__The Multiple-Algebraic Calculator__

__Created
By____
David@TechForText.com,
©2010__

__To contact the author use the
above email address, or__

__left click on these words for a website
communication form.__

__This website
provides a free calculation device, called the ____Multiple-Algebraic Calculator,
available in the JavaScript and spreadsheet formats.__ This Calculator demonstrates
that a number of complex calculations (involving 24 unknowns, calculus, and
trigonometry) can be carried out simultaneously, by devices created with
specialized design concepts in the JavaScript or spreadsheet formats. The
website also provides over 4500 words of interesting and useful information for
advanced spreadsheet users, and for individuals that create software-based
calculation devices. This includes a number of specialized techniques and
concepts that were used to create the __Multiple-Algebraic
Calculator.__

** **

The JavaScript version of the
__Multiple-Algebraic Calculator__ functions online and it is embedded in this
webpage, four paragraphs below, after the instructions. If you want the
spreadsheet version, or additional information, scroll all the way down beneath
the __online Calculator and the table of contents.__ Alternatively, you can go directly to the hyperlink table of
contents of this website, by left clicking on these words.

__Instructions for the
____Multiple-Algebraic
Calculator__

When the user enters 12 numbers, and left clicks with the mouse, the Multiple-Algebraic Calculator initially solves: AX+BY+KZ=M, DX+EY+FZ=N,GX+HY+JZ=W for X, Y and Z. With the calculated values for X, Y, and Z, and the numbers entered by the user, additional sets of calculations are automatically carried out involving six double integrals, ratios, algebra and trigonometry.

To use the Multiple-Algebraic Calculator enter TWELVE numbers in the twelve white input boxes. (To enter or delete numbers, left click with the mouse on the relevant white input box.) For calculated results left click with the mouse on the background. Alternatively, with the online JavaScript version you can press the calculation button, and with the spreadsheet version you can press the enter key. Numbers and words displayed in red type are calculated results.

**Note:** __If less than 12 numbers are
entered the Multiple-Algebraic Calculator may not function properly. The
Calculator has a number of error-checking devices that may display error
messages while entering numbers, which may continue until all 12 numbers have
been entered.__

__If you
entered a set of numbers that do not satisfy all of the equations in the
Multiple-Algebraic Calculator, you may get error messages that relate to one or
more specific equations.__

The Multiple-Algebraic Calculator is equivalent to a 15 page document. Thus, if you want to see all the equations, double integrals, trigonometry and calculated results you must scroll through 15 pages.

