__The
Algebraic-Calculator-For-Two-Unknowns __

__This website
provides free calculation devices in the online JavaScript__

__and spreadsheet
formats for solving two linear equations, for two unknowns,__

__ and 6280 words
of useful information for math students and mathematicians.__

__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.__

Below this brief introduction is
an online JavaScript calculation device (* Algebraic Calculator*) for solving two linear
equations, for two unknowns, X and Y. If you
want the Algebraic Calculator in the spreadsheet format, (Microsoft Excel, or
OpenOffice.Cal) or additional information, scroll all the way down, beneath the
online Calculator. Alternatively, you can use the hyperlink table of
contents of this website, which is located just below the online
calculator.

**To
go directly to the table of contents 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****,
****which can be done by left clicking on
these words. **

__Instructions, and the
Algebraic Calculator____ __

**The Algebraic
Calculator**
solves equations similar to

If one or both of the equations you want to solve has less terms than the above, and looks like this: BY=V, or this KX=W, enter a zero in the input box that corresponds to the missing coefficient. With the first example, A=0, and with the second example D=0.

When there are no apparent coefficients multiplying a variable, such as X+Y=V, you must enter 1 in the related input box. With the above example, A=1, and B=1.

Below, is the online version of
the **Algebraic Calculator**, embedded in this webpage. Note, if you
print your calculated results from this Calculator all of the text on this
website will be printed. **If
you do not want to print the entire website with your results use the printer
friendly version of the Algebraic Calculator by left clicking on these
words.**

__The
Online Version of the Algebraic Calculator__

__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.__

**Instructions, and the Algebraic Calculator** 2

**The Online
Version of the Algebraic Calculator** 3

**System
Requirements for The Downloads**
13

**The
Software You Need for the Downloads**
13

**Download
Links For The Algebraic Calculator In The**. 13

**Microsoft
Excel, OpenOffice Cal, And Javascript Formats**. 14

**Number
Handling Capacity and Error Messages of** 15

**The
Algebraic-Calculator-For-Two-Unknowns**. 15

**The
Algebraic-Calculator Can Handle**. 15

**Very Large
and Very Small Numbers,** 15

**General
Error Messages Displayed by the Algebraic Calculator** 16

**How the
****Algebraic Calculator**** Was Created**. 17

**Introductory Note About The General Utility of
the**.
17

**Methods
and Concepts Presented on this Website**. 17

**You Can
Use Formulas From This Website for Your Own Needs**. 18

**Creating
the Javascript Version of the Algebraic Calculator** 19

**Creating
the Calculation Mechanism**.. 19

**For the
Algebraic Calculator** 19

**To Use a
Conventional Formula in a Spreadsheet** 20

**By
Substituting Cell Designations for the Variables**. 20

**An
Alternative Method for Using**. 22

**Conventional Formulas in Spreadsheets**
22

**Involves
Defining the Variables in Terms of Cell Designations**. 22

**Symbols
Used in a Spreadsheet for Multiplication,** 24

**Division,
Addition, Subtraction, Square Roots, Cube Roots, Etc.** 24

**Creating
the Two Display Boxes**. 26

**In the
Algebraic Calculator** 26

**The Small
Gray, and Large Green Display Boxes: A Challenge**. 26

**For The
Display Boxes: Display Formulas**
29

**Simplified
English Interpretations of the**
30

**Computer
Code Comprising the Display Formulas**. 30

**Formulas
that Control the Display of AX+BY=V**. 31

**In the
Small Gray Display Box**. 31

**Formulas
For The Display Of KX+DY=W,** 34

**In The
Small Gray Display Box**. 34

**Formulas
for the Display of AX+BY=V**. 36

**In the
Large Green Display Box**. 36

**Formulas
for the Display of KX+DY=W**.. 39

**In the
Large Green Display Box**. 39

**Computer
Code in the Form of Spreadsheet Formulas**. 41

**for
Rounding Functions, and Error Checking**. 41

**A User
Controllable Rounding Mechanism**
41

**Computer
Code in the Form of** 45

**Spreadsheet Formulas For Error Checking**
45

**Services
Offered by the Author** 49

**Services
Offered by the Author David@TechForText.com**.. 51

** **

__System Requirements
for The Downloads__

__The
Software You Need for the Downloads__

The spreadsheet versions of the Algebraic Calculator require either Microsoft Excel, or the OpenOffice.org software package. In addition, Microsoft Windows is required for the spreadsheet versions.

