ksu.cis.calculator.defaultmodel. - An RPN calculator
using our LargeInteger class. This is the default.
- EDU.ksu.cis.calculator.javamodel. - An RPN calculator
using Sun's BigInteger class.
java -jar calculator.jar ModelLocation UIName
will run the calculator using the calculator model defined by
ModelLocation (described above) and the user interface
defined by UIName, which must be a Java class
The currently-implemented possibilities are:
- EDU.ksu.cis.calculator.DefaultUI - The user interface
with "Edit" and "Options" menus. This is the default.
- EDU.ksu.cis.calculator.JNLPCompatibleUI - The user
interface used by the applets above.
Please send any bug reports to Rod
Howell. Please include the following information:
- The operating system you are running under (e.g., Windows 98).
- Whether you were running the program as an applet or a
- If you were running the program as an applet, which browser you
were using (e.g., Internet Explorer).
- If you were running the program as a stand-alone application, the
version of Java you were using and any command-line arguments (if
applicable). You can obtain the version information by entering
from the command-line.
- A description of the erroneous behavior with enough details so
that I can reproduce it.
Copyright © Rod Howell, 2001, 2002. All rights reserved.
Last updated November 25, 2010.
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