__Online Multiple-Algebraic
Calculator Embedded in the Webpage__

__THE HYPERLINK TABLE OF CONTENTS OF
THIS WEBSITE__

__Left click with the
mouse, on the upper portion of the blue__

__words that relate to
the topic or subtopic you want to read.__

** **

__The
Multiple-Algebraic Calculator
1__

^{^Topic^ (This
is the Top of the Webpage)
1}

Instructions for the Multiple-Algebraic Calculator. 2

Online Multiple-Algebraic Calculator Embedded in the Webpage. 5

The Multiple-Algebraic Calculator, 12

And Links to Obtain the Free Downloads. 12

The Software You Need for the. 12

Multiple-Algebraic Calculator. 12

Multiple-Algebraic Calculator In the 13

Microsoft Excel, OpenOffice Calc, and Javascript Formats. 13

The Multiple-Algebraic Calculator 16

The Primary Purpose of The. 16

Multiple-Algebraic Calculator. 16

The Calculations of the Multiple-Algebraic Calculator. 17

Creating the Multiple-Algebraic Calculator, and. 21

Useful Concepts for Advanced Spreadsheet Users. 21

Creating the Multiple-Algebraic Calculator, 22

In the Spreadsheet and JavaScript Formats. 22

The Formulas, Symbolic Logic, Formatting Code, 23

Notation, and other Aspects of Spreadsheets, can be. 23

Conceptualized as a Higher-Level Computer Language. 23

The Spreadsheet Formulas for. 30

The Multiple-Algebraic Calculator, and 30

Useful Techniques for Advanced Spreadsheet Users. 30

Converting Conventional Formulas to Spreadsheet Formulas. 31

By Replacing Letters in the Formula with Cell Designations. 31

Converting Conventional Formulas 32

To Spreadsheet Formulas by Renaming 32

the Relevant Cells, with the Letters in the Formula. 32

The following Example will Clarify the Two Methods. 33

Described Above. (Converting Formulas with. 33

Cell Designations, or Renaming Cells) 33

Specialized Design Concepts. 35

For Calculation Devices That Perform 35

Complex Multiple Calculations Simultaneously. 35

Creating a Calculation Device That Performs. 35

Multiple Calculations in a Chain Like Sequence. 35

How Are Connections Between Formulas Created?. 36

A Number of Spreadsheet Formulas 40

Connected to the Same Data Source 40

Spreadsheet Formula Connections Resemble the. 42

Series and Parallel Connections used in Electronics. 42

Services Offered by the Author 46

Services Offered by the Author David@TechForText.com... 48

__The Multiple-Algebraic
Calculator,__

__And
Links____ to Obtain the Free
Downloads__

The spreadsheet versions of the
__Multiple-Algebraic Calculator__ require either Microsoft Excel, or the free
OpenOffice.org software package, (available from www.OpenOffice.org). In addition,
Microsoft Windows is required for the spreadsheet versions.

The JavaScript version of the Multiple-Algebraic Calculator can run on any operating system that has JavaScript support, but I only tested it with Microsoft Windows. In addition, the JavaScript version requires a browser that supports JavaScript. Almost all modern operating systems and browsers support JavaScript.

__Multiple-Algebraic Calculator In
the____
__

__Microsoft Excel, OpenOffice Calc,
and Javascript Formats__

**If you want the Multiple-Algebraic
Calculator in the Microsoft Excel format, left click on these
words.**

__Note: If you do not have Microsoft Excel,
download the OpenOffice.org software package, because it is free and it is
almost as good as the Microsoft Office suite. To download go to www.OpenOffice.org or left click
on these words.__** **

**If you want a diverse assortment of
calculation devices for algebra and trigonometry, go to the main website at
www.TechForText.com/Algebra****
****This can be done by left clicking on
these words. **

__The Multiple-Algebraic
Calculator__

The primary purpose of the Multiple-Algebraic Calculator is to demonstrate that calculation devices can be designed in the JavaScript or spreadsheet formats that perform a relatively large number of complex calculations *simultaneously, based on one set of numbers entered by the user. Specialized design concepts were used to create this functionality, which is discussed in the following two topics.

*The Multiple-Algebraic Calculator calculates most of the results sequentially, but from the perception of the user it appears simultaneous. This actually involves a number of calculations, carried out in a chain like sequence, in a fraction of a second, where one mathematical result is used to calculate another. For example, the Multiple-Algebraic Calculator calculates the value of Z first, which is required to calculate the value for Y. Then with the values of Z and Y it calculates the value of X.

__The
Calculations of the Multiple-Algebraic Calculator__

When the user enters 12 numbers, and left clicks with the mouse, the Multiple-Algebraic Calculator solves 24 unknowns, and carries out over 75 calculations, involving algebra, calculus, and trigonometry. This includes the following nine sets of calculations.

1) The Multiple-Algebraic Calculator initially solves the following equations: AX+BY+KZ=M, DX+EY+FZ=N, GX+HY+JZ=W. This involves solving first for, Z, then Y, followed by X. With the calculated results for X, Y, and Z and the numbers entered by the user the Multiple-Algebraic Calculator automatically carries out the sets of calculations presented below.

2) It carries out a set of calculations to check the calculated values for X, Y, and Z. This determines if the numbers the user entered satisfies the three equations that were presented above.

3) It performs calculations involving six double integrals

4) It converts the initial calculations involving the three equations (AX+BY+KZ=M, DX+EY+FZ=N, GX+HY+JZ=W) into a ratio problem, with only one unknown. To do this the Calculator solves two equations X*V=Y and X*P=Z for V and P. (Keep in mind that the values for Y and Z have been calculated.) This involves replacing X*V for Y, and X*P for Z in the above equations.

5) It carries out a set of error-checking calculations for the above.

6) The Multiple-Algebraic
Calculator solves 11 algebraic equations, (besides the equations and
calculations previously mentioned) for the following unknowns: S, T, Q, U, Aa,
Ba, Ca, Da, Ea, Fa, Ga, and Ha (**Note, I use a
capital letter and a lowercase letter, to represent one unknown, such as
Ba. This should not be confused with B multiplied by a.**)

7) The Calculator carries out a set of error-checking calculations for the above.

