Logarithms
A logarithm is an exponent. The logarithm (to the base 10) of 100 is 2 because 102 = 100. This can be abbreviated log10100 = 2.
Because logarithms are exponents, they have an intimate connection with exponential functions and with the laws of exponents.
The basic relationship is bx = y if and only if x = logb y. Since 23 = 8, log2 8 = 3. Since, according to a table of logarithms, log10 2 = .301, 10.301 = 2.
The major laws of logarithms and the exponential laws from which they are derived are as follows:
I. logb (xy) = logb x + logb y | bn · bm = bn+m
II. logb (x/y) = logb x - logb y | bn/bm = bn-m
III. logb xy = y · logb x | (bn)m = b(nm)
IV. logb x = (logb a)(logb x) | If x = br; b = ap, then x = apr
V. logb bn = n | If bn = bm, then n = m
VI. Log 1 = 0 (any base) | b0 = 1
In all these rules, the bases a and b and the arguments x and y are limited to positive numbers. The exponents m, n, p, and r and the logarithms can be positive, negative, or zero.
Because logarithms depend on the base that is being used, the base must be clearly identified. It is usually shown as a subscript. There are two exceptions. When the base is 10, the logarithm can be written without a subscript. Thus log 1000 means log10 1000. Logarithms with 10 as a base are called " common" or " Briggsian." The other exception is when the base is the number e (which equals 2.718282...). Such logarithms are written ln x and are called " natural" or " Napierian" logarithms.
In order to use logarithms one must be able to evaluate them. The simplest way to do this is to use a "scientific" calculator. Such a calculator will ordinarily have two keys, one marked "LOG," which will give the common logarithm of the entered number, and the other "LN," which will give the natural logarithm.
Lacking such a calculator, one can turn to the tables of common logarithms which are to be found in various handbooks or as appendices to various statistical and mathematical texts. In using such tables one must know that they contain logarithms in the range 0 to 1 only. These are the logarithms of numbers in the range 1 to 10. If one is seeking the logarithm of a number, say 112 or .0035, outside that range, some accommodation must be made.
The easiest way to do this is to write the number in scientific notation:
112 = 1.12 x 102
.0035 = 3.5 x 10-3
Then, using law I
log 112 = log 1.12 + log 102
log .0035 = log 3.5 + log 10-3
Log 1.12 and log 3.5 can be found in the table. They are .0492 and .5441 respectively. Log 102 and log 10-3 are simply 2 and -3 according to law V: therefore
log 112 = .0492 + 2 = 2.0492
log .0035 = .5442 -3 = -2.4559
The two parts of the resulting logarithms are called the "mantissa" and the "characteristic." The mantissa is the decimal part, and the characteristic, the integral part. Since tables of logarithms show positive mantissas only, a logarithm such as -5.8111 must be converted to .1889 - 6 before a table can be used to find the "antilogarithm," which is the name given to the number whose logarithm it is. A calculator will show the antilogarithm without such a conversion.
Tables for natural logarithms also exist. Since for natural logarithms, there is no easy way of determining the characteristic, the table will show both characteristic and mantissa. It will also cover a greater range of numbers, perhaps 0 to 1000 or more. An alternative is a table of common logarithms, converting them to natural logarithms with the formula (from law IV) ln x = 2.30285 x log x.Logarithms are used for a variety of purposes. One significant use-the use for which they were first invented-is to simplify calculations. Laws I and II enable one to multiply or divide numbers by adding or subtracting their logarithms. When numbers have a large number of digits, adding or subtracting is usually easier. Law III enables one to raise a number to a power by multiplying its logarithm. This is a much simpler operation than doing the exponentiation, especially if the exponent is not 0, 1, or 2.
At one time logarithms were widely used for computation. Astronomers relied on them for the extensive computations their work requires. Engineers did a majority of their computations with slide rules, which are mechanical devices for adding and subtracting logarithms or, using log-log scales, for multiplying them. Modern electronic calculators have displaced slide rules and tables for computational purposes--they are quicker and far more precise--but an understanding of the properties of logarithms remains a valuable tool for anyone who uses numbers extensively.
If one draws a scale on which logarithms go up by uniform steps, the antilogarithms will crowd closer and closer together as their size increases. They do this in a very systematic way. On a logarithmic scale, as this is called, equal intervals correspond to equal ratios. The interval between 1 and 2, for example, is the same length as the interval between 4 and 8.
