Author: George Gesslein II
Description: The User Guide for the Mathomatic computer algebra system.

# Mathomatic User Guide

### Table of Contents

1. Introduction and Features
2. History
3. Developer Information
4. Startup
5. Equations and Expressions
   1. Equations
   2. Non-Equations
   3. Constants
   4. Variables
   5. Operators
      1. Order of Operations example
   6. Complex Numbers
6. Commands
7. Documentation License

Mathomatic Command Reference

## Introduction and Features

Mathomatic is an easy to use and colorful algebra calculator that can symbolically:

- combine and solve equations containing many variables,
- completely simplify and compare mathematical expressions and equations,
- do simple calculus transformations and series, sum (∑) and product (∏),
- perform generalized real number, complex number, modular, and polynomial arithmetic,
- generate efficient C, Java, or Python language code from simplified equations,
- plot expressions with gnuplot in two or three dimensions, etc.

The name "Mathomatic" is a portmanteau of "math" and "automatic". It is a unique computer algebra system (CAS), in that all constants are one or more floating point values. All numeric arithmetic is IEEE standard floating point arithmetic, which most computers do very quickly. Mathomatic is written entirely in C, which is like a CAS written in assembly language, running as fast as the computer allows without any high-level language overhead.

Mathomatic is exceptionally good at solving, differentiating, simplifying, calculating, and visualizing elementary algebra. It is a console mode application using a color command-line interface (CLI) with pretty-print output that runs in a terminal emulator under any UNIX-like operating system. The console interface is very simple and requires little more than learning the basic algebra notation to get started. All input and output is line at a time ASCII text. By default, input is standard input and output is standard output. Mathomatic can be compiled with GNU readline for easier input. The pretty-print output is even prettier in HTML output mode, which supports color too.

## History

Mathomatic has been developed and supported almost every day by George Gesslein II, with help from the Internet community.

Mathomatic development started in the year 1986 to test new ideas and as an experiment in computerized mathematics, originally using the Microsoft C compiler for DOS and the author's own text editor as the only development tools. Versions 1 and 2 were published by Dynacomp of Rochester, New York in 1987 and 1988 as a scientific software product for DOS. Afterwards it was released as shareware and then emailware, with a 2D equation graphing program that could find asymptotes, written in Microsoft C for DOS. At the turn of the century, Mathomatic was ported to the GNU C Compiler (gcc) under Linux and became free and open source software by publishing under the GNU Lesser General Public License (LGPL version 2.1). The graphing program was discontinued; 2D/3D graphing of equations is now accomplished with gnuplot.

Mathomatic is currently developed and maintained on a Linux x86-64-bit computer and now stands at 23,000 lines of code (including comments). It has been optimized for speed, by using Linux debugging utilities that tell where Mathomatic spends most of its time.

## Developer Information

Building Mathomatic from source requires a C compiler with the standard C libraries. If compiled with the GCC C compiler for a Unix-like operating system, no changes need to be made to the source code. See the file README.txt for compilation instructions.

Mathomatic can easily be ported to any computer with at least 1 megabyte of free RAM. In the standard distribution, found on the Mathomatic home page, the maximum memory usage defaults to 400 megabytes (the "version status" command tells this). Maximum memory usage is not reached unless all equation spaces are filled. The default maximum memory usage should be less than the amount of free RAM, and can be easily reduced by compiling with the "-DHANDHELD" compiler command line option, which removes the extra code just for desktops and reduces the memory requirements to 1/6th of the default size. Memory usage can also be dynamically changed at startup with the -m option.

The Mathomatic source code can also be compiled as a symbolic math library that is callable from any C compatible program. See the file lib/README.txt for more developer information and how to include Mathomatic in your program.

Very little disk space (a few megabytes) is required to compile, install, and run the Mathomatic application. A readline library must be installed to compile-in and use readline (package called "libreadline-dev"), which allows editing and history recall of all Mathomatic line input by pressing the cursor keys.

## Startup

SYNOPSIS
  mathomatic [ options ] [ input_files or input ]
  rmath [ input_files ]

To start the compiled, interactive Mathomatic application, run a terminal emulator which opens a shell window, and type "mathomatic" at the shell prompt (without double quotes). If m4 (macro) Mathomatic was installed, you may type "rmath" instead, to use Mathomatic with input of functions like `sin(x)` and `sqrt(x)` allowed and automatically expanded to equivalent algebraic expressions. Logarithm function input is currently not available, because the logarithm function has not yet been implemented in the Mathomatic symbolic math engine.

If you are wondering what to try first in Mathomatic, type "help examples" at the Mathomatic prompt.

