That will shorten he number of symbols significantly. This is interesting, but what about the inverse problem. And therefore easier to remember by a tiny margin. But 3. At least when you talk about rationals. Does pi squared qualify? In that case its an easy to remember 3 digit number divided by another easy to remember 3 digit number divided by a number which is three repeated numbers followed by two repeated numbers followed by a final number reoccuring. Trust me ; its pi correct to 10DP with the last one correctly rounded up.
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Work it out. Hence the great reduction of digit size. He based himself not on fractions, but on powers of e to a multiple of Pi. The whole idea is to be able to re-write that N-digit numerator, or denominator, or power in a way which drastically decreases its digit-size. Base 10 is infinitely inferior to other mathematical notations. We remember patterns far more easily than abstract numbers.
I think you should re-do all of these calculations based on entropy, rather than raw digits. The value of pi that that bits of entropy generates is about 23 bits of entropy — a huge savings. Also remember that each number you add is harder to remember than the last, which is kinda the point of the entropy calculation, so that 15 is a whole lot easier than that Plus, it looks a heck of a lot better as a signature than the number.
There is a gear train in the lathe that moves the thread cutting device a certain amount per revolution. To cut threads at diametral pitch, a pair of the normal gears are switched out with a pair that have teeth in the ratio of pi:n. One machine I am aware of uses gears in a ratio — which is x For comparison, 3. I think you owe Archimedes an apology. Of course I you, anyone can do better. Can you guess what one character I have in mind? Not that I am a fan to that ratio, but just an observation why the approximation lives. Name Please enter your name. Email Please enter a valid email address.
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We can use the natural logarithm and square roots? Are you Jon going ot be in the conference Wolfram will have at the headquarters? The product of a fraction and an integer is a fraction whose numerator is the numerator of the given fraction, and whose denominator is the quotient of the denominator of the given fraction and the integer. The quotient, when one fraction is divided by another, is obtained by inverting the fraction which is the divisor and then multiplying according to Rule The quotient, when a fraction is divided by an integer, is a fraction whose numerator is the numerator of the given fraction divided by the integer, and whose denominator is the denominator of the given fraction.
The quotient, when a fraction is divided by an integer, is a fraction whose numerator is the numerator of the given fraction, and whose de- nominator is the product of the integer and the denominator of the given fraction. In writing a decimal fraction it is convenient to omit the denominator and indicate us value by placing a point. The number of figures, including zeros, to the right of the decimal point are called decimal places It should be noted that when there are fewer figures in the numerator than there ace zeros in the denominator, zeros are inserted to the right hand side of the decimal point, beat eea the decimal pome and.
For example: It is apparent that the position of the decimal point is the factor controlling the numerical value of any given sequence of figures For each place the point is moved to the righr, the value of the decimal fraction is multiplied by 10; and for each place it is moved to the left, the value is divided by 10 Thus, 2 5 becomes 25 when the decimal point is moved one place to the right, and 0 25 when the point is moved one place to the left In the first case, 2.
For every place the decimal point is moved toward the right, the value of the given number is increased ten times, and for every place the decimal point is moved to the left, the value of the given number is decreased ten times. Therefore, the relative values of the various places to the right and left of the decimal point are as follows: W5 VJ 3 -3 o a. Zeros may be added to the numerator and the division continued until the desired degree of accuracy is attained.
The operation of changing a decimal fraction into a common fraction of the same value is not frequently required, but where necessary can be obtained by one of the following two rules. To change a decimal into a common fraction, write the given sequence of figures without any decimal point as the numerator. The denominator is the integer 1 with as many zeros annexed at its right as there are decimal places in the given decimal fraction.
In many cases, the result- ing fraction can be simplified by dividing both the numerator and denominator by the same number. To change a decimal into a common fraction having a specified denomi- nator, multiply the decimal by a common fraction which has both a numerator and denominator equal to the specified denominator. The product is the equivalent common fraction. To add decimals, place the numbers to be added in a column so that their decimal points are in line under one another.