8) The Multiple-Algebraic Calculator solves eight equations that involve trigonometry, which involve the following unknowns: Angle_A, Angle_B, Ja, Ka, La, Ma, Na, and Pa.

9) The Calculator performs a series of error-checking calculations for the above.

__Creating the
____Multiple-Algebraic
Calculator, and__

__Useful Concepts for
Advanced Spreadsheet Users__

Note the concepts discussed in the following paragraphs applies to Microsoft Excel, OpenOffice Calc, and other advanced spreadsheet software. There are brands of less complex spreadsheet software available, which may not have the versatility and functionality that are discussed in the following paragraphs.

__Creating the
____Multiple-Algebraic
Calculator,__

__In
the Spreadsheet and JavaScript Formats__

The __Multiple-Algebraic
Calculator__
was initially created
in the spreadsheet format. Then it was converted to JavaScript using
specialized software. The primary advantage of the JavaScript format is it
functions online directly from a website, in any browser and operating system
that has JavaScript functionality.

The spreadsheet version does not automatically open in a browser, and it requires Microsoft Excel, or OpenOffice Calc. However, the primary advantage of the spreadsheet version is mathematical calculations can be saved. With JavaScript calculation devices, any numbers or text entered by the user is not saved. That is when the browser is closed, all the data entered by the user disappears.

__The
Formulas, Symbolic Logic, Formatting Code,__

__Notation, and other Aspects of
Spreadsheets, can be____ __

__Conceptualized as a Higher-Level
Computer Language.__

A concept that can be useful for developing creative skills with spreadsheet software is to realize you are dealing with a computer language that is based on mathematics and *symbolic logic. I will call this the spreadsheet computer language. If you are entering a few formulas, from the toolbar, or creating a conventional spreadsheet, the concept of a computer language is not apparent, and not even relevant. However, when creating complex calculation devices, with sets of formulas that interact with each other in complex ways, it is helpful to understand the mathematics, logic and notation of spreadsheets as a computer language.

*The symbolic logic consists of logical statements written in an abbreviated form with spreadsheet functions, such as =IF( ) =AND(), =OR(). This also includes any statement that can be defined as true or false, such as =A=B, =C<D, and =P>G. The symbolic logic can involve long statements involving a number of spreadsheet functions, coupled with numbers and text. This resembles computer code.

Formulas also often look like computer code in complex calculation devices. This is especially the case when formulas are lengthy, or when many formulas are connected in complex configurations, to obtain a calculated result.

The formatting code is also essentially a type of computer code. (This is probably only familiar to very advanced spreadsheet users.) The formatting code controls how numbers and letters are interpreted and displayed on a spreadsheet. Numbers can be interpreted and displayed on a spreadsheet as a day, a month, a specific date, a time, a conventional number, scientific notation, a percentage, a fraction, as dollars, etc. For example, 1 can mean any of the following depending on the formatting code”

Sunday (the code is dddd)

January, (the code is mmmm)

January 1, 1900, (the code is mmmm d, YYY)

100% (the code is 0.00%)

$1.00 (the code is $#,##0.00)

1 (the code is General).

The spreadsheet computer language provides ways to create connections between input cells (where the user enters numbers), output cells (which display calculated results), and formulas. When using spreadsheet software in a conventional way, most of us probably do not think of creating connections between any of the above. However, when creating complex calculation devices, it is necessary to devise precise connections between input and output cells and formulas. This often involves very complex configurations, involving hundreds of connections, from input cells, to formulas, which may transmit their calculated results through connections to many other formulas. This is the case with the Multiple-Algebraic Calculator.

I have not come across any sources that called the above a computer a language. However, the notation, symbolic logic, the compact nature of formulas and formatting code, and a number of other aspects associated with spreadsheets, fit the definition of a higher-level computer language. It also provides the versatility of a computer language, which is probably not apparent to most people that use spreadsheets.

Incidentally, Visual Basic, in Microsoft Excel, is generally referred to as a computer language, and is useful for creating macros. Visual Basic is very different than the spreadsheet computer language that I discussed above. However, the spreadsheet computer language can be converted to Visual Basic, but it can also be converted to JavaScript, and many other computer languages. Just like with the languages used by humans, translation is not always perfect. However, usually statements from one language can be translated to another, with both human and computer languages.

