__This website approximately 85% completed, and
it contains 9396 words, and a hyperlinked table of contents. The material in
this e-book still requires some editing. Some additional text will also be
added before it is completed.__

__ __

__Calculation Devices for Algebra and Trigonometry,__

__With Related Theories and Techniques for Creating__

__Calculation Devices in the JavaScript and Spreadsheet Formats__

__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 number of free calculation devices for algebra and trigonometry in the JavaScript and spreadsheet formats. This is coupled with related information, including theories and techniques involved with the design of calculation devices. Much of this information is focused on the novel use of spreadsheet software, as a computer language, to create dedicated calculation devices in the JavaScript and spreadsheet formats.

** **

**For the links for
the JavaScript and spreadsheet calculation devices left on these words, or go
to: www.TechForText.com/C**

** **

__ __

Many of the calculation devices from this website function
online, and they are in the JavaScript format. Devices that do not function
online are in the spreadsheet format, and they will be available by
downloading. Most of these calculation devices are on separate websites that
are ** directly linked to this webpage.** These websites contain
calculation devices embedded in webpages, download links, and instructions. On
some of these websites I also provide information on theories, concepts and
techniques by using a specific calculation device as an example.

__With
a Series of Topics, Subtopics,____ __

__And
a Hyperlinked Table of Contents.__

Each ** topic** is
written as an independent article, with the assumption that the reader may not
have read previous topics in this e-book. Thus, relevant information may be
repeated in some of the topics. All of the topics are written primarily for
individuals with relatively advanced backgrounds in mathematics and spreadsheet
software. This is because creating calculation devices is an advanced topic,
involving a type of computer programming with mathematical concepts.

__Links
for Additional Information and Resources__

Links to websites, created by other authors, are provided in this e-book,, for additional information and resources, and a diverse perspective on spreadsheets, mathematics, and calculation devices. This includes links for instructional videos, and various types of free software available for download.

__ __

__ __

__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 are interested in.__

The Top of the Webpage. Error! Bookmark not defined.

** **

__Calculation Devices for Algebra and
Trigonometry,__

With Related Theories and Techniques for Creating

Calculation Devices in the JavaScript and Spreadsheet Formats

**With a
Series of Topics, Subtopics,**

**And a
Hyperlinked Table of Contents.**

**Links for
Additional Information and Resources**

System Requirements For the Dedicated

Calculation Devices Provided From this Website

**The
Software Requirements for the**

**JavaScript
Calculation Devices**

**Spreadsheet
Calculation Devices**

**The
Computer Requirements for the Calculation Devices**

General Concepts and Utility of

**What Are
Dedicated Calculation Devices?**

**What Are
the Advantages of Dedicated Calculation**

**Devices
Over Conventional Mathematics Software?**

**Dedicated
Calculation Devices?**

**There Are
Many Dedicated Calculation Devices**.

**On The
Internet, For Many Types Of Calculations**

Creating Dedicated Calculation Devices,

In the Spreadsheet and Javascript Formats

**How Are
Dedicated Calculation Devices Created? **^{^Subtopic^}

**Creating
Dedicated Calculation Devices**

**In
JavaScript to function online on the Internet.**

From A Mathematical Perspective,

How Are Dedicated Calculation Devices

Created in the Spreadsheet Format for Algebra?

**Two Basic
Methods of Creating Calculation Devices**

**With
Conventional: Algebraic Equations and Formulas**

**In the
Spreadsheet Format, by Using**

**Cell
Designations to Represent the Letters in Formulas**

**Creating
Calculation Devices in the Spreadsheet Format,**

**By
Renaming Relevant Cells with the Letters in the Formulas**

**The Two
Methods Discussed Above :**

**Using
Cell Designations to Represent Letters in Formulas,**

**Or Renaming
Cells to Match Letters in Formulas**.

Step 1) Enter =2*A1, in cell B, as indicated above.

Step 2) Copy the formula =2*A1, in cell B with the copy function.

Step 3) Select the cells from B2 down the column to B100.

