Created By David@TechForText.com, ©2010
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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.
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.
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 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.
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.
*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.
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
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,
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.
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.
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.
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.
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.
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.
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.
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.
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:
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.
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.
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