# Mathomatic User's Manual

## Introduction

Mathomatic is a symbolic math interpreter that can:

- combine and solve algebraic equations,
- completely simplify and compare expressions,
- perform calculus transformations,
- help with series analysis,
- perform standard, complex number, and polynomial arithmetic,
- generate efficient C, Java, or Python code from equations, etc.

Mathomatic was originally written in Microsoft C for MS-DOS and released as shareware. It was made a better product under the GNU/Linux operating system, using only the GCC C compiler. The C source code is highly portable and will compile and run correctly under any operating system.

## Startup

SYNOPSIS
  mathomatic [ options ] [ input_files ]

To start the Mathomatic interpreter, type "mathomatic" at the shell prompt.

Color mode is toggled by the "-c" option. ANSI color mode outputs ANSI escape sequences to make each level of parentheses a different color, to improve readability.

The other options are described in the man page.

After any options, file names may be specified that will be read in with the read command.

## Equations and Expressions

Mathematical equations and expressions are entered into equation spaces. The number of available equation spaces is displayed every time Mathomatic starts up.

The Mathomatic prompt contains the number of the current equation space (origin 1).

To enter an equation into the first available equation space, simply type the equation at the prompt. Equations consist of a Left Hand Side (LHS) and a Right Hand Side (RHS). The equation sides are separated by one, and only one, equals sign (=). An equation side consists of an algebraic mathematical expression, which is a mix of variables, constants, and operators, mostly in infix notation. Parentheses are used to override operator precedence and group things together.

Shown below 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 variables b and c. b and c are called independent variables. In Mathomatic, any variable can be made the dependent variable by simply typing the variable name in at the prompt. This will solve the equation for that variable and, if successful, make that variable the LHS.

Here is the above equation entered into Mathomatic and solved for b:

    1-> a=b-(c+2)

    #1: a = b - c - 2

    1-> b

    #1: b = 2 + c + a

    1->

### Constants

Constants are IEEE 64-bit (8 bytes) double precision (14 decimal digits) floating point numbers. They may be entered in normal, scientific, or hexadecimal notation. They are displayed in decimal (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.

Constants always start with a digit (0..9) or a period. To enter a constant in hexadecimal, prepend it with "0x".

Examples of equivalent constants follow:

Normal Notation | Scientific Notation | Hexadecimal Notation
--- | --- | ---
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 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 0 produces infinity, not ±infinity which would be more correct.

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 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 converts it back to a fraction for display, after all floating point arithmetic has been done.

Irrational numbers, such as 2^(1/2), are preserved and simplified, if possible. This can be turned off with the command "set no preserve_roots", 2^(1/2) will then always be approximated as 1.4142135623731.

### Variables

Variables are what Mathomatic is all about. That is where the term "symbolic" comes from, because variables are symbolic in nature. They can represent known or unknown values, or any expression.

Variables consist of any combination of letters, digits, underscores (_), and backslashes (\). They never start with a digit.

The following variables are predefined and are not normal variables:

    e# - the universal constant e (2.718281828...).
    pi - the universal constant pi (3.1415926...).
    i# - imaginary number (square root of -1).
    sign, sign1, sign2, sign3, ... - may only be +1 or -1.
    integer - may be any integer.

By default, letters in variable names are case sensitive.

To automatically enter a unique sign variable, precede any expression with "+/-".

### Operators

Operators have precedence decreasing as indicated:

    - negate                               (unary prefix operator)
    ! factorial (gamma function)           (unary postfix operator)
    ** or ^ power (exponentiation)         (binary infix operator)
    * multiply    / divide      % modulus  (binary infix operators)
    + add         - subtract               (binary infix operators)
    = equate                               (signifies equivalence)

Operators in the same precedence level are evaluated left to right.

The negate operator (-) may precede any expression. The negate operator has the highest precedence of all operators. This is different from many math programs, where negate has been erroneously given the same precedence as times/divide. So things like -2^x will give the expected -2 to the power of x, and not -1*(2^x), which are completely different.

The default operator for variables and constants is multiply (*). If a variable or constant is entered when an operator is expected, a multiply operator is automatically inserted.

The modulo operator a % b gives the remainder of the division a / b.

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

Standard absolute value notation is allowed, |x| is converted to (x^2)^.5. This doesn't always work as expected, but works enough to be useful.

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

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

Mathomatic will first parenthesize the highest precedence operator (power) and then the lower precedence operators (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))

Mathomatic will evaluate this 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 then 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 can perform complex number addition, subtraction, multiplication, and division. It will also approximate roots and powers of complex numbers.

Complex numbers are in the form:

    a + b*i#

where a is the real part and b is the imaginary part. a and b are constants. i# represents the square root of -1 ("-1^.5" in Mathomatic notation).

The imaginary number i# may appear anywhere within an expression, as many times as you want. Mathomatic will handle 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, will be approximated, and only a single root will be produced, even though there may be many roots. This may produce inaccurate, unexpected, or even incorrect results, therefore a warning is displayed when this is done.

Conjugation of complex numbers is accomplished by typing the following command:

    replace i# with -i#

## Commands

Please consult the Mathomatic Command Reference, for detailed information on commands.

Copyright © 1987-2006 George Gesslein II

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