Step 4) Click on the paste function, or press Ctrl and V

**Calculation
Devices for Free Download, or Online Viewing**

**Created
By Substituting Cell Designations for Variables.**

**Calculation
Devices for Free Download, or Online Viewing**

**Created
By Renaming Cells to Match the Letter in Formulas**

**What Are
Error-Checking Devices**

**What Are
Advantages of Error-Checking Devices.**

**Creating
A Simple Error Checking Device**

**SPREADSHEET:
Formulas, Symbols, and Cell Designations,**

The Formulas, Symbolic Logic, Formatting Code,

Notation, and other Aspects of Spreadsheets, can be

Conceptualized as a Higher-Level Computer Language.

**The
Utility, Concepts and Supporting Arguments**

**Of A
Spreadsheet Computer Language**

**The
Multiple-Algebraic Calculator, and**

**Useful
Techniques for Advanced Spreadsheet Users**

**Converting
Conventional Formulas to Spreadsheet Formulas**

**By
Replacing Letters in the Formula with Cell Designations**

**Converting
Conventional Formulas**

**To
Spreadsheet Formulas by Renaming**

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

**The
following Example will Clarify the Two Methods**

**Described
Above. (Converting Formulas with**

**Cell
Designations, or Renaming Cells)**

For Calculation Devices That Perform

Complex Multiple Calculations Simultaneously

**Creating
a Calculation Device That Performs**.

**Multiple
Calculations in a Chain Like Sequence**

**How Are
Connections Between Formulas Created?**

**A Number
of Spreadsheet Formulas**

**Connected
to the Same Data Source**

**Spreadsheet
Formula Connections Resemble the**

**Series
and Parallel Connections used in Electronics**

**Two Types
Of Computing Sequential (Series) Non-Sequential Parallel**

Algebraic Calculation Devices, For Linear Equations

Algebraic Calculation Devices For Nonlinear Equations

**A
Calculation Device To Solve Quadratic Equations**

Trigonometric Calculation Devices

Algebraic Calculation Devices Two Unknowns

Algebraic Calculation Devices that Perform a Number of Calculations Simultaneously

Websites Created By Other Authors, for

Additional Information and Resources, and

__System Requirements For the Dedicated__

__Calculation Devices Provided From this Website__

__The
Software Requirements for the____ __

__JavaScript
Calculation Devices__

The JavaScript calculation devices I created will function
with any modern operating system and Internet browser that have ** JavaScript functionality.**
This includes the most popular operating systems, such as Microsoft Windows,
Macintosh, and Linux, but I have only tested the calculation devices with
Microsoft Windows.

Most, if not all, modern browsers
have JavaScript functionality. If you do not have a browser that supports
JavaScript, or want to upgrade, you can obtain browsers for free, by
downloading them from the websites listed at the end of this paragraph. To go
to any of the websites listed, left click on a blue web link. If a link fails
to function, it is most likely because the web address was changed, by the
manufacture of the browser. If this happens, you should do an Internet search,
preferably with Google, using the name of the browser, and the word download,
such as __Internet Explorer download__.

Internet Explorer http://www.microsoft.com/windows/internet-explorer/

Opera http://www.opera.com/

Firefox http://en-us.www.mozilla.com/en-US/

Google Chrome http://www.google.com/chrome/

__Spreadsheet
Calculation Devices__

The calculation devices that I created in the spreadsheet
format require Microsoft Excel, or the OpenOffice.org software package, and the
Windows operating system. If you do not have Microsoft Excel, you should
download the OpenOffice.org software package, because it is free. This
software package is almost as good as Microsoft Office, and it contains
spreadsheet software (__OpenOffice Calc__) that has most of the
functionality of Microsoft Excel.

**To download the free
OpenOffice software package, left click on these words, or go to the following
website: www.OpenOffice.org**

__ __

__The
Computer Requirements for the Calculation Devices____ __

The JavaScript and spreadsheet calculation devices I created for this website do not require much computer resources. Any computer that has enough memory and processing power to run a modern operating system, and spreadsheet software, will have enough power for all of the calculation devices provided from this website.

Incidentally, highly complex calculation devices, involving thousands of formulas, may require a relatively powerful computer, to function optimally. However, the most complex devices created for this website involve less than 150 formulas.

__General Concepts and Utility of__

__What
Are Dedicated Calculation Devices?__

__Based on the way I am using the terminology__, dedicated calculation devices
are software based devices that are designed to carry out a **specific type**,
**category** or **set of calculations.** The dedicated devices__ are
very different__ than conventional mathematics software that can perform many
types of calculations, such as conventional spreadsheets, the Windows
calculator, MathCAD, Mathematica, etc.