Color mode is toggled by the -c option on the shell command line, like this:

    $ mathomatic -c

ANSI color mode is the default, which outputs ANSI terminal escape sequences to make each level of parentheses a different color, improving readability. If ANSI color mode is on, an ANSI compatible terminal emulator is required. If the colors are hard to see, use the -b option instead, which will always turn on bold color mode, increasing the color brightness.

The other options are described in the Unix/Linux man page for Mathomatic. After any options, text files may be specified on the shell command line that will be automatically read in with the read command, unless the -e option is specified, in which case mathematical expressions and Mathomatic commands are expected, separated by unquoted space characters.

It is recommended that the name mathomatic be shortened to am and e for quicker and easier access from the shell command line. This can be done in the Bash shell by adding the following two lines to your ~/.bashrc file:

    alias am=mathomatic
    alias e="mathomatic -e --"

Then just typing "am" at the shell prompt will run Mathomatic as an interactive application. Typing "e" followed by a quoted mathematical expression at the shell prompt will quickly and silently bring up Mathomatic and calculate and display the result. "am" stands for "algebraic manipulator", and "e" stands for "evaluate".

## Equations and Expressions

On the World Wide Web are some high quality pages with basic algebra definitions.

Mathematical equations and expressions are entered into Mathomatic equation spaces by typing, pasting, or reading them in at the main prompt. The maximum number and size of available equation spaces is displayed every time Mathomatic starts up. When an expression grows larger than half the equation space size, processing stops and the "Expression too large" message is displayed, returning you to the main prompt. The reason it does this is because each equation space consists of two equation sides, which are fixed size arrays that are easily manipulated by the software.

Each equation space is successively numbered with an equation number (starting at 1). The main prompt "`1—>` " contains the equation number of the current equation space. The current equation can be changed by typing a valid equation number at the main prompt (called selecting an equation space), or by entering another equation or expression, which becomes the current equation.

Any previously entered expression can be automatically entered again by entering a "#" followed by the relative or absolute equation space number of that expression. If this is entered first thing at the main prompt, it means something entirely different, that you are selecting that equation space; Otherwise, the RHS or expression at that equation space is substituted, for your convenience.

### Equations

To enter an equation into the first available equation space and make it the current equation, simply type it in at the main prompt. Each equation space consists of two equation sides, called the Left-Hand Side (LHS) and the Right-Hand Side (RHS), separated by an equals sign (=). Each equation side consists of a mathematical expression, which is a mix of constants, variables, and operators, mostly in standard algebraic infix notation. Parentheses are used to override operator precedence and group things together. Valid parentheses characters are () and {}.

Note that the equals sign does not make an assignment to any variables, it only signifies equality (sameness) between the results of evaluating the LHS and RHS. Shown here is a valid equation with its parts labeled:

           equation
    -----------------------
    | variables   constant|
    |--------------     | |
    ||     |      |     | |
     a  =  b  -  (c  +  2)
    | |   |   |      |    |
    | |   |   --------    |
    | |   |   operators   |
    ---   -----------------
    LHS          RHS

In the above equation, the variable a is called the dependent variable because its value depends on the independent variables b and c. In Mathomatic, any variable can be made the dependent variable by simply typing the variable name in at the prompt. This will solve the current equation for that variable and, if successful, make that variable the LHS.

Here is the above equation entered into Mathomatic and solved for b, then calculated for the values a=1 and c=1:

    1—> a=b-(c+2)

    #1: a = b − c − 2

    1—> b

    #1: b = 2 + c + a

    1—> calculate
    Enter a: 1
    Enter c: 1

     b = 4

    1—>

The "`#1:`" listed in front of each displayed equation always indicates the equation space number it is stored in.

Mathomatic automatically does both symbolic and numeric mathematics computations during any manipulations. This means that it can handle algebraic formulas, as well as numbers. What follows is a simple example of the result of both types of computations working together during equation simplification and solving:

    1—> 3*(x-1) + 1 = 2x + 1

    #1: (3·(x − 1)) + 1 = (2·x) + 1

    1—> simplify

    #1: (3·x) − 2 = (2·x) + 1

    1—> solve verify x

    #1: x = 3

    Solution verified.
    1—>

The "solve verify" command, used above, solves the current equation and then verifies the result by plugging the result into the original equation and simplifying. If an identity results (the LHS is identical to the RHS), a "Solution verified" message is displayed, otherwise "Solution might be incorrect" is displayed.

### Expressions that are not equations

Non-equations, that is any mathematical expression without an equals sign, may be entered into equation spaces too. However, if the expression entered at the main prompt contains no variables, it will be calculated and displayed with the calculate command, unless the autocalc or auto option is turned off.