Then add as if they were whole numbers, and place the decimal point in the sum direcdy under the decimal points of the numbers being added. Then subtract as if they were whole numbers, and place the decimal point m the difference directly under the decimal points of the numbers above. Zeros may be inserted between the decimal point and the figures obtained as a product to make the required number of decimal places. The position of the decimal point in the quotient can usually be found by inspection The assumed position of the decimal point can be checked for validity by multiplying the quotient by the divisor to obtain the original dividend In division of decimals, u is often convenient to change the divisor to a whole number by moving the decimal point to the right as many places as there are figures in the decimal The decimal point in the dividend must be moved an equal number of places to the right, counting from the original position If there are fewer figures to the right of the deci- mal point in the dividend than there are to the right of the decimal point in the divisor, annex enough zeros to the righc of the dividend to rake care of the new decimal point.
Such numbers as those mentioned, which have exact square roots, are termed perfect squares. The square root of any number which is not a perfect square, such as 37, 11, etc. The explanation and solution of several problems in square root, given below, indi- cates the procedure in finding the square root of any number, whether it be a perfect square or not. For imperfect squares, the root should be found to one more decimal place than is desired in the answer, so that the final figure of the root may be in- creased or left unaltered according to whether the additional figure obtained in the root is greater or less than 5.
First separate by actual indications as shown, or mentally the num- ber into periods of two figures each, commencing at the decimal point and going both to the right and to the left. The extreme left period may consist of two digits, as in this example, or only one digit as in Example 2.
The extreme right period should consist of two digits, as a zero can always be annexed where an additional figure is necessary to complete the period. Now bring down die second period 90 and annex, not add , it to the re- mainder 2 , thus obtaining , termed the first remainder. This number 4 is placed above the second period and becomes the second figure of the required root. This same number 4 is also placed to the right of the trial divisor, making the true ditisor As before, bring down and annex the next period 25 to this remainder, producing a second remainder of First separate by actual indication as shown, or mentally the number into periods of two figures each, commencing at the decimal point and going both to the right and to the left The extreme left period, as In this example, may consisc of only one digit, bur is nevertheless treated as a regular two digit period; the extreme right period should consist of two digits, as a zero can always be annexed where an additional figure is necessary to complete the period.
ALGEBRA 17 The number of decimal places in the root of a number which is not a perfect square is controlled by the number of periods of zeros annexed at the right of the number. The decimal point in the root will be placed so that there are as many digits to the left of the decimal point as there are whole number periods in the number of which the root is de- sired. In the example, the decimal point in the root will be located so as to provide two figures to the left of the decimal point, since the given number contains two whole number periods.
Now bring down the second period 89 and annex not add it to the remainder 1 , thus obtaining , termed first remainder. Step 3. Find how many times this trial divisor 4 is contained in the first r ema inder without its right-hand figure 9. Obviously, 4 is contained in 18 four times. This same number 4 is also placed to the right of the trial divisor, making the true divisor As before, bring down and annex the next period 32 to this remainder, producing a second remainder of It is obviously larger than the second remainder , and cannot be subtracted from it with a positive result.
Three is therefore not the third figure of the root, and this step must be repeated using a smaller number than 3. Whenever the product of the trial divisor and the last placed figure in the root exceeds the corresponding remainder, the root number chosen is too great, and a repetition of this step in the process is necessary using a smaller figure Step 5.
Continue the process until all periods of the number have been brought dov.
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The number 5 therefore, represents some value actually greater than 5. Con- sequently, if the answer is to be written to two places to the right of the decimal point, it will be expressed as Example 3: V It sometimes appears that a step in the solution for the root is in error, as for example in the finding of the second figure of the square root of This is impos- sible as there can never be a two-digit number placed in the root as one of the figures, and 9 is therefore the largest number which it is possible to use.
To obtain the square root of decimal fractions, first move the decimal point an even number of places to the right so that the number is expressed as some whole number between 1 and Find the square root of this number and then move the decimal point half as many places to the left as it was moved to the right in the first place This is the square root of the given decimal fraction. This will be the square root of the given number.
The rule as first stated is recommended in preference to this latter method since slightly less labor is involved. If only the numerator of the fraction is a perfect square, then the logical procedure is not so evident as in the previous example. However, it is recommended that the square root of the numerator and the denominator be separately found, the square root of the perfect square being found mentally, and the square root of the other term being found by slide rule, logarithms, or by arithmetical calculation depending upon the degree of accuracy required.