However, dedicated calculation devices, have some unique advantages over conventional mathematics software. This includes practical utility that can only be obtained from dedicated calculation devices. This is explained in the following paragraphs.

__What
Are the Advantages of Dedicated Calculation__

__Devices
Over Conventional Mathematics Software?____ __

There are several very important uses and advantages that dedicated calculation devices have over conventional mathematics software, which include the following:

1) Dedicated calculation devices can be created to perform a large number of calculations simultaneously, when the same input data applies to a set of calculations. This can involve dozens, hundreds, or even thousands of calculations performed simultaneously, by one specially designed calculation device. This can save time and work, and money for business. With a large set of highly complex calculations that are frequently carried out, several weeks of work might be saved each year.

2) Dedicated calculation devices are usually very user-friendly. These devices can be designed for users without mathematical backgrounds, (such as clerks, and administrative assistants) so that they can perform very complex calculations, just by entering data and clicking with the mouse. This can give managers the ability to obtain complex mathematical feedback, with out delay and with out waiting for the assistance of mathematicians. This also can save time and work, and the expense of hiring experts to carry out complex calculations that are frequently performed.

3) Dedicated calculation devices can be designed for advertising and sales. This can involve calculation devices that function over the Internet that relate to the potential cost of products or services. Examples are calculation devices that calculate the cost of loans, and mortgage versus rent. Especially useful for sales over the Internet are calculation devices with submit buttons. These devices add up the total cost of the items the customer enters. Then the customer finalizes the purchase by entering a credit card number, and pressing a submit button.

4) Dedicated calculation devices can be designed for educational purposes. This can involve the illustration of a series of steps needed to solve a math problem. Testing of mathematical skills can be carried out over the Internet with devices designed with submit buttons, in the JavaScript format. Dedicated calculation devices can be designed for practicing mathematics. This can involve a device that presents the same math problem with a new set of numbers each time a button is pressed.

** **

__Dedicated
Calculation Devices?__

Dedicated calculation devices are time consuming to
create, and they must be designed for a specific type, category or set of
calculations. Thus, creating these devices is only practical when the same
type of calculation is required on a regular basis. If this is not the case,
using conventional mathematics software, such as spreadsheets, MathCAD, or even
a hand held calculator is the best option, __unless you can find a dedicated
calculation device that already exists.__

__There
Are Many Dedicated Calculation Devices__

__On
The Internet, For Many Types Of Calculations__

There are many dedicated calculation devices on the Internet, especially for commonly performed calculations. Some of these devices I have created myself, and they are listed at www.TechForText.com/Math. You can find calculation devices on the Internet created by many other authors by using the search phrase that relates to the calculations you want to perform. This is of course a trial and error type of search. For common mathematical problems, you will probably find an appropriate online dedicated calculation device on the Internet.

However, for highly unusual or highly specialized calculations the best option for an individual is to use a hand-held calculator, or conventional mathematics software. The best option for a large or medium-size business is to hire someone to create the required dedicated calculation device.

__Creating Dedicated Calculation Devices,__

__In the Spreadsheet and Javascript Formats____ __

** **

__How
Are Dedicated Calculation Devices Created?
__

Almost any programming language can be used to create dedicated calculation devices. However, writing computer code for these devices can be very time-consuming. Some of the devices I have created have hundreds or even thousands of lines of code. One of the devices that I created had 82 pages of computer code. It can take many days, weeks, or even months to create these devices by writing code.

However, I **DO NOT** create
calculation devices by writing conventional computer code. I create them using
spreadsheet software, but these devices generally do not look like or function
like conventional spreadsheets. They have clearly delineated input boxes for
data, and clearly defined sections for the calculated results. They are
designed so that data can only be entered in the input boxes, and the
calculated results are presented in relatively large display boxes. __Just
like conventional software, these calculation devices can be reused
indefinitely, and they are not consumed like a conventional spreadsheet.__
However, they cannot operate without the spreadsheet software that was used to
create them, and they cannot function over the Internet.

__Creating
Dedicated Calculation Devices__

__In
JavaScript to function online on the Internet.__

When I create calculation devices for the Internet (for online functionality from a website) I also start with spreadsheet software. This involves creating a dedicated calculation device in the spreadsheet format, and converting it to JavaScript and HTML code with specialized software. The calculation mechanism is in JavaScript, and the HTML code is for the text, color and graphics.