Non-equations cannot be solved, unless you set them equal to zero or something, then they become an equation. Non-equations are stored in the first equation side (LHS) for that equation space. The RHS will be empty (size 0).

### Constants

In Mathomatic, numeric arithmetic is double precision floating point with about 14 decimal digits accuracy. Many results will be exact, because symbolic math is an exact math, and because multiple floating point numbers can be combined for a single mathematical value; for example: 2^(1/3), which is the cube root of 2 exactly.

Constants are approximated real numbers stored internally as IEEE 754 standard 64-bit (8 bytes) double precision floating point values. They may be entered as decimal (base 10) numbers in normal notation or in scientific notation (also called exponential notation). Constants may also be entered in hexadecimal (base 16) by starting them with "0x".

Constants are displayed in decimal (base 10, rounded to 14 digits) using either normal or scientific notation, whichever is shortest. Results are usually accurate from 12 to 14 digits, due to accumulated round-off error, because all constants are stored internally as double precision (rounded to 15 decimal digits) floats. And the amount of round-off error is not tracked, making Mathomatic unsuitable for applications requiring high precision, like astronomical calculations.

Excepting constants with a name (like "inf" for the infinity constant), constants always start with a decimal digit (0..9), a period, or a dash (-).

Examples of equivalent constants follow:

Normal Notation (base 10) | Scientific Notation (base 10) | Hexadecimal Notation (base 16)
--- | --- | ---
10 | 1e1 (1.0 times 101) | 0xa
.125 | 1.25e-1 (1.25 times 10-1) | 0x.2
255 | 2.55e2 (2.55 times 102) | 0xff

The exact syntax to enter constants as above may be found by looking up the C library function `strtod(3)`. In the Unix shell, `"man strtod"` will do that.

Double precision floating point limits:

- The largest valid constant is ±1.797693e+308 (slightly less than 2^1024).
- The smallest valid constant is ±2.225074e-308 or 0.

The infinity constant is entered by typing "inf". Positive and negative infinity are distinct and understood, however division by zero produces one infinity value, not the two-valued ±infinity which would be more correct. Also, floating point overflow produces either positive or negative infinity.

    1—> 1/0
    Warning: Division by zero.

     answer = inf

    1—> 0/0
    Warning: Division by zero.

     answer = nan

    1—>

nan or NaN stands for Not a Number and it means an invalid or indeterminate floating point arithmetic result. NaN cannot be directly entered into Mathomatic. The appearance of the constant NaN in an expression means the expression is unusable.

Fractions (such as 100/101) are preserved if the numerator and denominator are not large. Fractions are always presented in fully reduced form; for example, 6/9 is converted to the irreducible fraction 2/3. Constants which are exactly equal to a fraction are converted and displayed as fully reduced fractions; for example, 0.5 converts to 1/2. Mathomatic internally converts a fraction to a single floating point value, then may convert it back to a fraction for display after all floating point arithmetic has been done, if the result is equal to a fraction.

Irrational numbers, such as the square root of two (2^(1/2)) and pi, are preserved and simplified for exactness, unless explicitly approximated.

Denominators of fractions are usually rationalized in Mathomatic; for example, 1/(2^(1/2)) becomes the equivalent (2^(1/2))/2 upon simplification. This can be turned off with the command "set no rationalize_denominators".

### Variables

Variables are what Mathomatic is all about. That is where the term "symbolic" comes from, because variables are symbolic in nature. They are symbols that can represent known or unknown values, or any expression. Variables need not be defined in Mathomatic, just entering the variable name is enough.

Variable names consist of any combination of letters (a..z), digits (0..9), and underscores (_). They never start with a digit. By using the "set special_variable_characters" command, you can add to the allowed variable name characters. By default, letters in variable names are case sensitive, meaning the alphabetic case of each letter in the variable name is important. For example, variables named "A1" and "a1" represent two different variables, unless "set no case_sensitive" is entered beforehand.

The following variables are predefined and are not normal variables:

e, ê, or e# - the universal constant e (2.718281828…)
pi, π, or pi# - the universal constant pi (3.1415926…)
i, î, or i# - the imaginary unit (square root of -1)
sign, sign1, sign2, … - may only be +1 or -1
integer, integer1, … - may be any integer value

The above can be used anywhere variables are required.

To automatically enter multiplication by a unique, two-valued "sign" variable, precede any expression with "+/−".