The JavaScript version of the 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.

__Download Links For The Algebraic
Calculator In The____ __

__Microsoft Excel,
____OpenOffice____ Cal____, And Javascript
Formats__

**If you want the 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.__** **

__Number Handling
Capacity and Error Messages of__

__The
Algebraic-Calculator-For-Two-Unknowns__

__The
Algebraic-Calculator Can Handle__

__Very Large and Very Small
Numbers,__

**The**** Algebraic
Calculator** can handle very large and very
small numbers with well OVER 250 digits. However, when approximately 14
digits are involved, the numbers are displayed in scientific notation. An
uppercase

An example, of a large number is 10000000000000000000, and it is displayed by the Algebraic Calculator, in scientific notation, in this format: 1.00E+19. An example of a very small number is 0.0000000000000000001, and it is displayed by the Calculator in scientific notation in this format: 1.00E-19. (Note: sometimes numbers that are extremely small are rounded down to zero, by the Calculator.)

__General Error Messages Displayed
by the Algebraic Calculator__

This calculator shows an error message when there are no solutions for the numbers that were entered. For example, if both equations have the same slope, (when graphed the lines from the equations are parallel), there will be no solution. A good example is for AX+BY=V, A=500, B=600, V=1100, and for KX+DY=W, K=5, D=6, W=11. Another example, is when a sub-calculation involves division by zero. This happens if B=0, or if (KB-DA))=0.

The calculator also shows an error message if a sub-calculation or the values of X or Y exceed 307 digits. This can happen, when entered numbers have great differences in magnitude, and they are involved in a sub-calculation that involves division.

__How the
____Algebraic
Calculator____ Was
Created__

__Introductory Note About The
General Utility of the__

__Methods and Concepts Presented on
this Website__

The concepts and techniques used
to create the **Algebraic Calculator **have **general** utility. They are useful for creating calculation
devices in a number of formats, including JavaScript and spreadsheet
formats. They can also increase the versatility of spreadsheet
software.

Spreadsheet software, generally have a limited set of built-in formulas, which are sometimes called functions. These functions number in the hundreds, but there are millions of formulas in mathematics, science, and industry. With the techniques discussed in this section, most if not all of these formulas can be used in a spreadsheet, or to create dedicated calculation devices.

__You Can Use Formulas
From This Website for Your Own Needs__

Note: The Algebraic Calculator contains over 30 formulas, which are presented in the following paragraphs. You can use any of the formulas on this website, but you will probably have to modify them to meet your specific needs. You can use the Windows copy and paste mechanism to transfer complex formulas to your spreadsheet.

__Creating the Javascript Version of
the Algebraic Calculator__

The JavaScript version of the algebraic calculator was created from the spreadsheet version. This involves specialized software to convert Microsoft Excel to the JavaScript format. Thus, almost all of the following material on this website applies to both spreadsheet and JavaScript versions.

__Creating the Calculation
Mechanism__

The first step for the creation of
a calculation device is to find or create the formulas needed to perform the
required calculations. For the __Algebraic Calculator__, I created the
formulas, by simply solving the equations: **AX+BY=V** and **KX+DY=W** for
**X** and **Y**. This resulted in the two formulas presented below:

__X=(WB-DV)/(KB-DA)__**, **and __Y=(V-AX)/B
__

However, the computer cannot read the above formulas. The meaning of the letters and implied mathematical instructions in the formulas had to be translated into a form (or language) that the software would understand. With the software I was using, Microsoft Excel, there are two ways that letters from formulas can be translated for the computer. These methods are discussed below, and they are also available in OpenOffice Calc, and other spreadsheet software.

__To
Use a Conventional Formula in a Spreadsheet__

__By
Substituting Cell Designations for the Variables____ __

The method that I often use to
convert formulas for used with spreadsheet software, involves __replacing each
letter__ in a formula with specific cell references, which serve as inputs
cells. After this, the rewritten formula is placed in any convenient cell,
which will display the calculated results, when numbers are entered into the
input cells.

