History of Math for the Liberal Arts CHAPTER 6. Two Great Achievements: Logarithms & Cubic Equations

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1 1 2 4 History of Math for the Liberal Arts 5 6 CHAPTER Two Great Achievements: Logarithms & Cubic Equations Lawrence Morales Seattle Central Community College 2001, Lawrence Morales; MAT107 Chapter 6 Page 1

2 TABLE OF CONTENTS TABLE OF CONTENTS... 2 PART 1: Introduction and Non-Western Math, and Evolving Calculating Methods... 4 Historical Background... 4 Multiplication from India... 5 Gelosia Multiplication... 6 PART 2: The of the Cubic and Quartic Equations Historical and Mathematical Background Cardano and His Gang of Italian Algebraists Ars Magna and the of the Cubic... 1 Solving the General Cubic Equation The of the Quartic Equation and Beyond PART : Modern Calculations Decimal Fractions PART 4: Napier and The Emergence of Logarithms The Idea and Use of Logarithms The Idea Behind Logs Rule of Logs One Step Closer to the Tables The Log Tables... 4 Larger Numbers and the Log Tables... 6 Logs of Very Large Numbers... 8 The Antilog Tables... 8 The Evil Twins Meet Each Other Finally Complex Calculations PART 5: New Calculating Devices The Slide Rule Napier and Other Calculating Devices PART 6: Appendix A of the Quartic PART 7: Appendix B Log and Antilog Tables PART 8: Appendix C Napier s Rods , Lawrence Morales; MAT107 Chapter 6 Page 2

3 PART 9: Homework Multiplication with the Gelosia Grid System Checking s of Cubic Equations Using Cardano s Formula on Depressed Cubics Using Cardano s Formula on Non-Depressed Cubics Stevin s Notation Basic Logarithm Rules Using the Log Tables Using the Antilog Tables Using Log and Antilog Tables to Do Calculations Applications of Logarithms Writing Blank Gelosia Grid...Error! Bookmark not defined. PART 10: Endnotes , Lawrence Morales; MAT107 Chapter 6 Page

4 PART 1: Introduction and Non-Western Math, and Evolving Calculating Methods Historical Background After the Greeks, a variety of cultures spread previous mathematical accomplishments and also created many of their own. For example, in the last three centuries B.C.E., China worked on ideas of square and cube roots as well as methods of solving systems of linear equations. They also tackled mathematical surveying techniques during this period. In Egypt, Apollonius developed a theory of conic sections. From 0 to 400 C.E., Ptolemy made inroads into astronomy while Hypatia, the famous female mathematician, made a name for herself by commenting and lecturing on the work of Apollonius and by being a gifted and inspiring teacher of mathematics and philosophy. From , the Italian, Boethius, contributed to the field by writing arithmetic books that were used by Europe during a time when mathematics in that part of the world was in decline. His book, Arithmetic taught others about Pythagorean number theory. On the other side of the world, the Mayan numeration system was developed and used for astronomical purposes. And in India, a major mathematical movement emerged: Aryabhata advanced trigonometry, while Brahmagupta made major contributions in mathematics and astronomy. It was also during this time that the Hindu Arabic decimal place value number system began to emerge and gain popularity. From , India continued its work as it developed algebraic techniques. In what is modern day Iraq, Al Khwarizmi wrote an influential text on algebra whose title actually gives us the word algebra. (The title was Hisab al-jabr w al-muqabala. ) Other Islamic mathematicians were also hard at work, not only preserving and translating ancient Greek texts, but making their own advances, especially in the area of algebra. During this time, Spain became a passageway for the Hindu Arabic numbers into Europe. From , Islamic mathematics continued to develop with work on what is now known as Pascal s triangle, found geometric solutions to certain cubic equations (important in this chapter), and explore sums of powers. In India, Al Biruni advanced spherical trigonometry, while in China, Pascal s triangle (as it is now called) was used to solve equations. In Spain, Arabic works were translated into Latin, which would be important in the coming resurgence in European mathematics. Toward that end, Leonardo of Pisa advocated the use of Hindu Arabic numbers. In Italy, the rich world of Islamic mathematics was introduced and began to spur interest. From , major contributions continued to flow from the Islamic world, China, and India. In England, new algebra and trigonometry texts emerged. In France, Viète pushed a new decimal fraction system. During this time, Copernicus proposed a new heliocentric theory that would greatly affect how mathematics developed from that time onwards. In Italy, a group called the Italian Algebraists conquered the problem of finding an equation for solving cubic (third degree) and quartic (fourth degree) polynomial equations, which we will study in more detail later in this chapter. 2001, Lawrence Morales; MAT107 Chapter 6 Page 4

5 Many of these intermediate topics are worthy of their own independent treatment. Due to time considerations, we will look at just one interesting contribution by Bhaskara. (Future versions of this text will have more information on Islamic, Hindu, and Chinese mathematics). Later, we will look at two major achievements in the history of mathematics. The first topic we will explore in more detail is the algebraic solution of cubic and quartic equations. The second achievement we will examine is the set of tools that were developed which greatly simplified the process of doing long, complex computations. These tools would enable mathematicians and scientists to make great inroads into their fields of studies during the 1600 s and 1700 s. Specifically, we will focus on the invention of logarithms by John Napier and the dramatic impact they had on the mathematical landscape. Multiplication from India In the 12 th century, the Indian mathematician Bhaskara (also known as Bhasharacharaya in India) represented the peak of mathematical knowledge. He was the head of an astronomy observatory at Ujjain, which was the prominent center for mathematics in India at the time. He had a thorough understanding of the number 0, negative numbers, and methods of solving equations centuries ahead of the Europeans. He established his reputation (in part) based on a work titled Lilavati ( The Beautiful ), which had 1 chapters and covered topics in arithmetic, geometry, and algebra. One of the interesting contributions he included in his work was a proof of the Pythagorean theorem. He started with a square that was cut into four triangles and one square, and then rearranged them to create two squares that were positioned immediately next to each other. His picture was accompanied by the simple phrase, Behold! Think About It Why is this a proof of the Pythagorean Theorem? 2001, Lawrence Morales; MAT107 Chapter 6 Page 5

6 This appears to have been inspired by the Chinese and one of their own proofs of the Pythagorean Theorem. Recall the hsuan thu from the Pythagorean chapter. You can see the corresponding inner squares embedded in the middle of the larger squares with side c. This smaller square gets moved into the upper right hand corner of the two larger squares that are positioned side by side, above. Gelosia Multiplication In Lilavati, Bhaskara also provides five different methods for multiplication. One of these is interesting because it later emerged in the Middle Ages and was eventually adapted by the inventor of logarithms, John Napier. It is now called the gelosia method of multiplication. The word gelosia means lattice or grating. In this method a grid is set up, several simple multiplications are placed into the grid, and then 161 diagonals on the grid are added together to get a final result. For 162 example, to do , a three by three grid would be drawn 16 with the two numbers to be multiplied written on the top and right 164 side From the first picture (left), which looks a lot like a grid or lattice, 167 you can see where the method gets its name. To fill in the grid, you 168 take the number in a column and the number in a row and multiply 169 them. You place the result in the square where that row and column 170 intersect each other. For example, column 8 and row give a 171 product of 24, so 24 goes in that square. Note that the tens digit 172 goes to the left of the diagonal of that cell 17 while the ones goes to the right of the 174 diagonal. If the product of a column and 175 row is less than ten, we put a 0 in the tens 176 position all spaces should be filled in to 177 avoid confusion. The picture (left) shows 178 all the cells filled in This method allows you to do all the multiplication at once it s only after this is done that we move to addition. The addition is done along diagonals. Starting in the lower right corner, which represents the ones place, we add up all numbers in that diagonal. If we get more than ten for the sum, we carry any groups of ten into the next diagonal up. Hence, in the second diagonal, we have 8+2+2=12, so we carry 1 up to the next diagonal and keep the 2 (just like when we add vertically). The third diagonal has a sum of = 19. Again, that 1 will carry up and the 9 is kept and recorded for that row. Continuing in this manner will produce a digit for each 2001, Lawrence Morales; MAT107 Chapter 6 Page 6