The translation from the spreadsheet language to another computer language can be done manually or in many cases electronically, using specialized software. Manual translation may not be feasible in many cases. For example, the JavaScript version of the Multiple-Algebraic Calculator has over 4,000 lines of computer code, consisting of over 240,000 characters. When placed in a Microsoft Word document this code required over 95 pages. It would probably take an experienced JavaScript programmer many weeks if not months to translate the spreadsheet version of the Multiple-Algebraic Calculator to JavaScript manually. This would also require a programmer with expertise in algebra, trigonometry and calculus.

Electronic translation of a calculation device created in the spreadsheet format to another computer language usually takes the computer about a minute or two. (The Multiple-Algebraic Calculator was converted to JavaScript in less than two minutes.) However, computer code obtained from electronic translation usually requires a little editing to improve the appearance or functionality of a calculation device.

__The
Multiple-Algebraic Calculator, and____ __

__Useful Techniques for Advanced
Spreadsheet Users__

Most of the spreadsheet formulas I used to create the multiple algebraic calculator, I created by solving conventional equations, for the variable I needed. This essentially results in a conventional formula, which cannot be used in spreadsheets. I then converted these formulas into spreadsheet formulas by making a few modifications. This included using an asterisk (*) for multiplication, slashes (/) for division, and a carrot (^) with an appropriate number for square roots, cube roots, squares, etc. However, the resulting formulas required further modification, because they contained letters that spreadsheet software CANNOT identify, in mathematical terms.

__Converting Conventional Formulas
to Spreadsheet Formulas__

__By
Replacing Letters in the Formula with Cell Designations__

There are two basic ways to make
the letters in a conventional formula meaningful to spreadsheet software.
One method involves the use of cell designations, __which are the default names
of spreadsheet cells, based on the way I am using the terminology__.
Three examples of cell designations are, A5, B3, and G9. Cell designations
are also called cell references.

The basic technique of converting a conventional formula to a spreadsheet formula is to replace the letters in the formula with relevant cell designations. (This must be coupled with the modifications discussed in the previous subtopic.) The relevant cells are usually the input cells where the user enters numbers for the formula. Sometimes relevant cells are cells that contain calculated results from other formulas.

__Converting Conventional
Formulas__

__To
Spreadsheet Formulas by Renaming__

__the
Relevant Cells, with the Letters in the Formula.__

Conventional formulas that have been rewritten with cell designations can be confusing to work with, and even more confusing when they are presented to other people for study. The alternative to this technique is to define the letters in a formula in terms of cell designations. This essentially involves renaming the cells on the spreadsheet to match the letters in the formula. There is a mechanism in Microsoft Word, and OpenOffice Calc that provides this functionality. With this technique the letters in the formula are not changed.

__The
following Example will Clarify the Two Methods____ __

__Described Above. (Converting
Formulas with__

__Cell Designations, or Renaming
Cells)__

For illustration purposes I will use a simple formula for area, which is length multiplied by width equals the area. This formula can be represented as L*W=A. Now if we want to convert this into a spreadsheet formula, we must use two input cells, one for the user to enter the length and the other for the width. For this example let us assume we are using cell B3 for length, and cell B4 for width. With the first technique described above, the area formula must be rewritten in terms of cell designations, which in this case is =B3*B4. Let us assume we are placing this formula in cell B10, which will display the calculated results.

With the alternative method described above, cell B3 is renamed to cell L, and cell B4 is renamed to cell W, and the letters in the formula are not changed, but the equal sign is always placed on the left side, as such: =L*W. If we place this formula in cell B10, our calculated results will be displayed in cell B10.

Cell B10 can also be renamed if
necessary, which can involve a letter, such as __A__, or even a word such as
__area__. This renaming would make it quite easy to transfer the
calculated results to another formula. This is explained in more detail in
the following topic.