For example, the formula: G+H=U
can be converted for spreadsheet use, by ** replacing** the

The method discussed above, rewriting formulas in terms of cell references, 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 references to others, because each term has a letter and number. For example, 2+X, can be rewritten for a spreadsheet as 2+ B5, and it can be misinterpreted by a person as 2 +5 times B. There is another method of converting formulas for spreadsheet use, which avoids all of the difficulties mentioned above.

__An
Alternative Method for Using____ __

__Conventional Formulas in
Spreadsheets__

__Involves Defining the Variables in
Terms of Cell Designations__

This method involves interpreting
(or defining) the meaning of each letter in a formula for the software, in terms
of specific input cells. This can be done with the **name **mechanism in Microsoft Excel, or
OpenOffice Calc. With this method the letters from the original formula
remain the same in the spreadsheet version.

For example, by using the **name
**mechanism, G+H=U can be converted for spreadsheet use, by defining G as cell
A1, and defining H as cell B1. Then the spreadsheet
formula is =G+H. This looks
similar to the original formula, except it has an equal sign on the
left. This formula =G+H can be placed in any convenient cell, which we
will assume to be cell D3. Cell D3 represents the U from the original
formula D3=U: G+H=U. With this example, when numbers are entered in cells
A1 and B1, cell D3 will display the calculated results.

For additional information on using the name mechanism in Microsoft Excel see the following websites, which can be done by left clicking on the web addresses.

Words on website:”**Define and
use names in formulas” ****http://office.microsoft.com/en-us/excel/HA101471201033.aspx#backtotop**

Words on website: “Introduction To Defined Names”

**http://www.cpearson.com/excel/DefinedNames.aspx**

__Symbols Used in a Spreadsheet for
Multiplication,__

__Division, Addition, Subtraction,
Square Roots, Cube Roots, Etc.__

The methods described above interpret the meaning of letters in a formula, in terms of specific cells on the spreadsheet. However, the spreadsheet software still will not be able to understand the mathematical instructions implied by a conventional formula. The software does not understand the human conventions for multiplication, square roots, cube roots, and many other mathematical operations. This necessitates the following conversions for Microsoft Excel, OpenOffice Calc, and other spreadsheets software:

Multiplication requires an asterisk ( *) XY must be converted to =X*Y, and B(X+Y) must be converted to =B*(X+Y)

Division requires a slash (
**/** ) such as X divided by Y is =X/Y. This also applies to fractions,
such as one fifth is written as 1/5. Fractions can also be written as
decimals.

Squares, square roots, cube roots,
etc, requires a carrot sign (**^**). For spreadsheets **x ^{n}**

The above (**=X^n**) and
**=POWER(X, n)** also applies to negative exponents, such as when n=-2:
10^-2=0.01, 4^-2=0.0625.

The plus (+) and minus sign (-) are recognized by spreadsheet software, without any modification.

When adding a series of numbers
the word **=SUM** and **()** is used. For example, for adding vertically
in a column **=SUM(B6:B12)** and for adding horizontally in a row**
=SUM(A6:E6)**

(BASED ON THE ABOVE PRINCIPLES) To
create the algebraic calculator the formulas, ** X=(WB-DV)/(KB-DA),
**and

** **

**=(W*B-D*V)/(K*B-D*A)** and **=(V-A*H2)/B**

These formulas were placed in two
cells on the spreadsheet. They solve the equations that were presented in
the beginning of this section, which are **AX+BY=V** and
**KX+DY=W**. The calculated values for **X** and **Y** are
displayed in the cells that the formulas **=(W*B-D*V)/(K*B-D*A)** and **=(V-A*H2)/B **have been
placed in.

__Creating the Two
Display Boxes__

** **

__The
Small Gray, and Large Green Display Boxes: A Challenge__

A more challenging task, then creating the calculation mechanism of the Algebraic Calculator, was to create the two display boxes. The reason for this will become apparent in the following paragraphs.

The smaller display box is gray,
and it displays equations (**AX+BY=V** and **KX+DY=W**) when they have
single-digit numbers for A and B, or K and D, excluding decimals. All the
numbers represented by the A, B, V, K, D, and W are rounded to remove all
decimal places when they are displayed in the gray box. This does not
affect calculated results, because the numbers rounded for this relatively small
display box are not used for calculations.