After the conversion process is completed, it is usually necessary to slightly edit the resulting computer code, in an HTML editor, to maximize functionality and aesthetics of the calculation device. After this editing process is complete, I usually copy the code and paste it into a webpage I design for the device.

The above can sometimes be time-consuming. This is because it is necessary to create a device in a spreadsheet format that successfully converts to JavaScript without aesthetic imperfections. This often involves a trial and error process. However creating a JavaScript device from the spreadsheet format usually requires anywhere from 5 to 150 hours, instead of weeks or months that can be required for writing conventional code. When aesthetics are not important, and the calculation device is relatively simple, the whole process of designing and building a calculation device in the spreadsheet format, and converting it to JavaScript may require only a few minutes.

** **

__From A Mathematical Perspective,__

__How Are Dedicated Calculation Devices__

__Created in the Spreadsheet Format for Algebra?__

__ __

__Two
Basic Methods of Creating Calculation Devices__

__With
Conventional: Algebraic Equations and Formulas____ __

In this topic I am presenting two methods of creating dedicated calculation devices in the spreadsheet format, with conventional algebraic equations and formulas. Both of these methods result in devices that can be directly converted to JavaScript with appropriate software.

The first technique involves __using
cell designations to represent the letters in conventional equations and
formulas__. The second technique involves __renaming cells with the letters
in the equations and formulas.__ (That is the default cell designations are
replaced with the letters in the equations and formulas.) With both of these
techniques, it is necessary to write algebraic equations in terms of letters.
Then the equations can be solved for the unknowns in terms of letters. This
results in formulas for the unknowns. These formulas must be converted to
spreadsheet formulas, using the techniques explained in the following
subtopics.

Note: Another method of creating calculation devices involves using the Goal Seek mechanism available in Microsoft Excel, OpenOffice Calc, and probably some other brands of spreadsheet software. With the Goal Seek mechanism the spreadsheet software solves equations, using a type of trial and error process. This is especially useful for algebraic equations that cannot be solved with conventional techniques, or equations that are time consuming or difficult to solve with conventional methods. However, to use the Goal Seek mechanism it is necessary to convert algebraic equations into a format that can be used in a spreadsheet. This can be done with either of the techniques mentioned above.

Unfortunately, calculation devices created with the goal seek function cannot be converted into the JavaScript format, with currently available technology. Thus, I am not presenting this technique in any further detail in this text.

__In
the Spreadsheet Format, by Using__

__Cell
Designations to Represent the Letters in Formulas__

__ __

__Using cells to represent variables__ is one of the most practical ways
of creating calculation devices. It can be useful for all types of
calculations, including algebra. This method is described below in a
step-by-step way.

The first step involves solving an
equation that relates to the calculation device that you want to create. This
equation should be general enough to define the set of equations you want to
solve. I will use the following simple equation as an **example**: **AX+BX+CX
=N.** Keep in mind that this equation represents a set of many equations,
(an infinite set) where the letters A, B, C and N can have many different
values. For example, all of the following equations belong to the set:

5X+3X+3X =100,

-7X-2X-9X =666

999X+333X-4X =10000

-3X-7X =100.

(when A=0) 3X+9X =36,

(when A and B=0) 10X =1000,

This equation **AX+BX+CX=N **can
easily be solved for X. as follows** X(A+B+C)=N, **and then **X=N/(A+B+C) **The result for X, highlighted
in yellow, can be used as a spreadsheet formula, with the modifications
described below.

The next step, with this method, is to use the cells in a spreadsheet to represent the values of A, B, C, and N. We can actually use any set of cells we choose, but for convenience, I will use the following:

** **

**A=cell D11 , **

** **

**B= cell D12, **

** **

**C= cell D13**

** **

**N= cell D14**

** **

The next step is to write the
results of the equation we solved above X=N/(A+B+C)
in terms of the **above cells**. When this is done we have
X=D14/(D11+D12+D13). Now we must assign a cell to represent the calculated
result for X. I will use cell D16=X. The formula we derived above
(X=D14/(D11+D12+D13) provides the calculated value of X. **If we place the
formula =D14/(D11+D12+D13) in cell D16, for example, the D16 will display the
calculated value of X. Note the letter X is not part of the formula that is put
into cell D16, and the same applies to A, B, C, N. **