7 diagonal and we can read the final answer by starting from the upper left and reading down and around the corner. In this case, we get the result that =209, Let s compare this with a modern method of multiplication that is taught in U.S schools (but not everywhere). With this method, each digit in the first number is multiplied with each digit in the second. However, carrying must take place for each multiplication where more than 10 is produced Any carrying leftovers must be added to the next product. Thus, multiplication, carrying, and addition are all intermingled during the 197 multiplication process. Finally, everything can be added up, assuming that zeros have been put into the proper locations so that all the place values line up correctly. Example 1 : Think About It How are the two systems of multiplication alike? How are they different? What are the advantages and disadvantages of each? If you showed each to someone unfamiliar with either method, which do you think would be easier to figure out? Use the gelosia method to calculate For this product we need a 4 by grid. Multiplying first and then adding gives: So our result is 1,670, , Lawrence Morales; MAT107 Chapter 6 Page 7

8 Check Point A Multiply : Example 2 : See endnotes to check your answer. 1 Use the gelosia method to compute This method works perfectly fine when we have numbers with fractional parts. We simply multiply as usual, ignoring the decimal point for the time being. Think About It Why does moving the decimal point three places fix everything in this example? We get a result of 109,745, which is obviously too large for Since we have three decimal places to compensate for, we simply move the decimal point three places to the left. We get ,which is correct. Think About It How would you do 55 4 with this method? This method found it s way into 15 th century Europe (via Islamic mathematicians and their arithmetic books) when Luca Pacioli ( ) included it in his popular book on arithmetic. His comment on the origin of the method s name is interesting not only because it tells us where the word may come from but also because of the insight it gives into a part of the culture of Italy at the time. 2001, Lawrence Morales; MAT107 Chapter 6 Page 8

9 By gelosia we understand the grating which is the custom to place at the windows of houses where ladies and nuns reside, so that they cannot be early seen. Many such abound in the noble city of Venice. 2 The method did not survive for long after the fifteenth century. When printing presses were invented it is likely that the method, with all of its grids and lines, was too demanding on the first printing presses, so it gave way to other algorithms that were more easily typeset. Think About It Our word jealousy comes from the word gelosia. Can you think of a reason why these two words would be connected? 2001, Lawrence Morales; MAT107 Chapter 6 Page 9

10 PART 2: The of the Cubic and Quartic Equations Historical and Mathematical Background For many hundreds of years, mathematicians had methods of solving quadratic equations. Eventually, they developed techniques that are equivalent to the modern quadratic formula. 2 Today, we know that if we have an equation of the form ax + bx + c = 0, we can find the solutions of this equation algebraically, if they exist, with the quadratic formula: b ± b 2 4ac x = 2a By finding the solution algebraically, we mean that we can find numbers that satisfy the equation by using the only basic operations of addition, subtraction, multiplication, division, powers and roots. Various mathematicians had some very creative solutions to quadratics that were not algebraic solutions, but they were valid nonetheless. However, there was something very alluring about finding the equivalent of a formula for these kinds of equations. This allure led many mathematicians to look for solutions to other kinds of polynomial equations. In particular, the third and fourth-degree polynomial equations and their solutions were pursued long and hard by mathematicians. The algebraic solution of the cubic and quartic equations is one of the great highlights in all of the history of mathematics. A cubic equation is one of the form: 2 ax bx cx d = 0 The highest power of x is three. A quartic equation has four as the highest power of x and has the form: 4 2 ax + bx + cx + dx + e = 0 By 1500, algebraic solutions to third and fourth degree equations had not been found, however, despite valiant attempts to do so. In the 12 th century, al Khayyami had a method of solving an equation of the form x + cx = d, but his method rested on seeing the equation as an equation between solids. His solution was very geometric and today feels very Greek in nature. Al Khayyami actually examined 14 different kinds of cubic equations, described the physical objects needed to solve each one, and then proved that each solution was correct. But this approach was very different than the kind that was pursued later in history. Mathematicians were not satisfied with a geometric approach; they wanted one that was more algebraic in nature. 2001, Lawrence Morales; MAT107 Chapter 6 Page 10

11 294 Cardano and His Gang of Italian Algebraists In 1494, Luca Pacioli, an Italian mathematician wrote a book called Summa de Arithmetica. In 297 this work, Pacioli discussed the solution of linear and quadratic equations. In this book, he 298 started using the word co, short for cosa (thing) to represent an unknown quantity. This was an 299 early version of our symbolic algebra system where a single letter, typically x, stands for an 00 unknown quantity. Pacioli also discussed the challenges of solving the cubic equation with an 01 algebraic approach and decided that it was probably NOT possible to do. In the next century, 02 however, many of his Italian counterparts did not share his opinion and attacked this problem 0 with great vigor At the University of Bologna, Scipione del Ferro ( ) ignored Pacioli s opinion and 06 discovered a formula for the solution to a cubic of the form: x + mx = n Pacioli did not publish his result. Instead, he kept it secret! To understand why he did this, it is 11 worthwhile to understand the Renaissance university of the time. This was a time when the 12 modern implementation of tenure did not exist. Jobs at universities were not secure like they are 1 today under tenure. To keep a post, you had to have not only political influence and the ability 14 to shmooze the appropriate people, but you also had to have the intellectual force to withstand 15 public challenges to your post These public challenges could occur at any time. In these scholarly battles, the current holder of 18 an academic post and his challenger would meet in public and match wits against each other. If 19 the challenger won, he would bring public humiliation to his counterpart, who would often have 20 to resign his post to his challenger. This was not a positive development in one s career! A new discovery like del Ferro had found was something to save in case of a challenge. If an 2 opponent appeared on the scene and had a list of problems to pose, del Ferro felt confident 24 enough that he could get at least some of them correct. On the other hand, if he was reasonably 25 sure that he was the only one with a solution to a cubic, he could counter his challenger s list of 26 problems with a smattering of cubic equations to solve, thereby 27 giving him a decent chance of surviving the challenge As it turned out, del Ferro never needed to use his secret weapon. It is 0 reported that just before his death he passed on his solution to one of 1 his students, Antonio Fior. Unfortunately, Fior was not as prudent as 2 his master. Upon hearing that another mathematician, Niccolo Fontana 4, had boasted that he had found a solution to another form of 4 the cubic equation ( x + mx 2 = n), Fior immediately challenged 5 Fontana to a public contest. (Fontana was and is currently also 6 known as Tartaglia the stammerer due to the fact that he was 7 unable to speak clearly. Fontana had suffered a sword wound to his 8 face in 1512 when a French soldier attacked his hometown.) 9 Tartaglia 2001, Lawrence Morales; MAT107 Chapter 6 Page 11

12 Tartaglia sent Fior a list of 0 varied problems. Fior, on the other hand, sent Tartaglia 0 cubic equations, all of the form x + mx = n. As the story goes, Tartaglia worked furiously on these problems, presumably with his previous knowledge of how to solve equations of the form 2 x + mx = n available to help him. On February 1, 155, Tartaglia cracked the solution and was able to solve all 0 of Fior s problems. Fior, of course, could not solve all of Tartaglia s problems as they were chosen more carefully. Gracefully, Tartaglia relieved Fior s obligation as the loser to shower Tartaglia with 0 banquets. However, the damage was done. Fior is no longer remembered except for the foolishness he displayed during this challenge. As word spread that Tartaglia had solved the solution to the cubic, it eventually reached the ears of Gerolamo Cardano 5 ( ), one of the most interesting characters in the entire history of mathematics. Cardano was a doctor by trade, at one point serving the Pope, but Cardano also worked extensively on mathematics. Cardano s own autobiography, De Vita Propria Liber (The Book of My Life), gives us some idea of this man s personality. He was a man who was consumed with superstition and tragedy. There is much that could be written about him, but readers are encouraged to do some basic research and reading to get more information. (See Homework Problem (85) for that opportunity.) Cardano Cardano wrote to Tartaglia and asked him for the solution to the cubic equation. Tartaglia, of course, initially refused. Why would he give away such a valuable secret? But Cardano did not give up. He continued to write Tartaglia until he wore him down. Finally, on March 25, 159, Tartaglia revealed the method to Cardano in the form of a coded cipher after Cardano agreed to take the following oath: 6 I swear to you by the Sacred Gospel, and on my faith as a gentleman, not only never to publish your discoveries, if you tell them to me, but I also promise and pledge my faith as a true Christian to put them down in cipher so that after my death no one shall be able to understand them. Along with a student, Lodovico Ferrari ( ), Cardano unraveled the secret of the cubic equation in the form x + mx = n, making significant progress. Not only did they master the techniques given to them by Tartaglia, but they also extended his work to apply to cubic equations of any form (a huge accomplishment) and used Tartaglia s techniques to solve polynomial equations of degree four! Cardano and Ferrari were bound by Cardano s oath, however. Even though they had pushed Tartaglia s work far beyond where Tartaglia had ever taken it, all of their progress was based on the oath-protected secrets of Tartaglia. Cardano, though, was anxious to publish his results. He did not need to protect an academic post since he was a doctor by trade. Looking for a way out of this predicament, Cardano and Ferrari traveled to Bologna, where our story began. There, they studied the private papers of del Ferro and saw the exact solution of Tartaglia written in del Ferro s own handwriting, but well before Tartaglia had discovered them 2001, Lawrence Morales; MAT107 Chapter 6 Page 12