The value of the resulting fraction is then found by dividing the numerator by the denominator. This is true because the product of any number of positive numbers is positive in sign, and the product of any even number of negative numbers, by themselves or together with positive numbers, is positive in sign. A check on the accuracy of the absolute value of the root found from any calculation may be accomplished by squaring the computed value to see if it produces the original number.
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In the special case where a number ending in five is to be squared, a useful method is available which simplifies the necessary arithmetical work. The steps in the application of this method are: Step 1. Write the given sequence of digits omitting the 5. Call the resulting number V. Step 3- Annex 25 to the right of the product obtained in Step 2.
The number thus obtained is the square of the given number. However, in many cases the word power is extended to include fractional exponents. This usage of the term root is not to be confused with the root of an equation defined elsewhere. Exponents may be integers, fractions, or a combination of the two, or they also may be algebraic expressions including the logarithms of quantities. Regardless of nature, they are placed to the right and above the term which they affect. Where possible' their size should be less than the size of the term itself.
Because of their great importance in algebra and in logarithms Section IV the appli- cation of these rules should be thoroughly understood. An expression within parenthesis, brackets, braces, a radical sign, or overscored or underscored with a horizontal line is to be treated as a single quantity. This rule is frequently applied when dealing with ex- ponents, but is equally important in all algebraic operations.
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The product of two or more unlike quantities, each having the same exponent, is equal to the product of these quantities with the same exponent as the exponent of the factors being multiplied together. The quotient of two like quantities which have either like or unlike exponents is equal to this quantity with the difference between the exponents of the dividend minus the divisor as an exponent.
The quotient of two unlike quantities, each having the same exponent, is equal to the quotient of these quantities with the same exponent as the exponents of the quantities being divided. The fraction can then be simplified according to the rules applying to multiplication. Rules 29 and Such a fraction excludes any terms in addition or sub- traction. It is common knowledge that any quantity divided by itself is one. Tilings equal to the same thing, or equal things, must be equal to each other.
Such forms may be classed as either expressions or equations. An equation is defined as a statement of equality be- tween two expressions, that is, that two expressions are equal. An equation that is true for only certain values of the unknown involved is termed a conditional equation. At the opposite extreme from identities stand descriptions which ate not true equations be- cause they cannot be satisfied by any number. When the word equation is used with- out further qualification, it is a conditional equation that is implied. Equations are also classified according to their degree.
The degree of an equation containing only one unknown is the same as the numerical value of the largest exponent of that unknown The equation x — 4 is an equation of the first degree. Such equations are also termed linear equations since all of its plotted points will fall on a straight Jme. An equation containing x 3 is called a cubic equation or an equation of the third degree Whenever an unknown quantity is to be represented in either an expression or an equation it is customary to assign one of the last letters of the alphabet, as x, y, or z to indicate this quantity Numerical quantities of known value are sometimes repre- sented by the first letters of the alphabet, a, b, c,.
This form also presents less possibility for misin- terpretation of facts. The greatest advantage of the algebraic representation is that it allows a rapid and precise solution of the problem in question. Rules cannot be definitely set forth in the writing of equations as they can in the methods of solution.
To write an equation, the conditions of the problem must be understood, and also, the person writing the equation must know the signs and symbols —the algebraic language. This second requisite will be acquired with the study of the subjects throughout the text. The following suggestions may be helpful as. Step 1. Read carefully the statement of the problem as given in words.
Determine what is the unknown quantity or quantities, and represent these by letters of the alphabet, always using the minimum number of letters. For instance, if one term B is twice as large as another term A, it is represented as 2 A. If all of the unknown quantities cannot be expressed in terms of one of the unknowns as suggested above then as many equations must be written, as there are different letters used to represent the unknowns.
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Each of these equations must represent a separate, independent relation- ship. Consequendy erms roots, variables, and unknowns are used interchangeably. It is also important to know that the number of values which can be found to satisfy a smgle equation of one unknown will be equal to the highest power of the unknown appearing in that equation. This one equation is solv- able and two answers may be expected as a result Unforrunatcly , the detailed description of many operations used in solving equations makes it appear that the labor involved is considerable, when actually, the method is surprising!