With the __method that I am
describing here__, the spreadsheet cannot recognize the meaning or value of
letters. However, it is necessary to place the letters A, B, C, N, and X next
to the cells that we used to represent them, so that the user will know where
to enter the numbers, and where to find the value of X. The most convenient
place is to put these letters to the *left of the numbers within an equal sign,
as follows:

** **

__A=____ is placed in cell C11 because we used cell D11 to
represent A__

__ __

__B= ____is placed in cell C12 because we used cell D12 to represent
B __

__ __

__C= ____is placed in cell C13 because we used cell D13 to
represent C __

__ __

__N=____ is placed in cell C14 because we used cell D14 to
represent N __

__ __

__X=____ is placed in cell C16 because we used cell D16 for the
calculated result which is the value of X. __

** **

*NOTE, the above input boxes can be labeled many other ways, such as by placing the letters A, B, K, N, on top of the input box. When an input box is labeled with a word, I often place the label on top, or sometimes on the bottom of the input box.

The technique described above, is probably the simplest way of creating a calculation device. However, this method has some disadvantages. Rewriting formulas in terms of cell designations can result in errors, and it can be tedious when a formula has many terms. It is also difficult to explain a formula written in terms of cell designations to others, because each term has a letter and number. For example, 6+Y can be rewritten for a spreadsheet as 6+ D10, and it can be misinterpreted by a person as 6+ 10 times D. All of these difficulties can be eliminated with the technique described in the following subtopic.

__ __

__ __

__Creating
Calculation Devices in the Spreadsheet Format,__

__By
Renaming Relevant Cells with the Letters in the Formulas____ __

The first two or three steps with this technique are the same as the alternative method described above. That is, write an equation that you want to solve, in terms of letters. Then, solve the equation for the unknown.

I am using the same equation that
I used in the previous subtopic as an example, but I am replacing the letter **C**
with **K**, because of technical reasons. When this equation **AX+BX+KX=N **is
solved for the unknown, X, the result is **X=N/(A+B+K). **

** **

The next step is to choose a set
of cells on the spreadsheet to serve as input boxes, where the user will enter
the values of: A, B, K, and N. (For this example, I am using **Cell B2, Cell
B3, Cell B4, and Cell B5.) ** The input boxes should be labeled by placing
the appropriate letter, with an equal sign, to the *left of the input boxes,
such as:

** **

**A=[Input Box Cell B2]**

** **

**B=[Input Box Cell B3]**

** **

**K=[Input Box Cell B4]**

** **

**N=[Input Box Cell B5]**

*NOTE, the above input boxes can be labeled many other ways, such as by placing the letters A, B, K, N, on top of the input box. When an input box is labeled with a word, I often place the label on top, or sometimes on the bottom of the input box.

Here is the difference between
this technique, and the method described in the previous subtopic. After the
above has been completed, the variables A, B, K, and N are defined in terms of
the input boxes shown above. ** This simply means that the four cells (B2,
B3, B4, B5) are renamed to: A, B, K, and N.** This renaming technique is
done with a mechanism designed for the purpose, which is available in Microsoft
Word, OpenOffice Calc, and some other brands of spreadsheet software.

__Specifically, the four cells are
renamed as shown below, which results in the
indicated relationships between the cells and letters.__

**Cell B2 is renamed to A**** As a result of this renaming:
wherever =A is placed on the spreadsheet, it will display the number or letters
that the user enters in Cell B2.**

**Cell B3 is renamed to B**** As a result of this renaming:
wherever =B is placed on the spreadsheet it will display the number or letters
that the user enters in Cell B3.**

**Cell B4 is renamed to K**** As a result of this renaming:
wherever =K is placed on the spreadsheet it will display the number or letters
that the user enters in Cell B4.**

**Cell B5 is renamed to N**** As a result of this renaming:
wherever =N is placed on the spreadsheet it will display the number or letters
that the user enters in Cell B5.**

For example, if the number 5 is entered
in cell B2, which is the input box for **A, A **would be equal to five. If **A**
was multiplied by **2**, such as **=2*A**, and placed in any cell on the
spreadsheet, the result would be 10.