__For Calculation
Devices That Perform__

__Complex Multiple
Calculations Simultaneously__

** **

__Creating a Calculation Device That
Performs__

__Multiple Calculations in a Chain
Like Sequence__

If several formulas are placed on a spreadsheet, they will not result in a calculation device that performs multiple calculations simultaneously. The formulas must be connected to each other in a logical configuration. This generally involves creating a structure where the output (calculated result) of one formula, is fed into one or more other formulas. This is similar to connecting electronic components to each other on a circuit board. The connections lead from one component to another in predetermined pathways through wires. With spreadsheet formulas there are no wires, but we can think of the connections as virtual wires, or imaginary wires. However, the important idea to understand is the connections between formulas are real.

__How
Are Connections Between Formulas Created?__

How are connections between formulas created for a set of multiple calculations that are carried out simultaneously? The answer is with cell designations. Specifically, connections between formulas are created when the cell designation of a calculated result from one formula, is used in another formula. For an example, let us assume that there is a formula in cell B10, and we want to multiply this result by 2, and place it in cell D10. To do this, we create the following formula =2*B10, and place it in cell D10. This can be continued, in a repeating chain like sequence, to additional formulas. If we want to add 100 to the result from cell D10 and place it in cell D20 we write the following formula =D10+100, and place it in cell D20. If we want to divide the results from cell D20 by 100, and place it in cell D30, we write the formula =D20/100, and place it in cell D30. This sequence can continue for hundreds of formulas in JavaScript and thousands in the spreadsheet format. (This is an estimate based on the computer and software I own.)

Note, sometimes it is convenient to replace the default cell designations, with names, such as a letter or word. When this is the case, the names can be used in the same way that the cell designations are used to connect a series of formulas in a chain like sequence. Keep in mind that cell designations are actually the default names. If a cell is renamed A, B, area, time, mass, money, taxes, or payroll, all of the techniques, principles, and examples that apply to the cell designations also apply to the renamed cells.

When formulas are connected in a chain like configuration as described above, the calculated result of the first formula in the chain is needed to calculate the second result. The third result cannot be calculated until the second result has been calculated. This sequence continues throughout the chain. Thus, when formulas are connected in the chain configuration, the computer must calculate the results in sequence. That is it must calculate the result from the first formula in the chain, then the result from the second formula, followed by the third, fourth, fifth, etc.

Generally a sequential chain of calculations involving a number of formulas, are carried out at a very rapid rate, and from the perception of the user, all the calculations appear to be calculated simultaneously. I use the words simultaneously or simultaneous from the perception of the user in this text, unless otherwise noted.

A sequential chain of formulas on a spreadsheet is not the only configuration that will result in multiple calculations that are carried out simultaneously. This is explained in the next subsection.

__A
Number of Spreadsheet Formulas____ __

__Connected to the Same Data
Source__

One or more input cells can be
connected to many formulas for multiple and simultaneous calculations.
**For example**, let us assume that A3 and A4 are input cells. When the
user enters numbers in A3 and A4, and presses the enter key, all of the
following formulas will calculate results simultaneously. With this
example, all the results are determined by the numbers entered into cells A3 and
A4:

=A3*A4, =A3+A4, =A3^A4, =A3/A4, =A3^2, =A3^3, =2*A3, =2+A3, =A3/2, =A4^2, =A4^3, =2*A4, =2+A4, =A4/2.

I am continuing with the above example. If we have spreadsheet formulas that have additional input sources, besides cells A3 and A4, they will also be calculated simultaneously, but there calculated results will not be totally determined by the numbers in cells A3 and A4. The following set of formulas is an example:

=B2*A3*A4, =C4+A3+A4, =(C5+A3)^A4, =B3/A4, =(B2+A3)^2, =(C2+A3)^3, =2*B7*A3, =2*B4+A3,

With the examples presented above, the spreadsheet formulas were connected to input cells, where the user enters numbers. However, the above configuration, or something similar to it, can involve a number of formulas that are connected to one or more formulas. That is in stead of input cells, the above formulas could have been connected to the calculated results from one or more other formulas. This concept is important for creating complex calculation devices, which often require many formulas connected to each other in complex configurations.