13 independently for himself. As far as Cardano was concerned, he was no longer bound by his oath. He would publish del Ferro s findings, not Tartaglia s, even though they were the same. And that he did. In 1545, he published Ars Magna (Great Art), which is considered one of the great masterpieces in mathematical history. (It s still in print!) He prefaced his work with the following attribution: Scipio Ferro of Bologna well-nigh thirty years ago discovered this rule and handed it on to Antonio Maria Fior of Venice, whose contest with Niccolo Tartaglia of Brescia gave Niccolo occasion to discover it. He gave it to me in response to my entreaties, though withholding the demonstration. Armed with this assistance, I sought out its demonstration in [various] forms. This was very difficult. 7 Tartaglia was furious, of course. He accused Cardano of deceit, sending nasty letters to Cardano with the charges clearly spelled out. Cardano basically ignored them. His student, however, was more easily drawn into the debate. In 1548, Tartaglia and Ferrari squared off in a public, mathematical challenge in Milan, Ferrari s stomping grounds. Tartaglia lost, blaming his performance on the rowdiness and partisanship of the crowd. 8 (A rowdy crowd at a math contest?) Tartaglia went back home, defeated once again, and Ferrari was proclaimed the winner. Ars Magna and the of the Cubic In Ars Magna (1545), Cardano detailed del Ferro s and Tartaglia s technique for finding the solution of a cubic equation in the form: x + mx = n 2 This is called a depressed cubic because the x term is missing. Cardano s solution was given entirely in words. There were no modern algebraic symbols yet present to help him express his results. In modern notation, Cardano s verbal solution of this kind of cubic takes on the following modern form: Cardano s Formula n n 2 2 x = + R + R 2 n m where R = , Lawrence Morales; MAT107 Chapter 6 Page 1

14 You can guess why this equation is not introduced into basic algebra courses. There is relatively little that is basic about it. Prior to the advent of handheld calculators, using this formula would prove to be almost impossible in most cases. The demonstration of this formula is not an easy task to undertake. 9 However, it is based on basic algebraic principles and so if you are patient and diligent enough, you can follow how it was derived. In this text, we will skip an explanation of where the equation comes from and focus our efforts on learning how to use it. (Note: In order to use this formula, you need to really know how to use your scientific or graphing calculator. To take cube roots on your calculator you can check to see if your model has a x button. If it does not, you can take a cube root by raising a number to the power of 1/. For example, 8 1 / x y = 2. On a calculator, the power button looks usually like y, x, or the ^ symbol. To do 8 1/ would require a key sequence like: 8^(1/), or 8 y x (1/). Play with it on your model until get 2 for 8 1/ before you proceed.) Before we do any specific examples, it might be helpful to propose some steps that can be followed to use this formula correctly: Steps in Using Cardano s Formula Step 1: Make sure that the equation given is in the form x + mx = n. Step 2: Identify the proper values of m and n. Step : Calculate the value of R and also R. Simplify if possible. Step 4: Plug in the value for R into Cardano s equation and carefully compute the final value. We will start with the equation that Cardano used to illustrate his method. He stated his problem in words (since he did not have variables to work with) in a form that might look like this: Cube plus six times a number is twenty. What is the number? A translation of this into modern algebraic notation would be the following: x + 6x = 20. Cube plus six times a number is twenty. What is the number? 2001, Lawrence Morales; MAT107 Chapter 6 Page 14

15 Example : Use Cardano s formula to solve x + 6x = 20. Step 1: Make sure that the equation given is in the form x + mx = n. This equation is already in this form so we can proceed. Step 2: Identify the proper values of m and n. The value of m is 6 and the value of n is 20. Step : Calculate the value of R and also R. Simplify if possible. 2 n m R = = = = = 108 This means that R = 108 Step 4: Plug in the value for R into Cardano s equation and carefully compute the final value of x: n n x = + R + R = = + + This is what we could call the exact solution. If we want a decimal approximation to this, we can use a calculator to do this: x = It s amazing that these two are actually equal. 2001, Lawrence Morales; MAT107 Chapter 6 Page 15

16 We get an estimate of This is pretty close to 2, so we ll round it off to 2 exactly and see how close we are. If we check this in the original equation, we get: x + 6x = 2 = = We do get the correct value. Note that this equation gives one solution for this equation. We can look at the graphs of y = x + 6x and y = 20, to see where they intersect since the intersection point(s) represent solutions of the original equation. The graph here shows that there is indeed one solution. You can see that when x = 2, the curve of y = x + 6x crosses the line y = 20, just as we expect it to. Obviously, Cardano did not have the advantage of seeing such a graph the x y coordinate system was not yet invented, but it does give us some insight into what is going on here. Example 4 : Important Note In order to be as accurate as possible with these calculations, it is best to keep as many decimal places as possible. If you know how, it is best to try to do the entire sequence of calculations without clearing our your calculator or using the clear button. (By not using the clear button, my graphing calculator gives me the exact value of 2!) You should experiment with your calculator and try to get an exact value without having to use the clear button. If you cannot get it to work, use four decimal places throughout your computations. Use the cubic formula to solve x + x = 10 Step 1: Make sure that the equation given is in the form x + mx = n. This equation is already in this form so we can proceed. Step 2: Identify the proper values of m and n. The value of m is and the value of n is (2,20) 2001, Lawrence Morales; MAT107 Chapter 6 Page 16

17 Check Point B : Step : Calculate the value of R and also R. Simplify if possible. 2 n m R = = = = = 26 This means that R = 26 Step 4: Plug in the value for R into Cardano s equation and carefully compute the final value n n x = + R + R = = + + This is we call the exact solution; the decimal approximation is: x = A check of this shows a close match, although the decimal rounding causes a little error. Use the cubic formula to solve See endnote for an answer. 10 x + 10x= , Lawrence Morales; MAT107 Chapter 6 Page 17

18 Check Point C Use the cubic formula to solve x : Seen endnote for an answer. 11 Solving the General Cubic Equation + 5x= 15 Cardano next tackled the non depressed equation. To do this, he came up with a clever way of temporarily converting a non depressed equation into one that is depressed. He then used his formula on the depressed equation and then adjusted his result to reflect the fact that he had changed the original equation. The general method is as follows: Solving the Non-Depressed Cubic Equation: 2 Given the equation ax + bx + cx + d = 0 b Step1 Let x = y. a b Step2 Substitute x = y into the original equation to get an equation a in y. The result should be a depressed cubic. Step Use Cardano s formula to solve the equation for y. b Step4 Find the original, desired values of x using x = y. a Example 5 2 Solve 2x 0x + 162x 50 = 0 : b 0 Step 1: Let x = y = y = y + 5. Remember this for later: x= y+ 5 a (2) Step 2: Substitute x = y + 5 into the original equation to get a depressed cubic. x = y x 0x + 162x 50 = 2( y+ 5) 0( y+ 5) + 162( y + 5) 50 = ( y 15y 75y 125) 0( y 10y 25) 162y = y 0y 150y 250 0y 00y y = + 2y 12y , Lawrence Morales; MAT107 Chapter 6 Page 18