Now, we can return to the equation
that was obtained by solving **AX+BX+KX=N **for X, which is: **X=N/(A+B+K)**. This equation, highlighted
in yellow, now can be used as a spreadsheet formula, assuming that the cells
indicated above were appropriately renamed. This simply involves removing the
X. and placing the formula =N/(A+B+K) in a convenient cell on the spreadsheet.
**For example, if we place =N/(A+B+K) in cell B6, B6 will display the
calculated results.** We can also rename cell **B6 to X. **

Renaming cells that display calculated results, with an appropriate letter, or word, can be quite useful. It can prevent confusion and errors when the calculated results must be transmitted to other formulas, or to error-checking devices, which are discussed in the next topic.

__The
Two Methods Discussed Above :__

__Using
Cell Designations to Represent Letters in Formulas,__

__Or
Renaming Cells to Match Letters in Formulas__

Step 1) Enter =2*A1, in cell B, as indicated above.

Step 2) Copy the formula =2*A1, in cell B with the copy function.

Step 3) Select the cells from B2 down the column to B100.

Step 4) Click on the paste function, or press Ctrl and V

Another situation using the default cell designations may be preferable, for some people, is when the calculation devices involves one or two formulas, or when you are entering the default formulas provided with the spreadsheet software. This technique might save a little time, and complexity, for some people, because it eliminates the step of renaming spreadsheet cells.

__ __

** **

__Calculation
Devices for Free Download, or Online Viewing__

__Created
By Substituting Cell Designations for Variables.__

If you want a calculation device that was created by __Substituting
Cell Designations for the letters in the equation:__ AX+BX+CX=N, left click
on one or more of the following links. (All of the following calculation
devices solve the equation: AX+BX+CX=N, for X.):

** **

If you want all of the above formats, in a single zip folder, left click on these words.

__ __

__Calculation
Devices for Free Download, or Online Viewing__

__Created
By Renaming Cells to Match the Letter in Formulas__

If you want calculation devices created by renaming spreadsheet cells to match the letters in AX+BX+KX=N left click on one or more of the following blue hyperlinks. (All of the following calculation devices solve the equation: AX+BX+KX=N, for X.):

** **

** **

** **

__What
Are Error-Checking Devices__

__What
Are Advantages of Error-Checking Devices.____ __

The error checking devices can detect if malfunctions, in hardware, software, or the calculation device itself are causing calculation errors. Errors of this type are relatively rare. Sometimes a calculation device can become corrupt, over time. This can occasionally happen if the user inadvertently deletes a formula. However, this is usually not a possibility, because most well-made calculation devices have

The error checking device can indicate when numbers entered by the user do not satisfy the equation. There are default error checking devices in spreadsheet software, and also in JavaScript software that was created by converting a spreadsheet. The default devices indicate when the numbers entered by the user result in division by zero, or if the numbers are too large for the calculation device.

If the fault error-checking does not display an error
message. That is there are default mechanism built into spreadsheet software
that may display errors message when numbers do not satisfy an equation, **IN
SOME CASES.** For example, if the user entered numbers that result in
division by zero, you will see a fault error message.

The error checking device can spot mathematical errors while a calculation device is under construction, and after it has been completed. This can save a considerable amount of time and effort, when creating complex calculation devices, especially if there are many formulas, and/or formulas that have many terms.

This device might also spot certain types of software and computer malfunctions that result in calculation errors, which are extremely rare.

If there are errors that relate to the way the letters in the formula were defined, with the name mechanism, you will see the default error message in Microsoft Word or OpenOffice Calc, which is #NAME?

** **

** **

__Creating A Simple Error Checking
Device__

Note: if you did not read the
previous subtopic thoroughly you may find the following confusing. This is
because, I am still using the above example, with the equation: **AX+BX+KX=N.**
The formula for the calculated value of X, is** =N/(A+B+K),** which is in
cell B6. This means cell B6 displays the calculated value for X.

We can expand the utility of the
principles, and technique described in the subtopic presented above to create
an error checking device. The additional utility becomes apparent when we
rename cell B6 to X. Keep in mind that with our example, cell B6 contains the
formula that calculates the value for X, which is the formula highlighted in
yellow **=N/(A+B+K). Thus, with this
example,** the value for X is displayed in cell B6.