### Operators

Mathomatic implements the standard rules of algebra for addition (+), subtraction (−), multiplication (*), division (/), modulus (%), and all forms of exponentiation (^ or **). An example of a rule of algebra is 2*x + 3*x being simplified to 5*x.

All available operators are at least numerically capable (most are symbolically capable too, see above paragraph) and have precedence decreasing as indicated:

    ! factorial   (same as gamma(x+1) function; highest precedence)
    ** or ^ power (exponentiation; high precedence)
    * multiply      / divide        % modulus       // integral divide
    + add           − subtract or negate
    = equate      (lowest precedence)

Higher precedence operators are grouped (or evaluated) first, then multiple operators of the same precedence level are grouped left to right. This is called the "order of operations".

The default operator is multiply (*). If an expression (operand) is entered when an operator is expected, a multiply operator is automatically inserted. For example, entering 2x, 2(x), and (2)(x) all result in the expression 2*x.

The modulus operator (a % b) (spoken as "a modulo b") gives the remainder of the division (a / b), which allows modular arithmetic; also known as clock arithmetic, because the numbers wrap around, after they reach the modulus. Mathomatic can simplify, calculate, and sometimes even solve modular arithmetic. Using "integer" variables allows further modulus simplification when using the simplify command. An integer variable is specified by using a variable name that starts with "integer", like "integer1", "integer_x", etc. The resulting sign of any numerical modulus operation depends upon the "set modulus_mode" option.

The integral divide operator (a // b) divides a by b and then truncates by zeroing the fractional part to make the result an integer. For example, (8 // 3) results in 2, which is useful when doing integer arithmetic. This operator currently implements no rules of algebra, and will not evaluate if an operand is a complex number.

Factorials x! use the gamma function gamma(x+1), so that the factorial operator works with any real number, not just the positive integers. The factorial operator currently implements no rules of algebra, and will not evaluate for complex numbers or if an overflow happens.

Absolute value notation is allowed, |x| is always converted to (x^2)^.5. This is not the same as standard absolute value where the real and imaginary parts of complex numbers are separated out and then plugged into the distance formula (the Pythagorean theorem), returning a real distance value, but it works the same when given real number values with no imaginary units. The absolute value operation |x| results in a positive value for any x value; that is, if -1 is a factor, it is removed. With Mathomatic, if x is imaginary, you may get an unwanted imaginary result, but it will be positive!

#### Order of Operations example

The following example shows why operator precedence is important. Given the numerical expression:

    64/(-2)^4 + 6*(3+1)

Mathomatic will parenthesize the highest precedence operators first: power, then times and divide. Addition and subtraction are the lowest precedence, so no need to parenthesize them. The result will be:

    (64/((-2)^4)) + (6*(3+1))

This is evaluated by combining constants from left to right on the same level of parentheses, deepest levels first. So the calculations are performed in the following order:

    (64/16) + (6*4)   Combine deepest level parentheses first.
    4 + 24            Divided 64 by 16 and multiplied 6 by 4.
    28                Added 24 to 4.

If the calculations were performed in a different order, the result would be different.

### Complex Numbers

Mathomatic automatically performs complex number addition, subtraction, multiplication, division, and exponentiation. It can also approximate roots of real and complex numbers, giving a single result; when multiple results are possible, the first real number result is chosen. To view all roots of a complex number, use the roots command.

Complex numbers are usually of the form:

    a + b*i

where a is the real part (a real number) and b is the imaginary part (an imaginary number). i is the "imaginary unit" and it represents the square root of -1 ("(-1)^.5" in Mathomatic notation; additionally "sqrt(-1)" in m4 Mathomatic notation).

    a + b*i

is the rectangular coordinate form of a complex number. To view the polar coordinate form, use the roots command.

The imaginary unit i may appear anywhere within an expression, as many times as you want, Mathomatic will handle and simplify it properly.

As an example of imaginary numbers being produced, (-2)^.5 will be converted to (2^.5)*i.

Roots of complex numbers, such as i^.5 and .5^i, will be approximated, and only a single root will be produced, even though there may be many roots (see the roots command). That single root is called the "principal value", which may be unexpected and will often be inexact.

Because 1/i correctly simplifies to -i, after significant manipulation different numerical results might be produced with complex numbers and division. This is normal, so the sooner the result of a complex expression is calculated, the better.

Conjugation of all complex numbers in the current equation is accomplished by typing the following command:

    replace i with -i

## Commands

Mathomatic has about 43 simple English commands that may be typed at the main prompt. Please consult the Mathomatic Command Reference, for detailed information on all commands.

## Documentation License

Mathomatic documentation copyright © 1987-2012 George Gesslein II

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included here in the Mathomatic documentation directory.

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