The larger display box is green,
and the software presents equations in this box when the A and B, or K and D
have more than one digit, (excluding decimals). The smaller gray box
displays equations in the conventional format. The larger green box
displays equations in a **vertical format**, which can accommodate equations
that have numbers with many digits. For example, the conventional format
is obviously **AX+BY=V. **What I mean by the vertical format is as
follows:

** **

**
+**

**
BY**

** =V**

An example, with numbers using the vertical format is presented below, and the X and Y variables are on the left, and there is plenty of room for many digits.

**12346678789900467899055555551133.9876X**

**+**

**23434544444444442123999223456331.0899Y**

** **

**=
234656678798990**

Note, I do not know if anyone else has used a vertical format, or something similar to it. However, I found it to be quite practical when dealing with equations that have coefficients with many digits, or equations that have many variables.

__For
The Display Boxes: Display Formulas__

The challenge with creating the
display boxes was not simply related to the conventional of vertical format,
discussed above. The display boxes required formulas (strings of computer
code) that would evaluate the number of digits in the values for **A**,
**B**, **K**, and **D,** and to determine if an equation should be
displayed in the small gray box or large green box. I am calling these
formulas ** display formulas**. (The number of digits for V and W
is not a problem in both boxes, because there is enough room to display many
digits on the right side of the equations:

Two display formulas were needed,
for **each symbol**, in the equations **AX+BY=V**, and
**KX+DY=W**. There are seven symbols in each equation, including the
plus sign (+) and equal sign (=). Thus, I had to create 28 formulas.
These formulas were not the same for each symbol. For example, the formula
for the plus sign in the small gray box needed the capability of presenting a
negative sign, when the value of B or D is negative.

I created all of the required
formulas using all of the following: The **If Function**, the **Or
Function**, the **ROUND Function**, the **ABS (absolute value
function)** and ***** **+**, **=**, **>**,
**<**, **TRUE**, **( ), “”, and the comma ( ,).** These
formulas are essentially a form of computer code, or computer language, and they
are **listed below** **with simplified
English interpretations.**

111111111111111

__Simplified English Interpretations
of the__

__Computer Code Comprising the
Display Formulas____ __

Note: The i__nterpretations__
of the ** Computer Code Comprising the Display Formulas** (for the
display boxes) are more detailed for the formulas in the beginning of this
discussion. I assumed that less detail would be needed after reading the
translations in the beginning of this subsection, because there is some
similarity between many of these formulas.

All of the following formulas display symbols in the cell that the formula has been placed in. For example, if a formula has been placed in the upper portion of the gray box, in cell F9, the formula will conceal or display a symbol in cell F9.

__Formulas that Control the Display
of AX+BY=V__

Notes:

The following formulas are for the gray display box, and they are highlighted in gray.

The locations indicated in the following interpretations are determined by the location of the cell that the formula was placed in

**=IF((OR(
ABS(A)>9, ABS(B)>9, ))=TRUE, "",ROUND(A, 0 )** If the absolute value of A
or B, is greater than 9, (=TRUE) then do not display the values of A or B, but
if this is not the case, (absolute values of A and B are less than or equal to
9) round down the value for A to zero decimal places, and display it in the
upper portion of the gray box.

**=IF((OR(ABS(A)>9,ABS(B)>9,))=TRUE,"","X")**** ** If the absolute values of A
or B is greater than 9, do not display anything, but if this is not the case,
(the absolute values of A **and** B are less than or equal to 9) then display
the X term in the upper portion of the gray box.

**=IF((OR(ABS(A)>9,
ABS(B)>9, ))=TRUE, "", IF(B<0, "-", "+"))**** **If the absolute value of A or B is
greater than 9, do not display anything, but if this is not the case, display
the plus or minus sign, in the upper portion of the gray box. If B is less
than zero display the minus sign, in the upper portion of the gray box, if this
is not the case, (B is greater than zero) display the plus sign, in the upper
portion of the gray box.