19 Recall that this is equal to 0, so we can divide by 2 to get: y y y + y = + 6y 20= 0 + 6y = 20 Step : This is a depressed cubic and we can now use Cardano s formula on it. But note that this is essentially the equation Cardano used to illustrate the solution to the cubic. We saw in Example that x + 6x = 20 has a solution of 2, therefore y + 6y = 20 has a solution of y=2. Step 4: Since x = y + 5, and y = 2, we can see that x = = 7. Thus, x = 7 2 is the solution to the original equation 2x 0x + 162x 50 = 0. Check: ( ) ( ) ( ) = = 0 This verifies the solution is correct. Using this process may seem cumbersome to some, but we should remember that Cardano did not have variables or equations to work with. Cardano had to mainly use words to solve these equations. Hence, the process that we have is actually a lot easier to use than Cardano s. Example 6 Solve x 6x x 18 = 0 : ( 6) Step1: Let x= y = y+ 2 Step2: Substitute x in to the equation: x 6x ( y + 2) y y 2 + y 4 = 0 + y = x 18 = 0 6( y + 2) ( y + 2) 18 = 0 This last equation is in the form we need to use Cardano s formula. 2001, Lawrence Morales; MAT107 Chapter 6 Page 19

20 Step: We have m =, n = 4. We first find the values of R and R. 2 n m R = = = 4+ 1 = 5 Thus, R = 5. Now find the value of y using Cardano s formula. n n y = + R + R = + + = Using a calculator, we can get an estimate for this number: (.6180) = 1 Therefore, y = 1. This means that y = 1 is a solution to y + y = 4. Keep in mind that our original equation was in terms of x and that we used the substitution x = y+ 2. We now use this substitution again to find the value of x that we are after. Step4: y = 1, so x = y + 2 = 1+ 2 =. Hence, x =. Check: We let x = and substitute in the original cubic equation. x = = = 0 6x x 18 = 6 This solution checks and we re done. 2 ( ) + 15( ) , Lawrence Morales; MAT107 Chapter 6 Page 20

21 Check Point D : Use the cubic formula to solve See the endnotes for a solution x x x = 0 The of the Quartic Equation and Beyond With the cubic equation conquered, Cardano and his student, Ferrari, moved on to the quartic 4 2 equation of the form ax + bx + cx + dx + e = 0. Could a formula be found for that as well? The answer was yes, as long as the equation could be reduced to a depressed cubic, in which case they could use their previous discovery. It seems as though the cubic formula was more useful than they originally thought. We won t explore the solution of the fourth degree equation in detail here due to its complexity. However, if you re interested in seeing a modern representation to the solution of the quartic, you can read Appendix A at the end of the chapter. Fifteen years after Cardano and Ferrari did their work, Rafael Bombelli ( ) wrote a more systematic text to help students master these techniques. His book, Algebra, marks the high point of the Italian algebra of the Renaissance. 1 In this text, Bombelli introduced equations and techniques that led to the establishment of what we call imaginary or complex numbers and gave rules for working with them. Complex numbers include numbers of the form a + bi where i = 1. That is, the square root of a negative number is allowed and is considered a valid number. (These are not part of the real number system. In fact, the real numbers are actually a subset of the complex numbers.) Complex numbers play important roles in higher mathematics, physics, and engineering. We make one last note about solving equations: It was only natural for Cardano and Ferrari to pursue the quintic (fifth degree) equation and its solution. A quintic equation has the following form: ax + bx + cx + dx + ex + f = 0 But they were thwarted. They could not find an equation to solve a quintic because there is none. It is impossible to express the solution of a fifth degree equation with a formula using roots, powers, and the four basic mathematical operations. This was first proved by Niels H. Abel ( ). Later Evariste Galois ( ) extended his work and gave general conditions for when equations are solvable. Galois has a whole field of mathematics named after his work Galois Theory. Unfortunately, both Abel (top) and Galois (bottom) died at very early ages. The story of Galois life and death is fit for a television movie of the week. As the 1500 s and 1600 s passed, mathematics was rediscovered in Europe again. It stands on the shoulders of many civilizations that preceded them. By the early 1600 s, Europe was ready for new techniques that would help them tackle modern, complex calculations. 2001, Lawrence Morales; MAT107 Chapter 6 Page 21

22 PART : Modern Calculations The power and extent of modern calculations rest on at least three important developments. 14 The introduction of the Hindu Arabic numbers, the use and acceptance of numbers with decimal/fractional parts, and the invention of logarithms all played important roles in how mathematicians and scientists could do computations. The introduction of decimal fractions and logarithms both took place in the early 1600 s and each of them came about because of efforts to take very laborious computational techniques and simplify them. (Prior to this, very complicated methods from trigonometry had been widely used instead.) The need for this kind of change was motivated by the needs of research in astronomy, navigation, and commerce, to name a few. Decimal Fractions Simon Stevin 15 ( ) was one of the leading advocates of decimal fractions. Just to clarify what we mean by decimal fractions, we are talking about numbers like These are base ten numbers where the whole part and fractional part of a number are taken together and computations are done on them without splitting up their individual parts. Prior to this time, the whole and fractional part were often split apart and computed separately. Not only that, but fractions were often expressed in base 60 rather than in base 10. Think About It Why Base 60? Stevin worked in commercial, military, and administrative environments, and eventually taught math at the Leiden School of Engineering. He was one of the first people to fully embrace the emerging Copernican theory. He gains a role in this discussion because he published a pamphlet called De Thiende (The Art of Tenths) where he introduced decimal fractions. His publication came at a time when others had started taking advantage of decimal numbers, and so it was received with open arms. It is interesting to see the difference between his notation and ours. He used the symbols,,,, to represent descending powers of 10. Corresponds to the whole part of a number (ones place and higher, basically) Corresponds to the tenths place; (1/10) 1 is tenths. Corresponds to the hundredths place; (1/10) 2 is hundredths Corresponds to the thousandths place; (1/10) is thousandths, etc. 2001, Lawrence Morales; MAT107 Chapter 6 Page 22

23 However, instead of using tenth, hundredth, or thousandth, Stevin uses prime, second, and third, respectively. If he wanted to express the number it would look similar to this: Here s a breakdown to clarify his notation: whole 6 in the tenths (prime) place 5 in the hundredths (second) place 7 in the thousandths (third) place Example 7 : Check Point E : Write in Stevin notation Write, in Stevin notation See endnotes for solution. 16 If Steven wanted to do arithmetic with decimal fractions, an addition problem would look like the picture here, which is actually a piece of Stevin s publication on this subject. Around this time, support for these numbers grew quickly 17, despite their clunky implementation. In 1592, Giovanni Antonio Magini, a mapmaker, introduced the decimal point to separate the whole parts from the fractional part of numbers. Despite its ongoing evolution and staunch support for the system from people like Stevin, it would be another 200 years before the system was adopted for use in currency, weights, measures, etc. It was not until the French Revolution that the metric system was adopted for such uses. Think About It Another important name that paved the road for decimal What numbers are 779 fractions and logarithms to emerge was François Viète 18 being added here 780 ( ). Viète, convinced that a geocentric model and what is the 781 was more reliable than the emerging Copernican theory, result? 782 made great efforts to publish trigonometry calculations 78 that would help in the study of astronomy. For the most part, scientists 784 before him were used to using sexigesimal (base-60) fractions that were 785 a product of the Babylonian number system and their own work in 786 astronomy. Viète, along with more and more of his contemporaries, was 787 advocating a new system of decimal (base 10) fractions. One way in which Viète and others tried to simplify multiplication and division was by reducing them to addition and subtraction. The obvious reason for this was that adding and subtracting numbers is generally much easier than multiplying or dividing them. This process of reduction is 2001, Lawrence Morales; MAT107 Chapter 6 Page 2