__As a
result of renaming cell B6 to X____, the spreadsheet software can read and evaluate the
unsolved equation that we started with, if we enter asterisks (*) to indicate
multiplication, as such: A*X+B*X+K*X=N.__

If we place this equation without
the **N, =A*X+B*X+K*X **in any cell
on the spreadsheet, the value of N will be calculated, and displayed. If we
place** =N **in any cell on the
spreadsheet the value the user entered for N will be accessed, and displayed.
If there are no errors the two values will be equal. That is the calculated
value for N will equal the value the user entered for N. In general, the left
side and right side of an equation must be equal. With our example, the left
side is (**A*X+B*X+K*X)** and right
side is **(N). **

** **

With the principles described
above, we can create an error checking device that displays TRUE when there are
no errors, and FALSE when there are errors. To do this the above equation is
placed in any convenient cell on the spreadsheet, with ** an additional
equal** sign on the left, as such

The simple error checking device
described above can save time when creating a calculation device. This is
especially the case, when a calculation device requires a large number of
complex equations, which must be converted to spreadsheet formulas. The error
checking device quickly checks formulas. For example, if the equation **=A*X+B*X+K*X=N was incorrectly
solved for X, such as =N/(B+K) **the
word **FALSE** will appear in the cell that contains the equation: **=A*X+B*X+K*X=N.
**If the correct formula was entered, which is: =N/(A+B+K), you will see the
word TRUE.

Keep in mind
that the above error checking device only works if all the letters in the
formula were properly defined, with the name mechanism in the spreadsheet
software, and numbers are entered for **A, B, K, and N.**

The simple error checking device
scribed above sometime display error messages, when there are no errors. This
can happen when the calculations are complex, especially if the numbers have
many digits. This happens as a result of rounding errors, which are usually
not significant. For example, FALSE will be displayed, indicating an error, if
the user enters 10 for N, and the calculated value for N is A*X+B*X+K*X=9.999999999999.** **This
difficulty can be eliminated by using the ROUND function, set to five or six
decimal places, as shown below:

**=ROUND(A*X+B*X+K*X, 5)=ROUND(N, 5)**

** **

** **

__SPREADSHEET: Formulas, Symbols, and Cell Designations,__

__The Formulas, Symbolic Logic, Formatting
Code,__

__Notation, and other Aspects of
Spreadsheets, can be____ __

__Conceptualized as a Higher-Level Computer
Language.__

** **

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.

__The
Utility, Concepts and Supporting Arguments__

__Of
A Spreadsheet 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.

Statements created with Spreadsheet functions for symbolic logic, resemble computer code

*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, conventional formulas modified for spreadsheet use, logical statements that define a set of numbers, as well as text. All of this certainly resembles computer code, and it is feasible to write a series of statements with this code that can perform many operations, including the following:

Evaluate calculated results, for accuracy, or inaccuracy, and display related text

Display specified text when a predefined condition exists. For example, a device that calculates money, indication wither or no

Channel numbers or text through various pathways to one or more cells, based on specified criteria. For example, sending even numbers to a set of spreadsheet cells on the left, and odd numbers to a set of spreadsheet cells on the right.

The type of calculations carried out on a set of numbers, can be controlled and written statements entered by the user.

Spreadsheet formulas 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.

Formatting code

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:

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.

^{ }

^{ }

^{ }

__ __

__Two
Types Of Computing Sequential (Series) Non-Sequential Parallel__

__ __

__Algebraic Calculation Devices, For Linear Equations__

__Algebraic Calculation Devices For Nonlinear Equations__

__ __

__A
Calculation Device To Solve Quadratic Equations__

** **

__ __

__Trigonometric Calculation Devices__

** **

** **

__Algebraic Calculation Devices Two Unknowns__

** **

__Algebraic Calculation Devices that Perform a Number of Calculations
Simultaneously__

** **

** **

** **

__Websites Created By Other Authors, for____ __

__Additional Information and Resources, and__

**Words on website: **University of Akron This is the review of Algebra in 10 lessons **http://www.math.uakron.edu/~dpstory/mpt_home.html**

This essentially a free review course on algebra, and it is provided in the PDF format.

Words on website: financial accounting

http://www.flatworldknowledge.com/pub/1.0/financial-accounting/89943

This is an e-book on accounting

__List of Websites for spreadsheets__

__ __

__words of website:\\__

__ __

__ __

__Words on website: A video instructional series on
statistics for college and high school classrooms and adult learners; 26
half-hour video programs and coordinated books http://www.learner.org/resources/series65.html#__

__ __

__ __

__ __