__Spreadsheet Formula Connections
Resemble the____ __

__Series and Parallel Connections
used in Electronics__

The spreadsheet formula connections (discussed above, in three subtopics) are similar to the series and parallel connections in electronics. See the following websites for information on this concept from the prospective of electronics:

** **

**http://physics.bu.edu/py106/notes/Circuits.html**

**http://www.allaboutcircuits.com/vol_1/chpt_7/1.html**

The first technique described above, (connecting formulas in a chain like sequence) is similar to electronic components connected in series. When electronic components are connected in this way, the electric current flows from one component to another, and if one component fails, it will affect the entire circuit. The same applies to a malfunctioning spreadsheet formula connected in this type of configuration. Small Christmas tree lights are often connected in series, on a long wire, and if one light malfunctions, it will cause the entire string of lights to malfunction.

The second method discussed above,
involved connecting spreadsheet formulas to the same data source. The
simplest example of this type of connection is one or more formulas connected
directly to a set of input cells. This __is similar to connecting
electronic components in parallel.__ Parallel connections are
essentially independent from other electronic components on the circuit.
For example, the lighting circuits used in homes and business establishments are
generally connected in parallel. If one light bulb malfunctions, it will
not affect other light bulbs on the circuit. The same applies to
spreadsheet formulas that are connected in parallel. If one formula fails,
it will not affect the calculated results of other formulas, assuming that the
connections truly fit the parallel configuration.

With both spreadsheet formulas and electronic components, the connections needed to create a complex device usually involve both series and parallel configurations. This can consist of a number of components connected in series, which branch off to one or more parallel connections. A parallel connection can also branch off to one or more sets of components that are connected in series. The out put of two or more connections are sometimes channeled into one component. For example, the calculated results of a number of formulas can be channeled into one formula for additional calculations. This can involve calculating the sum of several calculated results.

The concept of series and parallel
connections, as discussed above, is not limited to spreadsheets and
electronics. Formulas in JavaScript or any other computer language can be
connected in series
or in parallel, or a combination of both. However, I do not know of any
source that used the terminology: **series** or** parallel connections**
in relation to spreadsheets or any other computer language. Perhaps the
reason for this is there are no wires or any physical connections involved with
software based formulas. Nevertheless, the concept (of **series
**and** parallel connections**) can be quite useful for designing,
explaining, and studying, complex calculation devices, with intricate formula
configurations.

__Services Offered by
the Author__

This website was designed to
maximize efficiency and ease-of-use (usability, user-friendliness). The
text is presented with large fonts. The paragraphs are relatively short,
and the sentence structure and wording were written to maximize
*comprehension. The website has a very simple layout, on a single
page. This makes it easy to __navigate intuitively__, by scrolling down
or up, or by using the hyperlink table of contents. All the links for
downloads and other websites are also written with large fonts, and clearly
marked as links, such as with the following words: * left click on these
words*.

*Note: The material on this website and the Multiple-Algebraic Calculator are technical. A background in mathematics and spreadsheet software is required for maximum comprehension. However, most individuals, without the technical background, will probably understand portions of the text to varying degrees. Even for those that have an adequate background, to grasp the concepts on this website it is necessary to read the text slowly and carefully.

__Services Offered by
the Author David@TechForText.com__

I design and build user-friendly
software based calculation devices for __arithmetic__, __accounting__,
__currency exchange rates__,
__algebra__, __trigonometry__,
__correlations__, __calculus__, and databases with built-in calculation
devices. I also create attractive online calculation devices, and web
communication forms for websites. This includes website forms with
built-in calculation devices.

I generally make these devices in the Microsoft Excel, OpenOffice Calc, and the JavaScript formats, but I can work with other spreadsheet formats besides the above.

I write instructions for the devices I build. I can also write instructions for software and computer devices created by others. In addition, I can write advertising for your websites, products and services.

**For a list of websites with calculation
devices that I created, left click on these words, or go to the following
website: www.TechForText.com/Math**

For a list of all the services I
offer see **www.TechForText.com**

For a list of all my websites see
**www.David100.com**

My resume is online at: **www.David100.com/R**

I can provide the services
mentioned above on a fee-for-service basis, or possibly based on temporary or
permanent employment. If you are interested in my services, and want
additional contact information or more data on the services I offer, you can
email me at **David@TechForText.com
**or **use a website communication
form, by left clicking on these words.**

I am located in the USA. If you are a great distance from my locality or are in another country, this is not important. I can provide these services worldwide, because the software and websites I make can be delivered through the Internet to any locality, providing there are no governmental restrictions.

**To return to the top of this website left click on these
words**