Valid Operations with Equations The rules and operations 1 to 35 inclusive have been described as applying to expressions However, since equations are statements of equality between expressions, these same rules are also used in the solution of equations In addition, certain opera- tions are usually required in which the entire equation and not just one of its expres- sions arc involved Since equations consist of two sides separated by the equal sign, it is possible to define these additional rules and operations as applying to both sides of the equation, even though one of these sides is zero This emphasis on both sides helps to eliminate the possibility of applying a rule or operation to only one side of an equation, and leav ing the other side unaffected.
In the solution of equations, the following operations may be performed without destroying the equality of the relationship.
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Transposing and changing sign is often erroneously applied to equations in which the various terms appear as products or quotients. The given equations were not in addition and subtraction, but involved division and multiplication respectively, and hence the rule does not apply. Both sides of an equation may be multiplied by the same or by an equal quantity. Both sides of an equation may be divided by the same or by an equal quantity. Zero can never be used as a divisor as demonstrated on Page Both Sides of an equation may be raised by any identical power.
In some cases this operation may cause additional roots to be formed which will not check when substituted into the given equation. This is explained in the paragraphs ennded Radical Equations. Any identical root may be extracted from both sides of an equation. The odd roots of any number arc positive, and the even roots are positive 01 negative according to Rules 4 and 5. The logarithm may be taken of both sides of an equation. In calculus, the derivative and the integral of both sides of an equation may be taken.
Practical Check on Algebraic Operations Whenever the validity of an operation in the solution of an algebraic equation is questionable, it is a good practice to set up a simple example similar to the problem involved, but using small integers in place of more complicated terms. By using values such that the correct answer is apparent by inspection, the questionable operation per- formed in the simple example will set forth the procedure necessarily applied in the original more complicated equation.
The several examples shown below are but a few of the many applications of this method. In choosing integers for substitution, the use of the values 1 and 2 is to be avoided unless it is definitely known that these values will not satisfy an operation which for any other integers would be an erroneous process.
This operation is known by some as: In a proportion the product of the mean quantities is equal to the product of the extremes. I his relationship may be referred to as the diagonal rule, or simply that the cross- products of a proportion are equal. See page This simplification is frequently used to change the denominator to an integer and thus a more comenient divisor Is 6-v — 4 ,. In explaining the various methods of solution the term unknown and the expression degree of an equation are frequently used.
These have been previously defined. This term refers to the prefix of the unknown term written to indicate how many times the unknown is to be taken as a factor; thus 5x means 5 times x or the quantity 5x. As in this example, the coefficient is usually a numerical value although not necessarily an integer. It is sometimes represented by one of the first letters of the alphabet as a, b, c.
Other numerical values in an equation unattached to unknowns are called constant terms. For example, 5,v- -f lO. The name constant is appropriate to such terms since numerical values are of constant or unvarying magni- tude. In this same equation the coefficients 5 and 10 are also constants from this viewpoint, but should be considered as coefficients inasmuch as they are attached to unknowns. Equations varying from the simple to the complex are sometimes referred to as polynomials, meaning many numbers.
Another word, function, is also used to mean an equation. This latter term will be used in preference to the word polynomial when a substitute term for equation is desired. This will allow the meaning of polynomial to be restricted to the use of expressions only. The following methods of solution are applications of the rules and operations de- scribed in the preceding pages. In many cases the answer or answers are obtained by the performance of a single operation, but to the other extreme, a great deal of computation may be required.
Furthermore, for certain types of problems, the results may be obtained by following two or more distinct procedures. The applications and relative merits of each of these solutions are not all at first evident. Where a choice in the method of solution exists it is only by experience that the most con- venient and practical form or forms will become apparent. There is no fallacy in this as it is possible to use either side for these terms. In the example above this happens to be the right-hand side, and by so doing one step in the solution may be omitted.