24 called prosthaphaersis. 19 In his publication, Canon, Viète presents several equations from trigonometry that accomplish this. For example, he gives the following two rules: 2sin A sin B = sin( A + B) + sin( A B) 2cos A sin B = sin( A + B) sin( A B) The sin and cos symbols are trigonometry functions that are actually abbreviations for sine and cosine, respectively. The phrase cos( A + B) does not mean cos times (A+B). Instead, it means the cosine of (A+B). It s sort of like a square root. The symbol has no meaning by itself we have to take the square root of some particular number. A + B does have meaning (usually). While we will not study these trigonometry functions in this course, what we will point out is that each of these two equations reduces multiplication into addition (and/or subtraction). The first equation takes two sines that are multiplied together and reduces the value to two sines that are added together. With tables to give the values of these cosines and sines, computations could (and did) go much more smoothly. We will see that this idea of reducing multiplication and division to addition and subtraction is an important one in the next part of this chapter. 2001, Lawrence Morales; MAT107 Chapter 6 Page 24

25 PART 4: Napier and The Emergence of Logarithms The mere mention of the word logarithm can cause many algebra students eyes to gloss over. Most do not understand these tools when they first see them and rarely remember them past the final exam. I m convinced that a big part of the reason that the rules and uses of logarithms are so hard for students to grasp is that they are usually presented completely devoid of their historical context. Furthermore, with the advent of technology, their general use has been reduced significantly. In this section, we ll try to establish that context and hopefully come to see logarithms as an important and powerful development in the history of math (even if they have been replaced by the modern calculator). Keep in mind that we are considering a time in history when calculators were not present and calculations all had to be done by hand. As work in astronomy and other fields progressed, the need for an efficient method of calculation had emerged. As we stated before, in some circles this was done with prosthaphaersis, where multiplication and division were reduced to addition and subtraction. John Napier 20 ( ) spent about twenty years of his life developing a new and easy-to-use tool that would use this approach to make computations easier to do. He gave it the name logarithm. Napier was an ardent Scottish Presbyterian who was strongly opposed to Catholicism. In his book A Plaine Discovery of the Whole Revelation of Saint John: Set Downe in Two Treatises, he identifies the Pope as the antichrist, urges King James to rid his house of papists, and predicts the end of the world will take place between 1688 and He was also a resourceful landowner, devising a hydraulic screw similar to that of Archimedes to control water levels in coal pits. On the mathematical front, he used numerology to look for hidden prophecies in the Bible and he looked for a way to simplify computations in trigonometry. To do this, he used a well known trigonometric equation of the time: 2sin A sin B = cos( A B) cos( A + B) Although slightly different from the equations given previously, we can see that the multiplication on the left is reduced to addition and subtraction on the right. Napier used this identity to build a table of logarithms and antilogarithms that could be used to simplify calculations. In 1614, he announced and published his tables in Mirifici Logarithmorum Canonis Descriptio (A Description of the Marvelous Laws of Logarithms). Later, he published information on how he developed logarithms and the theory behind them in Mirifici logarithmorum canonis constructio (A Construction of the Marvelous Laws of Logarithms.) The word logos means ratio in this context and arithmos means number; Napier merged them to form the word 2001, Lawrence Morales; MAT107 Chapter 6 Page 25

26 logarithm (ratio of number). The reason he chose this particular word has to do with how he came up with the logarithm concept and later developed it. It is important to point out that Napier s logarithms are different than our own. It wasn t until after his tools had become widespread and had been refined and developed by others that they took the form that they are in today. To Napier, logarithms were tied to physical motion and the mathematics related to motion. Napier defines his logarithms as follows: 21 Take a line segment AB and an infinite ray DE, as shown below: A D x C F Let C be a point that starts at point A and moves along AB. Let F be a point that starts at point D and moves along ray DE. Points C and F both start moving at the same speed. From that point on, their speeds change according to the following rules: Think About It Point C always moves with speed equal to the distance CB = y. Point F always moves at the same speed as when it started, Is point C slowing where x is the total distance F has moved from point D. down or speeding up as it moves According to Napier, x is the logarithm of y. along AB? Why? This definition looks very little like the modern definition of logarithms because the modern idea of log didn t come along until the time of Euler ( ). We ll explore the modern definition more in a moment, but what we should try to keep in mind is that both versions of the tool are trying to reduce multiplication and division to addition and subtraction. Using this (or the modern) definition of logarithm, it is possible to build tables that will make these reductions for us. Napier spent 20 years of his life painstakingly developing this theory and building his tables to an incredible degree of accuracy. He made very few errors, which is amazing considering the number and complexity of calculations that he undertook. Astronomers and mathematicians alike were ecstatic about this new invention since it really did do what Napier had intended. One of the most famous quotes about logs comes from Pierre Simon who stated that [logarithms], by shortening labors [on computations], doubled the life of astronomers. The acceptance of Napier s new tool was widespread and rapid. Rarely in mathematics are inventions so quickly and universally adopted. y B E 2001, Lawrence Morales; MAT107 Chapter 6 Page 26

27 In 1616, Henry Briggs ( ) worked with Napier to refine his invention so that they were more convenient to use. Briggs is known for helping to construct tables where log 10 = 1, thus creating what we today call the common logarithm. This is the log button you see on your scientific or graphing calculator. Briggs published common log tables for values from 1 to Later, others provided tables up to 100,000 to ten places of accuracy! These were in use all the way until the early 1900 s when the tables were computed to twenty decimal places in Britain. This was still before the age of the electronic calculator or computer! Napier s work had long range implications for broader mathematics, navigation, astronomy, banking, and number theory. In an indirect route, the logarithm eventually led down a path that resulted in the final proof that the three great Greek construction problems could not be solved by straightedge and compass alone. 22 The Idea and Use of Logarithms In this section we will explore basic ideas of how logarithms work and why they were useful in the time of Napier. This will hopefully explain why they were so popular. Furthermore, it is my hope that the rules of logarithms that may have perplexed you in the past will be placed into a proper context and that you will see how and why they were useful. Recall that Napier (and others after him) published tables of logarithms and antilogarithms that were intended to make complex calculations easier. As we work our way through this section, try to imagine yourself in a world where you cannot reach for a calculator to do any number crunching for you. Furthermore, the modern algorithms for multiplication and division that you learned as a child do not exist. With these restrictions, how would you do calculations like the following? (4, ) (1,998,76, ) These are certainly not computations we want to undertake without our little button filled slabs of plastic. But Napier and others in his time had no choice. Logarithms gave them hope of doing these calculations more efficiently. The Idea Behind Logs Here s the main idea behind logarithms and how Napier intended them to be used: We start with a problem that involves complex arithmetic computations (like those given above). After applying logarithm rules we get a new statement that we can look up on the logarithm tables. We do some addition and subtraction operations on this statement (as needed) to get an intermediate result. This intermediate result can be looked up in an antilog table to get us back to a result that compensates for the fact that we reduced everything to addition and subtraction. Think of the antilog table as the evil twin of the logarithm. What the logarithm does, its evil twin (the , Lawrence Morales; MAT107 Chapter 6 Page 27

28 antilog) comes along and undoes. So while the log will reduce to addition and subtraction, the antilog will covert back to multiplication and division. Here s a diagram of what we mean: Rule of Logs One Step Closer to the Tables Well, we ve put it off long enough. We can t go any further until we talk about the rules of logs. We will be working with modern versions of these rules that were unknown to Napier. But the spirit of the task at hand remains the same. The modern definition of the logarithm is defined in a way so that exponents are undone. For example, if we have the equation x 2 = 25, we know that we can take the square root of both sides, as square roots undo powers of 2. If the equation looks like 2 x = 25, taking a square root no longer works since the exponent (x) is not a 2. In fact, it s a variable! In order to undo this exponent, we need a tool that will specifically deal with this kind of equation. Leonhard Euler ( ) is responsible for the definition of log that we will use: p logb n= p b = n The symbol here is used to imply interchangeability. Both sides of the double arrow lead to each other and allow you to go back and forth between the two expressions. We often say that the log base b of a number n is the power to which b must be raised to produce n. This statement tells us what taking the log of a number (n) means. The number b is the base. The number p is the result. You should observe that while the right side of the double arrow has a power present, the left side does not. The logarithm has undone the power. Some simple review examples are in order. Example 8 Ugly computation is converted to a statement using logs Via log table Log rules/tables are used so addition and subtraction yield an intermediate result Write 2 = 8 as a log statement. Via anti-log table Antilog tables are used to convert the intermediate result to base 10 numbers the final result 2 = b, the base; = p, the power; 8 = n, the number that is produced. So, log2 8 = We say that the log base 2 of 8 is. That is, the power to which 2 must be raised to get 8 is. 2001, Lawrence Morales; MAT107 Chapter 6 Page 28