**=IF((OR(ABS(A)>9,
ABS(B)>9, )0)=TRUE, "",ROUND(ABS(B), 0 ))**

If the absolute value of A or B is greater than 9, display nothing, if this is not the case, (absolute values of A and B is less than or equal to 9) round down the absolute value of B to zero decimal places, and display B, in the upper portion of the gray box. (The absolute value of B is displayed because, the plus or minus sign is produced by the formula above this one.)

**=IF((OR(ABS(A)>9,
ABS(B)>9,))=TRUE,"","Y")**** **If the absolute value of A or B is
greater than 9, display nothing, if this is not the case, (absolute values of A
and B are less than or equal to 9) display Y, in the upper portion of the gray
box.

**=IF((OR(ABS(A)>9,
ABS(B)>9,))=TRUE,"","=")**** **If the absolute value of A or B is
greater than 9, display nothing, if this is not the case, (absolute values of A
and B are less than or equal to 9) display the equal sign in the upper portion
of the gray box.

**=IF((OR(ABS(A)>9,
ABS(B)>9, ))=TRUE, "",ROUND(V, 0 ))**** **If the absolute value of A or B is
greater than 9, display nothing, but if this is not the case, (absolute values
of A and B are less than or equal to 9) round the value of V to zero decimal
places, and display it.

__Formulas For The Display Of
KX+DY=W,__

Notes:

Code for the gray display box is highlighted in gray.

The locations indicated in the following interpretations are determined by the location of the cell that the formula was placed in

**=IF((OR(
ABS(K)>9, ABS(D)>9, ))=TRUE, "",ROUND(K, 0 ))**** **If the absolute value of K or D is
greater than 9, display nothing, but if this is not the case, display the value
of K, rounded to zero decimal places, in the lower portion of the gray
box.

**=IF((OR(ABS(K)>9,
ABS(D)>9,))=TRUE,"","X"** If the absolute value of K
or D is greater than 9, display nothing, but if this is not the case display X,
in the lower portion of the gray display box.

**=IF((OR(ABS(K)>9,
ABS(D)>9, ))=TRUE, "", IF(D<0, "-", "+"))** If the absolute value of K
or D is greater than 9 display nothing, but if this is not the case, display in
the lower portion of the gray box: the minus sign if D is less than zero, but if
D is equal to or greater than 0, display the plus sign.

** **

**=IF((OR(
ABS(K)>9, ABS(D)>9, ))=TRUE, “”, “D")** If the absolute value of K
or D is greater than 9 display nothing, but if this is not the case, display D,
in the lower portion of the gray box.

**=IF((OR(K>ABS(9),
ABS(D)>9,))=TRUE,"","Y")**** **If the absolute value of K. or D
is greater than 9, display nothing, but if this is not the case display
Y.

**=IF((OR(ABS(K)>9,ABS(D)>9,))=TRUE,"","=")** If the absolute value of K
or D is greater than 9, display nothing, but if this is not the case display the
equal sign in the lower portion of the gray box.

**=IF((OR(ABS(K)>9,
ABS(D)>9, ))=TRUE, "",ROUND(W, 0 ))** If the absolute value of K
or D is greater than 9, display nothing, but if this is not the case, display in
the lower portion of the gray box, the value of W, rounded to zero decimal
places.

__Formulas for the Display of
AX+BY=V__

__In
the Large Green Display Box__

Notes:

The following formulas are for the green display box, and they are highlighted in green.

The locations indicated in the following interpretations are determined by the location of the cell that the formula was placed in

**=IF((OR(
ABS(A)>9, ABS(B)>9, ))=TRUE, A, "")**** ** If the absolute value of A
or B is greater than 9, display A, in the left section of the green box, but if
this is not the case display nothing.

**=IF((OR(
ABS(A)>9, ABS(B)>9, ))=TRUE, "X", "")** If the absolute value of A
or B is greater than 9, display X, in the left section of the green box, but if
this is not the case display nothing.

**=IF((OR(
ABS(A)>9, ABS(B)>9, ))=TRUE, "+", "")** If the absolute value of A
or B is greater than 9, display the plus sign, in the left section of the green
box, but if this is not the case display nothing.

**=IF((OR(
ABS(A)>9, ABS(B)>9, ))=TRUE, B, "")** If the absolute value of A
or B is greater than 9, display B, in the left section of the green box, but if
this is not the case, display nothing.