The presence of either a — sign or a negative factor preceding an expression enclosed within parentheses serves to reverse the sign of each term of the expression when the parentheses are removed The procedure involved in the removal of paren- theses is based directly on the rules governing the products of positive and negative numbers. The operation as described above is in effect the same as multiplying the original equation by a quantity known as the least common multiple, abbreviated, t C M The value of the LC.
M is the same as that of the L C D, and is found by an identical method. Find the prime factors of each of the denominators when each fraction is represented in its simplest form See second example. Tind the product of all the different prime factors, using each factor the greatest number of times it occurs in any one denominator.
This is the L C M. In determining the L. For any given equation these values are usually different in absolute value and often times in algebraic sign as well. But, in many cases they may have identical values in every respect. Regardless of the nature or number of the answer obtained, the values resulting from the solution of the equation should satisfy the given equation when each of such numbers is substituted into the equation in place of the unknown quantity.
If the answer obtained fulfills this requirement, it can be called a root of the equation. The distinction between a root and an answer is that a root is an answer known to be correct for the given equation. Many answers, even though determined by correct algebraic methods, will not be roots of the given equation. The roots of the equation x 2 — llx 4 The solution of this equation is accomplished by methods considered later, but the validity of the roots can be easily ascertained. An inspection of the first fraction of the given equation reveals that it is not in its simplest form since 4x — Note that the unknown quantity in this case is in the denominator, and that this is the only place where it occurs The importance of using each fraction in its simplest form when determining the LCM for the given equation is apparent from the example above From this it might be implied that the use of the LCM is essential to the solution of equations im oh mg fractions This is not true, however, because in most cases the roots of the equation resulting from clearing the original equation of fractions are the same as the roots of the original equation, whether the L CM.
How- ever, the elimination of fractions from an equation by multiplying each term of both sides by the LCM is the most direct solution and should be employed in all cases. Both methods are correct Equations taking the form of proportions described in the paragraphs titled Ratio, Proportion and Variation are so frequently encountered that it is advisable to show ALGEBRA 35 by an example the most practical solutions. The position of the unknown, x, may be in any one of the four possible positions. This relationship may be referred to as the diagonal rule. This operation in Geometry is stated: In a proportion the product of the mean quantities is equal to the product of the extremes.
Rule 39 An equation containing only one unknown with that unknown occurring only once and with an exponent other than unity either a root or a power can be solved by changing the entire equation to the desired power of the unknown. That this is true is apparent if the solution of the same problem by another method is examined. Many equations include a number of terms arranged to indicate that more than one operation is to be performed.
An unlimited number of such examples can be shown, but only too arbitrarily chosen types will demonstrate the fact that only basic principles included in Rules 1 to 45 inclusive are involved. Rule 43 The second example is of a more complex nature. The solution as given is not necessarily written in its briefest form, as it is intended to clearly show tr'ch operation involved Rule 11 , 38 ALGEBRA 37 This solution may be completed by alternate methods.
Trial and error methods are also used in establishing the factors of an equation as explained in the paragraphs entided Factoring. Trial and error consists of assigning an arbitrary number to the unknown and deter- mining, by substitution, if the assumed number is a root of the equation.
A root of an equation is a number which when substituted for the unknown quantity will make the two sides of the equation equal, that is, it satisfies the equation. The work involved in the solution of an equation by this method is reduced if the given equation is first simplified. Furthermore, it is advisable to transpose all terms of the equation to one side of the equal sign with the other side of the equation be- coming zero.
For the purpose of explanation, and not necessarily the most logical procedure, the value of the unknown is first assumed to be 2. This reasoning is not always valid as may be seen by an examination of the curve of the second example in the paragraphs entitled Graphical Solution, page The graphical solution has an additional benefit in that it indicates approximately the value of any irrational roots which may exist, a feature which be- comes exceedingly laborious using trial and error.
Graphical Solution There is no elementary algebraic method of solving equations of the third or higher degree unless some of the roots may be discovered by trial and error or by factoring, as explained in the paragraphs bearing these titles The application of either of these methods from a practical viewpoint is limited, and altogether impossible, in the case of factoring, if irrational roots are involved However, any equation in any degree, as ALGEBRA 39 long as only one unknown is involved, may have its real roots approximated by a simple although somewhat tedious graphical method.