29 Example 9 Check Point F Example 10 : Check Point G x Write 2 = 25 as a log statement. 2 = b, the base; x is the power; 25 = n, the number. So we have log2 25 = x. Write 4 y = 16 as a log statement. What is y? See the endnotes to check your answer. 2 Write log 7 49 = x as a statement with exponents. 7 = b, the base; 49 = n, the number produced; x is the power required. So we get 7 x = 49. We can actually find the value of x here, since we know that 7 2 = 49. Thus x = 2. Write log y z = w as a statement with exponents See the endnotes to check your answer. 24 So, what does this have to do with making multiplication and division easier? Good question. Unless otherwise specified, we will assume from now on that our base is 10. Hence we will be working with the common logarithm that Briggs developed with Napier. If you don t see a base on a log, you can assume the base is 10. (By the way, that s what your calculator log button assumes.) Let s take two numbers and multiply them. We ll take two special numbers that will help us demonstrate the first log rule, which reduces multiplication to addition. Let x A = 10 and let y B = 10 A and B are two numbers, which depend on x and y, of course. Let s see what happens when we multiply them: 2001, Lawrence Morales; MAT107 Chapter 6 Page 29

30 Operation Comment x y AB = Multiply our numbers x+ y AB = 10 Recall that by rules of exponents, when we ( x+ y) multiply the same base with different exponents, AB = 10 we add the exponents and keep the base. For example x x = x = x. The parentheses around x + y emphasize that we can consider this all one exponent. log 10 AB = ( x + y) We simply apply the modern definition of the log, where 10 is the base, (x+y) is the exponent, and AB is the number/result. If we knew what x and y were, we could proceed. Although we don t know what values they take on, we can use the modern log definition to find expressions for each. x A = 10 log10 A= x Apply the modern definition of the log to A and B. y B = 10 log10 B = y Now comes the cool part log10 AB = x + y Substitute our values for x and y into the new = log A+ log B equation This gives the first famous rule of logs: We ve omitted the base since this rule is true for all valid bases. No doubt you ve seen this before, unaware of where it came from or why it even showed up in the first place. It s often paraphrased as The log of a product is the sum of the logs. With a historical context in place, we can see its importance. The left side is the log of two numbers that are multiplied. On the right side, we have reduced it to two logs being added together! Aha. There it is again. We ve reduced multiplication to addition and subtraction, just like Napier was seeking to do. We can look at a simple example to see how this law works: Example 11 log AB = log A + log B Write log( 5 44) as the sum of logs. log( 5 44) = log4 + log 44 If we knew the value of log 4 and the value of log 44, we could add them to get the value of log(5 44). 2001, Lawrence Morales; MAT107 Chapter 6 Page 0

31 Check Point H Write log( ) as the sum of logs. See the endnotes for solution. 25 These examples so far don t truly allow us to peer into the power of this rule. Here s an example that will begin to do that. Example 12 What is z = ? Here we have a multiplication problem. It s not hard. We can do it in a snap. But let s just pretend that these two numbers are very hard to multiply. Let s let x = 1654 and let y = 298. Operation Comments z = This is what we want to compute, but we re assuming it s too hard or too timeconsuming to do by hand. log z = log( ) We take the log of both sides. In this step we are taking an ugly computation and converting it to a statement about logs. (See diagram above). log z = log log298 Apply the first log rule to the right side. The log of a product is the sum of the logs. log z = Use the log tables to look up log1654 and = log298. We then easily add up the results. What tables, you ask? You ll meet them soon enough. We ll finish this first. log z = Here is our final result, in logland. z = Use an antilog table to get z. Remember, the antilog is the evil twin of the log, so it undoes the what the log does. In this case, it removes the log from the z and returns the value of z. (Again, the antilog tables are on their way.) When we check our answer, we are just slightly off, but we only worked with about four decimal places. Recall that the British had tables of twenty places in place in the early 1900 s. 2001, Lawrence Morales; MAT107 Chapter 6 Page 1

32 Hopefully you see from this example the process we described in the diagram above. We start by creating a statement about logs. We then use log rules and tables to create results that are easily added. Finally, we can use an antilog table to get us out of logland and to a final answer. Before we introduce the log and antilog tables, we should mention two more log rules that are important. The first of these is: This is the rule that reduces division to subtraction. It is related to the familiar rule of exponents m x m n that says = x. That is, when we divide exponents with the same base, we subtract the n x 6 exponents. This is why x 8 = x instead of x 4 2 x. We don t divide the exponents. The proof of this law is left as an exercise. Example 1 Example 14 : Ugly computation is converted to a statement using logs Simplify 95.2 log log = log 95.2 log8. Hence, if we can compute log95.2 and log8, we 8 can subtract them to get the value of the original log. Simplify Via log table 85. log x. Log rules/tables are used so addition and subtraction yield an intermediate result A log = log A log B B 85. log = log85. log x x Via anti-log table Antilog tables are used to convert the intermediate result to base 10 numbers the final result 2001, Lawrence Morales; MAT107 Chapter 6 Page 2

33 Check Point I : z Simplify log.5. See endnotes for solution. 26 m mn The third log rule is also useful and is related to the familiar rule of exponents: ( ) example, ( ) n x = x. For x = x = x. This rule states that when we raise an exponent to a power, we multiply the powers. The related log rule is: Many people remember this rule by observing that all it says is that in a logarithm the power of the number can move to the front of the log. Example 15 Example 16 Check Point J Simplify log x log x. = log x Simplify log x. Recall that x = x 1 log x = log x 2 Simplify 2 log x. 1/2 See endnotes for solution. 27 c log n = clog b b n 2001, Lawrence Morales; MAT107 Chapter 6 Page

34 Let s prove this last log rule, c log n = clog b b n Operation Comments Let x = logb n We start in this unusual place but be patient and you ll see how things work out. b x = n Use the modern definition of log. We file this away for use later. c Consider logb n Now we look at the log we are seeking to simplify. c x c x log n = log ( b ) Substitute n = b from the step above. b xc = ( ) b log b b Multiply exponents according to rules for exponents. = xc This one is a little subtle. Remember that the log base b of a number is the power that b must be raised to get that number. In this case our base is b and we want to raise that to some = cx = clog b n c So log n = clog n b The Log Tables b xc power to get our number, b. Well, of course, xc to get b we need to raise b to the power of xc, so xc is the log. Here we just substitute the value of x from the very beginning to get our desired result. Most of you reading this have probably never seen a log table before. But before the relatively recent appearance of the handheld calculator and computer, log tables and their more sophisticated counterparts, slide rules, were the dominant tools for complex calculations. Using a log table, we can do sophisticated problems of multiplication and division, as well as powers and roots, without a calculator at our side. So for this section, you ll want to put away your calculator and journey back in time. For the most part, we will limit ourselves to computations with numbers that have three or four decimal places. This will make using tables a little easier than they might be otherwise. Also, we will be using base-10 logarithms for all of our computations. You should have the log tables in front of you for the next series of examples. They are in Appendix B. There are four tables. Two are regular log tables. The other two are antilog tables. We ll start with the log tables. Looking up logs is relatively easy with these tables. To find the log of 4.25, we look down the left side of the first table under the column labeled N. We locate the row that says 4.2, as shown below. 2001, Lawrence Morales; MAT107 Chapter 6 Page 4

35 log 10 N Ten Thousandth Parts N Then move along the row until you are in the column that is labeled with the 5. Ignore the Ten Thousandth Parts of the table for now. You should see the number Thus, log = What does this mean? We go back to the definition of log: ( = log ) (10 = 4.25) Hence, the log of 4.25 tells us to what power 10 must be raised to get This makes sense when we remember that 1 < 4.25 < 10. That is: 1< 4.25< 10 So 10 < 4.25 < The second inequality indicates that the power of 10 we require is somewhere between 0 and 1, and certainly fits the bill. Example 17 Example 18 Check Point K : What is log ? Looking at row 9. of the table, the number in column 2 is Hence, = 9.2, as desired. This makes sense since 9.2 is close to 10 = 10 1, so our log (0.9694) should be close to 1. What is log ? We look up row 7.4 and inside column 0 (since 7.4 = 7.40) we find What is log ? See endnotes for an answer , Lawrence Morales; MAT107 Chapter 6 Page 5