**=IF((OR(
ABS(A)>9, ABS(B)>9, ))=TRUE, "Y", "")** If the absolute value of A
or B is greater than 9, display Y in the left section of the green box, but if
this is not the case display nothing.

**=IF((OR(
ABS(A)>9, ABS(B)>9, ))=TRUE, "=", "")** If the absolute value of A
or B is greater than 9, display the equals sign, in the left section of the
green box, but if this is not the case display nothing.

**=IF((OR(
ABS(A)>9, ABS(B)>9, ))=TRUE, V, "")**** **If the absolute value of A or B is
greater than 9, display V in the left section of the green box, but if this is
not the case display nothing.

__Formulas for the Display of
KX+DY=W__

__In
the Large Green Display Box__

Notes:

Code for the green display box is highlighted in green.

**=IF((OR(
ABS(K)>9, ABS(D)>9, ))=TRUE, K, "")**** **If the absolute value of K or D is
greater than 9, display K, in the right section of the green box, but if this is
not the case display nothing.

**=IF((OR(
ABS(K)>9, ABS(D)>9, ))=TRUE, "X", "")** If the absolute value of K
or D is greater than 9, display X, in the right section of the green box, but if
this is not the case display nothing.

**=IF((OR(
ABS(K)>9, ABS(D)>9, ))=TRUE, "+", "")** If the absolute value of K or D
is greater than 9, display the plus sign, in the right section of the green box,
but if this is not the case display nothing.

**IF((OR(
ABS(K)>9, ABS(D)>9, ))=TRUE, D, "")** If the absolute value of K
or D is greater than 9, display D, in the right section of the green box, but if
this is not the case display nothing.

**=IF((OR(
ABS(K)>9, ABS(D)>9, ))=TRUE, "Y", "")**** **If the absolute value of K or D is
greater than 9, display Y, in the right section of the green box, but if this is
not the case display nothing.

**=IF((OR(
ABS(K)>9, ABS(D)>9, ))=TRUE, "=”,"")**** **If the absolute value of K or D is
greater than 9, display the equal sign, in the right section of the green box,
but if this is not the case display nothing.

**=IF((OR(
ABS(K)>9, ABS(D)>9, ))=TRUE, W, "")**** **If the absolute value of K or D is
greater than 9, display W, in the right section of the green box, but if this is
not the case display nothing.

** **

** **

__Computer Code in the
Form of Spreadsheet Formulas__

__for Rounding
Functions, and Error Checking____ __

__A
User Controllable Rounding Mechanism__

In general, it is often necessary to round down numbers to a specific number of decimal places, especially when the numbers are calculated results. Very often, the last few decimal places of a calculation are not needed, or not mathematically significant. This can happen when the calculations are based on measurements that are less precise than the calculated results.

Rounding down the number of decimal places can also prevent confusing rounding errors. That is computers often make slight calculation errors when converting to binary and back to the decimal system. This often involves a whole number, followed by a decimal point, and a series of nines. For example, when the correct calculation should result in 2, the computer calculates 1.999999999999.

Calculation devices often have
built-in mechanisms to round down calculated results to a predetermined number
of decimal places, such as six decimal places. The Algebraic Calculator
has such a mechanism, __but it is controllable by
the user____.__ **That is the user can enter a number in
an input box to delineate the number of decimal places needed for a calculation,
with the Algebraic Calculator.** This can be done before the
calculations are started, or after the calculated results have been
obtained.

Interestingly, the number in the
input boxes, to control the number of decimal places, can be equal to or
**less** than zero. That is the number
of decimal places can be rounded down to a negative number. To explain
this I will use the number **1234.123** as an example. When this number
is rounded to one decimal place it is **1234.1**. When it is rounded to
**zero** decimal places it is **1234**, **-1** decimal places it is
**1230**, **-2** decimal places it is **1200**, and **-3** decimal
places it is **1000**.

I created the rounding mechanism
described above, __with the input boxes, for the user to control the number of
decimal places.__ I did this by modifying the conventional Microsoft
Excel formula, for rounding numbers. The conventional formula is as
follows: =ROUND(number, number of decimal places). My
modification involved entering a cell designation instead of the number of
decimal places in the formula. For example, if X is the number, and 28E is
a cell designation, then the modified formula is =ROUND(X, 28E). With this example, cell
E28 is an input box. That is the number of decimal places of the
calculation is determined by the number entered in cell 28E, with this
example.