The term real root as introduced here is the same as the term used before simply as root. The term rsal differentiates such roots from imaginary roots. By an imaginary root or an imaginary number is meant a number involving the square root, or any even root, of a negative number. For ordinary mathematical purposes it is assumed that the square root of a negative number does not exist. For actual facts, investigate complex numbers in advanced algebra. If any of the roots of an equation are imaginary, they will not appear on a graph plotted of that equation.
All algebraic equations can be represented by a line or a curve plotted on a plane. Such a line or curve is established by first finding a number of points, each of which is known to be one point through which the curve of the equation must pass. A number of such points are plotted and through these the curve or line may be drawn which is called the curve or the graph of the equation. In graphical representation, the plane upon which a point may be located is divided into four zones, or quadrants, which are numbered in a counter-clockwise direction with the upper right-hand quadrant as number one.
The lines separating the quadrants are termed axes; the vertical line is designated as the Y-Y axis, and the horizontal line as the X-X axis. The point of intersection of these two axes is termed the origin, or zero point, and from this point all measure- ments of distance are made. Distance along, or parallel to, the X-X axis is termed the abscissa, and is considered positive if measured to the right of the Y-Y axis and nega- tive if measured to the left. Distance along, or parallel to, the Y-Y axis is termed the ordinate, and is positive if measured above the X-X axis, and negative if measured below.
If a point lies in a plane, its location may be determined by two measurements of distance, or coordinates, one showing its distance from the Y-Y axis, abscissa , and the other its distance from the X-X axis, ordinate. It is customary to write the coordinates of a point in parenthesis with the abscissa first. The graphical solution of an equation of the type named above consists of arbitrarily assigning numerical values to the unknown and computing the corresponding values of the equation The values assumed for the unknown are plotted as abscissas and the corresponding values of the equation as ordinates Each pair of values so determined establishes a point and through a number of such points a smooth curve can be drawn.
The intersection of this curve with the X-X axis establishes a value of the abscissa, which is a root of the given equation In solving equations by the graphical method, the term function is used to replace the word equation, and the term tunable is used in place of unknown. This termi- nology is appropriate inasmuch as the equation is a function of the unknown since it depends on this latter quantity for its value To find the value of an equation it is necessary to transpose ail of its terras to one side of the equal sign Consequently, the term function implies an equation in which all terms are arranged in this manner.
In the study of functions the unknown is often called the variable, since from this point of view the problem is not so much the finding of a value for the unknown as it is the study of the changes of a variable quantity Thus the terras function and variable more clearly express the relationship of the unknown to the entire equation than the words formerly employed The foregoing explanation may be condensed into several distinct operations and thus clarify the acrual procedure to be followed. Step I. Arrange all terms of rhe given equation on one side of the equal sign and simplify the resulting expression, now termed the function.
Step 2 Arbitrarily assign different values for the variable and determine the corresponding values of rhe function Plot a number of points corre- sponding to the number of pairs of values determined in this manner, using the value of the variable as abscissa and the corresponding value of the function as ordinate Obviously, if the value of the function be- comes zero with some assumed value for the variable, then that value of the variable is a root of the equation Step 3.
Through the number of points thus established draw a continuous smooth curve. This fact may be verified by substituting the value found into the given equation. The graphical method of solving any equation in any degree as long as only one un- known is involved, is nothing more than a pictorial representation of the trial and error solution in which the value of the equation, with all terms on one side of the equal sign, becomes zero upon assuming a value of the unknown corresponding to a root.
The examples on page 42 illustrate the method of graphical solution. The equation used for the first example is of the second degree and has roots which are small integers. Its roots have been already found by trial and error. In the second example the usefulness of the graphical solution is more apparent. In both examples the total number of possible roots have been found because in each case the curve crosses the X-X axis the number of times which corresponds to the degree of the equation.
Con- sequently no imaginary roots exist for these selected equations. It might be suggested at this point that a combination of the trial and error method together with the graphical method is a convenient means of finding either fractional or irrational roots. First use the trial and error method to find the approximate or boundary roots between which the precise root is known to exist, and then, by the graphical method, plot only this portion of the curve using several points adjacent to the axis.