36 The Ten Thousandth Parts of the tables is there for when the number you are taking the log of has three decimal places instead of two. To use this part of the table, you take the normal two decimal place log and then add the ten thousandth part to the very end of the result. Here s an example: Example 19 : What is log 6.875? For this problem we first compute log 6.87 as before: N log 10 Ten Thousandths Parts N Check Point L : log 6.87 = Then, we look under the 5 column of the ten thousandth parts of the table since the third decimal place of is a 5. This gives a, which really represents (That s why it s called a proportional part). Take this and add it to the end of the previous result. So = Hence, log = Since our calculators do not exist, we can t check this, but rest assured that it s very close to the actual log of this number. Find log 4.68 Check the table and make sure you can get Note that the log requires that numbers strictly between 1 and 10 be used as its inputs. Larger Numbers and the Log Tables Okay, what about larger numbers? Suppose we want log 152. The tables don t have a row with 152 in it, however. They only go up to 9.9. Here we have to rely on some common sense about powers of 10. We know that 152 > 100, so therefore 152 > Hence, our log must be bigger than 2 since we need at least 100. We also know that 152 < 10, since 152 < Thus, whatever power of 10 that gives us 152 must be between 2 and. Hence, log 152 must be between 2 and. To find how far past 2 we must go, we simply pretend that we are computing log When we look at row 1.5 of the table and move over to column 2, we find That s how far past two we must go. Hence, log 152 = , since is between 2 and. 2001, Lawrence Morales; MAT107 Chapter 6 Page 6

37 Here s another way to think about it: We can rewrite 152 as Thus, log152 = log( ) = log100+log1.52. (This last step uses one of the basic log rules.) We know that log 100 = 2 because 10 2 = 100. We can get log 1.52 from the table it s So we have log 152= = We call 2 (the integer part of the result) the characteristic and we call (the fraction part) the mantissa. The term characteristic was first used by Briggs. The term mantissa is of Latin origin, originally meaning an addition or appendix. It was probably first used by John Wallis around 169. We can streamline some of this by trying to give a series of steps to follow when working with the log tables. Step 1 Move the decimal point of the given number so that you have a new number that is strictly between 1 and 10. Step 2 Count the number of places that you move the decimal point. This is the characteristic. Step Look up the new number (that is between 1 and 10) in the log tables to get the mantissa. Step 4 Add the characteristic and the mantissa together to get the final result. Example 20 Find log If we write this as , we note that need to move it places to get Our characteristic, therefore, is. (We first note that 1,000<4869<10,000. Put another way, 10 < 4869 < Hence the power of ten required is more than but less than 4.) To find the mantissa, we look up on the table. We look up row 4.8, column 6, and ten thousandth part 9. We get = Combining our characteristic and mantissa gives log 4869 = Hence, = Check Point M Find log 5,60. Check the endnote for an answer , Lawrence Morales; MAT107 Chapter 6 Page 7

38 Logs of Very Large Numbers You have to try to imagine what this means for doing calculations. With these tables, we can now do sophisticated computations while at the same time getting three or four places of accuracy without using a calculator (which they did not have at the time anyway). Here s an example, however, of why more than three places of accuracy might be needed. Example 21 Find log 7,84,945. We start by finding the characteristic. We need to move the decimal place 6 places to get , so our characteristic is 6. To find the mantissa, we find on the log table. Looking up row 7.8, and down column we see But what about the ten thousandth part? Since there are more than decimal places we have to round to 7.85 and look in the 5 column for the ten thousandth part, which gives us, representing Thus, our mantissa is = Finally, we get a log of = The actual value is something like Even if we round this number to the fourth decimal place, we do not get This is due to the fact that we had to use an estimate to get the ten thousandth part of the mantissa. This may not seem like a big deal to us. After all, three decimal places of accuracy are pretty good for most applications. But remember that when working with very big or very small numbers, like astronomers were at that time, losing accuracy in the fourth decimal place could mean that your calculations end up being significantly off by the time you finish your calculations. Eventually, we want to use these tables to do complex calculations but we must first learn to use the antilog tables. The Antilog Tables The analogy of the antilog as the evil twin of the log who undoes what the log does is helpful here. Whereas the log tables tell us what power to raise 10 to in order to get a result, the antilog tables tell us the opposite. Hence, the antilog tables are really just power of 10 tables, as you will see here. Example 22 If log N = 0.68, what is N? Note that this is the opposite question than we had before. Here we know the log but are looking for the original number N. By the modern definition of logs, we have: 2001, Lawrence Morales; MAT107 Chapter 6 Page 8

39 log N = 0.68 log N = c c 0.68 = N Hence, we only need to compute Without a calculator, we turn to the antilog tables. To compute , we find the 0.68 row and then move over into column with as its heading. Here is a piece of the antilog table that applies here: ANTILOG TABLE: 10 P = N Thousandth Parts P The table indicates that = This makes some sense when we remember that since 0<0.68<1, then 10 0 < <10 1. Hence, we have 1<4.819<10, which is certainly true. To use the thousandths parts of the table, we follow the same basic idea as before. Example 2 If log N = 0.825, find N. By the definition of log, we have =N. We first find row 0.82 and then move over to column. This takes us to ANTILOG TABLE: 10 P = N Thousandth Parts P Then, we look up the 5 in the thousandth parts of the table and see and 8 there, which now represents We add these, = Therefore, we have = Note that the antilog requires numbers between 0 and 1 as inputs. 2001, Lawrence Morales; MAT107 Chapter 6 Page 9

40 Check Point N If logn = 0.184, find N. What does N represent as a power of 10? See the endnote for an answer. 0 So then what do we do about larger numbers? Example 24 If logn =.874, what is N? We rewrite this as = N. First we notice that = m+ n m n. (Rules of exponents x = x x ) Thus, = The power of 10 on the end can be obtained from the table = So we have = The actual value is , so we are reasonably close (four places of accuracy when rounded.). Example 25 If logn = , find N. First we write = N. Next, we write N = = 10,000, From the table we have: N = 10,000, = 16,440,000. There is an alternative way to approach antilog problems. Notice that in the previous few examples, the whole number part of the number we are given does not play a role when we use the tables. Only the fractional part has an impact. So, as a shortcut, we can look up the fractional part in the antilog tables and then move the decimal point in the result. The number of places that we move corresponds to the whole number part of the given number. In the previous example, note that you had to move the decimal point 7 places which corresponds to the 7 in Check Point O If logn = 5.62, find N. See the endnotes to check your answer , Lawrence Morales; MAT107 Chapter 6 Page 40

41 The Evil Twins Meet Each Other Finally We have been saying that antilogs and logs undo each other s work. Let s see a pair of examples that demonstrate that. Example 26 What is the value of x = log 76,540? We note that 10 4 < 76,540 < This means our characteristic is 4 and we need to find the mantissa from the table. Since the table requires an input between 1 and 10, we use From the log table, we get = Hence, log 76,540 = This means that = Now we ll do the opposite of this problem it s twin. Example 27 If logn = 4.889, what is N? This is linked to the previous example we should expect to get back 76,540. (Why?) Because logn = means that = N, we can use the antilog table to find N = = 10, = 10,000 ( ) = 10, = 76,520. Okay.we re off a little bit, but if we had more decimal places of accuracy in our table, we would get an even better result. What is important to observe from the last two examples is that applying log and antilog tables basically take you back to the original starting point. For this reason, we say that the log and antilog tables are inverses of each other in the same way that square roots and powers of two are inverses of each other. Complex Calculations Now imagine for a moment that you are living in the 1600 s and your astronomical calculations require that you multiply 874,200,000 7,85,00. You have no calculator. You don t have any easy method of multiplying these two numbers until Napier knocks on your door and introduces you to his logs. With our three basic log rules, and our four handy log/antilog tables, we can do this and many other nasty computations. 2001, Lawrence Morales; MAT107 Chapter 6 Page 41