Logarithmic scales are used for many purposes. The pH scale used to measure acidity and the decibel scale used to measure loudness are both logarithmic scales (that is, they are the logarithms of the acidity and loudness). As such, they stretch out the scale where the acidity or loudness is weak (and small variations noticeable) and compress it where it is strong (where big variations are needed for a noticeable effect). Another example of the advantage of a logarithmic scale can be seen in a scale which a sociologist might construct. If he were to draw an ordinary graph of family incomes, an increase of a dollar an hour in the minimum wage would seem to be of the same importance as a dollar-an-hour increase in the income of a corporation executive earning a half million dollars a year. Yet such an increase would be of far greater importance to the family whose earner or earners were working at the minimum-wage level. A logarithmic scale, where equal intervals reflect equal ratios rather than equal differences, would show this.
Logarithmic functions also show up as the inverses of exponential functions. If P = ket, where k is a constant, represents population as a function of time, then t = K + ln P, where K = -ln k, also a constant, represents time as a function of population. A demographer wanting to know how long it would take for the population to grow to a certain size would find the logarithmic form of the relationship the more useful one.
Because of this relationship logarithms are also used to solve exponential equations, such as 3 - = 2x as or 4e k = 15.
The invention of logarithms is attributed to John Napier, a Scottish mathematician who lived from 1550 to 1617. The logarithms he invented, however, were not the simple logarithms we use today (his logarithms were not what are now called "Napierian"), Shortly after Napier published his work, Briggs, an English mathematician met with him and together they worked out logarithms that much more closely resemble the common logarithms that we use today. Neither Napier nor Briggs related logarithms to exponents, however. They were invented before exponents were in use.
http://www.unc.edu/~sebailey/math10/Homework12.pdf
http://www.ibiblio.org/obp/books/socratic/output/loganalog.pdf
http://www.dtc.umn.edu/~odlyzko/doc/arch/discrete.logs.pdf
Some bacteria reproduce very quickly, as you may have discovered if you have ever had an infected cut or strep throat. Under the right circumstances, the number of bacteria in certain cultures will double in as little as an hour (or even less). In this section, we discuss some functions that can be used to model such rapid growth.
For instance, suppose that initially there are 100 bacteria at a given site, and the population doubles every hour. Call the population function P(t ), where t represents time (in hours) and start the clock running at time t = 0. The initial population of 100 means that P(0) = 100. After 1 hour, the population will double to 200, so that P(1) = 200. After another hour, the population will have doubled again to 400, making P(2) = 400. After a third hour, the population will have doubled again to 800, making P(3) = 800 and so on.
Let's compute the population after 10 hours. (Try guessing this now. Most people guess poorly on this type of problem.) You could calculate the population at 4 hours, 5 hours, and so on, or you could use the following shortcut. To find P(1), you double the initial population, so that P(1) = 2 100. To find P(2), you double the population at time t = 1, so that P(2) = 2 2 100 = 22 100. Similarly, P(3) = 23 100. This pattern leads us to
P(10) = 210 100 = 102, 400.
Depending on the organism, this is now a population that could cause some trouble!
The pattern discovered above suggests that the population can be modeled by the function
P(t ) = 2t 100.
We call P(t ) an exponential function because the variable t is in the exponent. There is a subtle question here: what is the domain of this function? We have so far used only integer values of t, but for what other values of t does P(t ) make sense? Certainly, rational powers make sense, as in P(1/2) = 21/2 100 , where you recognize that 21/2 = . This says that the number of bacteria in the culture after a half hour is
P(1/2) = 21/2 100 = 100 141.
In this context, the function is not intended to give the exact number of bacteria at a given time, but rather, to predict an approximate number.
In any case, it's a simple matter to interpret fractional powers as roots. For instance, x1/2 = , , , and so on. Further, we can write
and so on. But, what about irrational powers? These are somewhat harder to define, but they work exactly the way you would want them to. For instance, since is between 3.14 and 3.15, 2 is between 23.14 and 23.15. In this way, we define 2x for x irrational to fill in the gaps in the graph of y = 2x for x rational. That is, if x is irrational and a < x < b , for rational numbers a and b, then 2a < 2x < 2b. This is the logic behind the definition of irrational powers. (We completely resolve this question in Chapter 6.)
If for some reason, you wanted to find the population after hours, you can use your calculator or computer to obtain the approximate population:
P() = 2 100 882.
For your convenience, we summarize the usual rules of exponents below.
For any integers m and n,
For any real number p,
For any real numbers p and q, (xp)q = xp q.
For any real numbers p and q, xp xq = xp + q.
Throughout your calculus course, you will need to be able to quickly convert back and forth between exponential form and fractional or root form.
6.1
Converting Expressions to Exponential Form
Convert each to exponential form: (a) (b) (c) and (d) (2x 23 + x)2.
