# Mathomatic User's Manual

### Table of Contents

1. Introduction
2. History
3. Requirements
4. Startup
5. Equations and Expressions
   1. Constants
   2. Variables
   3. Operators
      1. Order of Operations example
   4. Complex Numbers
6. Commands

Mathomatic Command Reference

## Introduction

Mathomatic is an interactive symbolic-numeric math interpreter that can:

- combine and solve equations containing many variables
- completely simplify and compare mathematical expressions and equations
- do simple calculus transformations and series
- perform simultaneous real number, imaginary number, and polynomial arithmetic
- generate efficient C, Java, or Python language code from simplified equations
- carry out quick formula calculations, display expressions in 2D with color, etc.

It is a unique computer algebra system, in that all constants are one or more floating point values. All numerical arithmetic is floating point arithmetic, which most computers do very quickly.

Mathomatic is exceptionally good at solving, differentiating, and simplifying elementary algebra.

## History

Mathomatic development started in the year 1986, originally using the Microsoft C compiler for MS-DOS. Versions 1 and 2 were published by Dynacomp of Rochester, New York in 1987 as a scientific software product. It was later 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. It is now fully functional and available compiled and ready to run for Linux, Mac OS X, and Microsoft Windows.

## Requirements

Building 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.

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 200 megabytes (the version command tells this). Maximum memory usage is not reached unless all (100) equation spaces are filled. The default maximum memory usage should be less than the amount of free RAM, and is adjusted by changing the `DEFAULT_N_TOKENS` define in the C include file am.h and recompiling. Memory usage can also be adjusted at startup with the -m option.

Very little disk space (a few megabytes) is required to compile, install, and run Mathomatic. A readline library must be installed to use readline input editing and history.

## Startup

SYNOPSIS
  mathomatic [ options ] [ input_files ]
  rmath [ input_files ]

To start the interactive Mathomatic interpreter, run a terminal emulator which opens a shell window, and type "mathomatic" at the shell prompt. 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.

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

    $ mathomatic -c

ANSI color mode 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 (bold colors) instead, to enable color mode and increase the brightness.

The other options are described in the man page. After any options, text files may be specified on the shell command line that will be automatically read in with the read command.

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

    alias am=mathomatic

Then just typing "am" at the shell prompt will bring up Mathomatic. "am" stands for algebraic manipulator.

## Equations and Expressions

Algebraic equations and expressions are entered into equation spaces by typing, pasting, or reading them in. 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.

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. This current equation number can be changed by typing an existing equation number at the main prompt, or by entering another equation or expression.

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 (=). An equation side consists of an algebraic expression, which is a mix of constants, variables, and operators, mostly in standard 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 an 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 for x

    #1: x = 3

    1—>

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 only be calculated and displayed without storing, unless the autocalc or auto option is turned off. Non-equations cannot be solved.

### Constants

In Mathomatic, numerical arithmetic is double precision floating point with up to 14 decimal digits accuracy. Many results will be exact.

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 normal decimal (base 10) numbers or in scientific notation (also called exponential notation). Constants may also be entered using hexadecimal (base 16) notation by prepending it with "0x".

Constants are displayed in decimal (rounded to 14 digits) in 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 as double precision floats. The amount of round-off error is not tracked.

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

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 `strtod(3)` function. In the Unix shell, `"man strtod"` will do that.

- The largest value of a constant is ±1.7e+308.
- The smallest value of a constant is ±2.3e-308.

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 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 are usually rationalized in Mathomatic, so 1/(2^(1/2)) becomes the equivalent (2^(1/2))/2.

### 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" is entered beforehand.

The following variables are predefined and are not normal variables:

    e or e# - the universal constant e (2.718281828…)
    pi or pi# - the universal constant pi (3.1415926…)
    i or i# - the imaginary unit (square root of -1)
    sign, sign1, sign2, … - may only be +1 or -1
    integer - may only be an 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 (/), modulo division (%), 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 and have precedence decreasing as indicated:

    - negate (highest precedence)
    ! factorial (gamma function)
    ** or ^ power (exponentiation)
    * multiply      / divide        % modulus       // integral divide
    + add           − subtract
    = 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 negate operator (-x) may precede any expression and has the highest precedence of all operators, similar to the C programming language. This is different from other math programs, where negate has been given the same precedence as times and divide. So in Mathomatic, -2^x will give the expected (-2)^x, and not -1*(2^x), which are not equivalent expressions. This behavior is now configurable with the "set negate_highest_precedence" option.

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 modulo operator a % b (spoken as "a modulo b") gives the remainder of the division a / b. The sign of the result depends on the "set modulus_mode" option. Using the "integer" variable allows further simplification here.

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.

Factorials x! use the gamma function gamma(x+1), so that the factorial operator works with any real number, not just the positive integers.

Currently no rules of algebra are implemented for the integral divide operator (//) and factorials (!), and these two operators will not evaluate if an operand is a complex number.

Absolute value notation is allowed, |x| is 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 and then squared, 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.

#### 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 (negate, then 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 result is chosen.

Complex numbers are in the form:

    a + b*i#

where a is the real part (a real number) and b is the imaginary part (an imaginary number). i# represents the square root of -1 ("`-1^.5`" in Mathomatic notation), and may be entered into Mathomatic as "`i`"; it will be displayed as "`i#`".

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, therefore a warning is displayed when this is done.

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

    replace i with -i

## Commands

Mathomatic has about 40 simple English commands that may be typed at the main prompt. Please consult the Mathomatic Command Reference, for detailed information on all commands.

Copyright © 1987-2009 George Gesslein II

Up to the documentation index mathomatic.org