42 Recall: Example 28 Definition of log p log n = p b n b = Basic log Rules 1.) log AB = log A + log B A 2.) log = log A log B B c.) logn = clog n Multiply 874,200,000 7,85,000 We ll start all such problems by letting x equal the quantity we want to compute. Operation Comments x = 874,200,000 7,85,000 This is our computation log x = log(874,200,000 7,85,000) Take the log of both sides. This is going to transform our statement into one with logs, taking us into logland. It s a key step! log x = log874,200,000 + log 7,85,000 Apply log rule #1 (see above) 10 8 < 874,200,000 < 10 9, so Compute log 874,200,000 with the log 874,200,000 = 8 + log8.742 = table and save it for later. You can also think of 874,200,000 as to help find the characteristic < 7,85,000 < 10 7, so Compute log 7,85,000 with the table. log 7,85,000 = 6 + log 7.85 = You can also think of 7,85,000 as to help find the characteristic. log x = = Add these two logs together by hand. Addition is easy! This is not our answer, of course. We now need to call in the evil twin to take us out of logland log x = = x Rewrite using the definition of logs x = 10 = = Use the antilog table to compute what x is. Remember, x is the desired quantity. x = We ll leave this in scientific notation. 2001, Lawrence Morales; MAT107 Chapter 6 Page 42

43 Check Point P Example 29 If we cheat and take out our calculator, we ll find that 874,200,000 7,85,00 = This means that our log table was good to three decimal places, but missed the fourth. With tables that had more accuracy, we could get a better answer. Multiply 57,600,000 4,591,000,000,000 using the log and antilog tables. See the endnotes to check your answer. 2 Compute 89.5 x = with the log tables Although these numbers are not very ugly, they will demonstrate the process just fine. Operation Comments 89.5 x = Take the log of both sides log x = log log x = log89.5 log Apply log rule #2 log x = = Use the log tables to carefully find the two logs on the right. You should verify these before you proceed. Note that the characteristic of log89.5 is 2 since 10 2 <89.5<10. The characteristic of is 0. (Why?) log x = = x Definition of log x = 10 = = Rewrite and use the antilog table to compute the value of x. = x = This is our final answer. A calculator gives as the final answer. Not too bad (but not that great, either). Check Point Q Compute x = with the log tables See endnotes for a solution. 2001, Lawrence Morales; MAT107 Chapter 6 Page 4

44 Example 0 Operation Find the value of y = Many complex important calculations require that numbers be raised to powers like this. The tables can help us here as well. Comments y = log y = log84.65 Take the log of both sides. log y = 8log84.65 Use log rule # to move the power in front of the log. log y = 8 ( ) = Take the log of Note that the characteristic is 1 since 10 1 <84.65<10 2. At this point we have to do While we could use a log table to do this (nested logs, yuck), we observe that a simple multiplication like this is probably something Napier s contemporaries could have done they were concerned about the nasty ones. log y = Multiply by hand Now we know what log y is. We can undo the log by using the antilog table log y = = Apply the log definition and simplify so that we can use the antilog table log y = = Use the antilog table to do and we have a final answer. A calculator gives the result to be Check Point R This last example truly shows the power of logarithms. In the 1600 s, it was not easy at all to do this last kind of computation. By converting to logs, working with addition and subtraction (and perhaps some easy multiplication), and then converting back from logs to regular numbers, they could raise a decimal fraction to a power with relatively little work. Compute with the log tables. See endnotes for a solution. 4 We can even take roots of numbers with logs. We simply need to remember a basic rule of exponents: m / n n x = x m 2001, Lawrence Morales; MAT107 Chapter 6 Page 44

45 This tells us how to express roots with fractional powers. Here are two common identities: Example 1 Find the value of x = / 2 x = x x = x Operation Comments x = = ( 789.2) 1/ 2 Use the exponent rule to convert. log x = log( 789.2) 1/ 2 Take the log of both sides. 1 Use log rule # to move the power in front. log x = log Use the log table to compute log log x = ( ) 2 log x = Take half of the result by hand this is easy enough to do without invoking another layer of logs x = 10 = = = Rewrite x and then use the antilog table to compute The final answer of compares with a calculator result of Check Point S : 1/ Find the value of 1,42,000 using the log tables See endnotes for a solution. 5 With this capability in hand, astronomers, mathematicians, and other scientists could speed up their calculations and accelerate their discoveries. Of course, they still had to learn the basic skills of arithmetic, but that wasn t a major problem. It is analogous to when the handheld calculator infiltrated the classroom. They made calculations much faster and easier, but they are not the answer to learning math. Students of math still need to learn the basics so that they can check the reasonableness of answers and so that when confronted with a relatively simple problem, they don t have to go searching for a calculator to pound on. Unfortunately, it seems as though we are overly dependent on calculators. For some, it slows them down since every computation must be done with that wretched piece of plastic. (I ve even seen students reach for their calculator when asked what 2 is!). For most, it leads to a false sense of security about answers because their addiction to the calculating machine leaves them powerless to question whether or not their answers even make sense. 2001, Lawrence Morales; MAT107 Chapter 6 Page 45

46 PART 5: New Calculating Devices The Slide Rule As word of the logarithm spread, mathematicians took the idea and not only used it to help them with their work, but also extended it. Edmund Gunter, a faculty member at Gresham College where Briggs was a professor of geometry, invented a tool with log, tangent, and sine scales which was used by navigators. In 1621, William Oughtred invented a slide rule that helped with computations. The slide rule slowly evolved and became an important computing device until the handheld calculator and desktop computer emerged as the computing workhorses of the modern era. Napier and Other Calculating Devices Napier is also known for creating another, less powerful computing device called Napier s Bones. They are multiplication devices based on the gelosia method of multiplication that we discussed earlier. Napier noted that the columns on the gelosia grid were merely multiples of numbers at the top of the columns. He took the columns and placed them on vertical rods which could then be moved around easily and used as a mobile computing device. He called this invention Rabdologia which means a collection of rods in Greek. They eventually picked up the nickname Napier s Rods, or Napier s Bones. There is a full set of the rods in the appendix that you can cut out if you would like. We can see how these rods work with an example. Let s compute To do this: We take the 4, 8, and 6 rods and place them in order next to each other, and we place the index rod on the right of these. 2. To find the product 486 7, read along the row marked with a 7 on the index rod.. Add the numbers along the diagonals, just as before, to get 402. Ignore all other rows, as they are not needed (since we are multiplying by 7). 4 As you can imagine, this eliminates the need to write the grid down because you can rearrange rods and read off results rather quickly. Example Use the rods to find the product Here we have two steps to take. First multiply 569 by 7. Next, multiply 569 by 2 tens. Essentially, we are thinking of this problem as (569) (7 1)+569 (2 10) 2001, Lawrence Morales; MAT107 Chapter 6 Page

47 Example 569 (7 ones) = (569 7) 1 = 98 1 = (2 tens) = (569 2) 10 = = 1180 Result = 156 We are once again thinking of multiplication in terms of its relationship to addition. By adding together all the appropriate places, we get to a result. Use the rods to compute We think of this as 61 (4 1) + 61 (5 10) + 61 (8 100) 61 (4 1) = (61 4) 1 = (5 10) = (61 5) 10 = (8 100) = (61 8) 100 = Result = 58,874 A calculator check will show this to be true. Since addition was (and is) much easier than multiplication, this method 7 provided a nice way to be able to compute on the fly. In a time when 8 merchants and other professionals in fields such as commerce were required 9 to do increasingly complication computations, time saving devices like these were often cherished. It is reported that some people even jealousy guarded their secrets from others so that they could have a computational advantage. Check Point T Use your own set of the rods to find 542 8, and You can use a calculator to check your answers Think About It Could you adapt the rods so that they would compute ? How? 2001, Lawrence Morales; MAT107 Chapter 6 Page 47

48 The rods were eventually manufactured and carried around so that they could be used to mobile calculating. Here are some sets of Napier rods that show how they looked , Lawrence Morales; MAT107 Chapter 6 Page 48