The greater the number of points employed, the more exact the curve is represented, and hence the more accurate the approximation will be. In any graphical representation or solution, the scale of the drawing will be the con- trolling factor, not considering errors in solution, in the accuracy of the result.
Obvi- ously the larger the scale of the drawing, the more accurate can be the solution. For this reason two alternatives are employed to effect more accurate results. One of these alternatives was applied in die solution of example 2, and consists of using different scales for ordinate and abscissa in order that it may be easier to accu- rately estimate the point at which the curve crosses the X-X axis.
The second alternative is to plot a second curve, or set of curves, involving only those portions of the original curve which cross the X-X axis. These portion-curves should be drawn through several points located adjacent to the X-X axis, and to a scale several times as large as that of the original continuous curve from which the approximate points of intersection were discovered.
The application of these two expedients introduces no operation other than those already explained. There are several characteristics of the curves of equations which should be kept in mind when solving such equations by the method of plotting successive points.
These are not peculiarities of all equations, but must be recognized when encountered. The principle involved in graphically solving algebraic equations is that if the curve of the equation lies above the X-X axis for one value of the unknown and below for an- other value, there must be a root of the equation somewhere in between. This method is very effective when used in conjunction with certain methods of differential calculus. I low ever, when the only means of obtaining the curve is to plot successive points, the procedure is sometimes very misleading.
This is true because it is not always possible to determine the exact shape of the curve near the critical values. Not demonstrated in the previous examples is the occurrence of multiple roots de- ALGEBRA 4 3 fined as wo or more identical roots appearing in the same equation. If any equation containing multiple roots is plotted, the curve will be tangent touch but not cross to the X-X axis at a point whose abscissa is the value of the multiple roots.
For every such point of tangency it should be considered that two roots of the equation have been determined. Another factor of importance concerns the plotting of functions in which one or more fractions exist with the variable included in the denominators of one or more of such terms. In such cases the curve will not be continuous, but will break if such values of the variable are assumed which will make the denominator become zero with the numerator some other value.
The following sketches illustrate examples where the graphical method of solution either breaks down completely or must be used with great caution. In Case 2 it is tangent to the axis, and in Case 3 it crosses the axis at two near-by points. Case 1 can be detected only by advanced methods such as are used in solving the equation of Example 1 shown below. Similarly, Case 2 can be detected only by ad- vanced methods, unless the point of tangenq' is located by guessing the exact value for the variable. This is unlikely unless some of the roots are small integers. The condition represented by Case 3 can be detected by plotting points sufficiently close together, but is easily overlooked in practice as can be seen in Example 2 below, that is, it may be mistaken for Case 1, and two roots thereby lost.
In Case 4 the dotted curve is easily mistaken for the true one so that only one root is believed to exist when actually there are three. This can also be avoided by plotting points sufficiently close together. The curve in Example 1 misses the axis by approximately. The curve in Example 2 crosses the axis between 1. The first step, and one of several operations involved in the method of solving an equation by factoring, consists of transposing all terms to one side of the equal sign. The terms are at the same time arranged so that they are in an order in which the factors may be discovered by inspection in some simple cases, or by some method of computation for more complex problems as explained later.
Each factor thus found is further factored, if possible, so that the prime factors are obtained The prime factors are then written m place of the given equation producing an equivalent though different appearing relationship. The validity of these roots is established by substituting them into the given equation. From these relationships an important rule can be formulated which, for convenience, is stated in the reverse order to that just given ALGEBRA 45 The prime factors of many algebraic equations of the second degree are found by the trial and error method simply by finding two factors which when multiplied together will produce the given equation as a product.
Since the factors of such equations are usually polynomials it is essential to understand a method of multiplying polynomials together with a minimum of effort. The rule is as follows: The product of two polynomials is equal to the algebraic sum of the products obtained when all terms of one polynomial are multiplied by each term of the other polynomial. The application of this rule is demonstrated by two equivalent methods with the recommendation that the first form be employed until experience justifies the use of the second method.
This simplifies the operation of collecting terms. To facilitate solution only, and not a necessary operation, each polynomial is reduced to its least common denominator. Since the denominator of all products will then be 60 they are not written down in each instance.