A generalized version of the above formula is:

=ROUND(number, cell designation). This formula will also function in OpenOffice Calc, and probably in a number of other brands of spreadsheet software. The formula and the related concept have potential utility with many types of calculation devices. The general concept, (input boxes to allow users to control the number of decimal places of calculated results) can be incorporated into the design of any type of calculation device. However, the configuration of the formula as presented above only applies to spreadsheets, or any format that was created by converting a spreadsheet, such as JavaScript.

The Algebraic Calculator has two input boxes to control the number of decimal places, one is for the calculated results of X and Y, and the other is involved with error checking. Both of these boxes are set at six decimal places, which can be increased or decreased, as explained above.

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__Spreadsheet Formulas For Error
Checking__

The algebraic calculator has a built-in checking mechanism to determine if the numbers the user entered are valid for both equations AX+BY=V and KX+DY=W. This mechanism also checks for other types of errors, such as numbers that are out of range of the calculator, rounding errors, and errors that result from certain types of malfunctioning hardware and software.

Note: For most numbers entered into the algebraic calculator, with less than six digits the error will be zero. Errors are most likely to happen when numbers involve many digits, especially if some of the numbers are very small and others are extremely large. As will be explained later, the degree of error is calculated and displayed.

The algebraic calculator checking mechanism functions the same way that a human would check an algebraic calculation, which is by substituting the calculated results into the equations AX+BY=V and KX+DY=W. When this is done, if the substituted values for X and Y equals V for the first equation, and W for the second equation the calculated results are confirmed.

For example, a human would carry out the following to check the calculated results of X=10 and Y=-3 for the equations: 7X+8Y=46 and 6X+8Y=36:

To check the first equation:

7(X=10)+8(Y=-3)=**46**

7(10)+8(-3)=46

70-24=**46. **This result, of 46,

confirms the validity of the calculations.

To check the second equation:

6(X=10)+8(Y=-3)=**36**

6(10)+8(-3)=36

60-24=**36 **This result, of 36,

confirms the validity of the calculations.

The Algebraic Calculator checks the calculated results for X and Y in a way that looks somewhat complex, but it is based on the method described above. However, when software and computers perform even relatively simple processes, computer code is required. In this case the computer code is in the form of spreadsheet formulas, which are presented below with interpretations:

For AX+BY=V the following formulas are used (XR=X and YR=Y):

=ROUND((A*XR+B*YR), E28)

=ROUND(V, E28)

=IF((ABS(D24-D25))>(5/100), "NOT VALID", (D24-D25))

A simplified English translation
of the above is: The computer substitutes the calculated values for X and Y into
the equation AX+BY=V to determine the calculated value of V, which I will call
V_{calculated}. Then the computer rounds the value for
V_{calculated} to the number of decimal places indicated in cell
28. When this has been completed, a formula in another cell, obtain the
value that the user entered for V, which I will call V_{user}.
Then, V_{user} is rounded down to the number of decimal places indicated
in cell 28. In still another cell, the degree of error is calculated by
subtracting the above results, which are: V_{user }**-
**V_{calculated}=ERROR. If the absolute value of the ERROR is
greater then 5/100, the computer display the words: NOT VALID, but if this is
not the case, the computer display the degree of the calculated ERROR.

For KX+DY=W the following formulas are used (XR=X and YR=Y):

=ROUND((K*XR+D*YR), E28)

=ROUND(W, E28)

=IF((ABS(D26-D27))>(5/100), "NOT VALID", (D26-D27))

An English translation of this code is the same as above, except it is based on this formula: KX+DY=W, and the value the user entered for W and the calculated value for W.

Note that the error checking mechanism I created for the algebraic calculator has a controllable rounding mechanism, as described previously. This allows the user to control the number of decimal places involved with the error checking. This is necessary to prevent confusing rounding errors. For example, if the user entered 6 for W, and the correction mechanism calculated 5.99999999999 for W, which might give the false impression of a significant error, to some people. By rounding down the above to ten decimal places, or less, the rounding error is eliminated, and the number becomes 6.

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the Author__

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