For (a), simply leave the 3 alone and convert the power:
For (b), use a negative exponent to write x in the numerator:
For (c), first separate the constants from the variables and then simplify:
For (d), first work inside the parentheses and then square:
(2x 23 + x) 2 = (2x + 3 + x) 2 = (22x + 3) 2 = 24x + 6.
The function in part (d) of example 6.1 is called an exponential function with a base of 2.
6.1
For any constant b > 0 , the function f (x) = bx is called exponential function. Here, b is called the base and x is the exponent.
Be careful to distinguish between algebraic functions like f (x = x3) and g(x) = x2/3 and exponential functions. Confusing these types of functions is a very common error. Notice that for exponential functions like h(x) = 2x, the variable is in the exponent (hence the name), instead of in the base. Also, notice that the domain of an exponential is the entire real line, ( - , ), while the range is the open interval (0, ).
While any positive real number can be used as a base for an exponential function, three bases are the most commonly used in practice. Base 2 arises naturally when analyzing processes that double at regular intervals (such as the bacteria at the beginning of this section). Also, most computers perform their calculations using base 2 arithmetic. Our standard counting system is base 10, so this base is commonly used. However, far and away the most useful base is the irrational number e. Like , the number e has a surprising tendency to occur in important calculations. We define e by the following:
(6.1)
Note that equation (6.1) has at least two serious shortcomings. First, we have not yet said what the notation means. (In fact, we won't define this until Chapter 1.) Second, it's unclear why anyone would ever define a number in such a strange way. We will not be in a position to answer the second question until Chapter 6 (but the answer is worth the wait). It suffices for the moment to say that equation (6.1) means that e can be approximated by calculating values of (1 + 1/n)n for large values of n and that the larger the value of n, the closer the approximation will be to the actual value of e. In particular, if you look at the sequence of numbers (1 + 1/2)2, and so on, they will get progressively closer and closer to (i.e., home-in on) the irrational number e (named in honor of the famous mathematician Leonhard Euler).
To get an idea of the value of e, we compute several of these numbers:
and so on. You should compute enough of these values to convince yourself that the first few digits of the decimal representation of e (e 2.718281828459) are correct.
6.2
Computing Values of Exponentials
Approximate e4, e - 1/5 and e0.
From a calculator, we find that
e4 = e e e e 54.598.
From the usual rules of exponents,
(On a calculator, it is convenient to replace - 1/5 with -0.2.) Finally, e0 = 1.
The graphs of the exponential functions summarize many of their important properties.
6.3
Sketching Graphs of Exponentials
Sketch the graphs of the exponential functions y = 2x, and y = e - x.
Using a calculator or computer, you should get graphs similar to those that follow.
Figure 0.55a
y = 2x Figure 0.55b
y = ex
Figure 0.56a
y = e2x Figure 0.56b
y = ex/2
Figure 0.57a
y = (1/2)x Figure 0.57b
y = e - x
Notice that each of the graphs in Figures 0.55a, 0.55b, 0.56a, and 0.56b start very near the x - axis (reading left to right), pass through the point (0, 1), and then rise steeply. This is true for all exponentials with base greater than 1 and with a positive coefficient in the exponent. Note that the larger the base (e > 2) or the larger the coefficient in the exponent (2 > 1 > 1/2) , the more quickly the graph rises to the right (and drops to the left). Note that the graphs in Figures 0.57a and 0.57b are the mirror images in the y - axis of Figures 0.55a and 0.55b, respectively. The graphs rise as you move to the left and drop toward the x - axis as you move to the right. It's worth noting that by the rules of exponents, (1/2)x = 2 - x and (1/e)x = e - x.
The exponential functions are closely related to another family of functions called the logarithmic functions.
6.2
For any positive number b 1 , the logarithm function with base b, written log bx, is defined by
y = log bxif and only if x = by.
That is, the logarithm log bx gives the exponent to which you must raise the base b to get the given number x. For example,
log1010 = 1 (since 101 = 10),
log10100 = 2 (since 102 = 100),
log101000 = 3 (since 103 = 1000)
and so on. The value of log 1045 is less clear than the preceding three values, but the idea is the same: you need to find the number y such that 10y = 45. The answer is between 1 and 2 (why?), but to be more precise, you will need to employ trial and error. (Of course, you can always use your calculator or computer to compute an approximate value, but that won't help you to understand what the log is all about.) You should get log 1045 1.6532. Later in the course, we introduce you to a powerful method for accurately approximating the values of such functions.
Observe from Definition 6.2 that for any base b > 0 (b 1) , if y = logb x, then x = That is, the domain of f (x) = logb x is the interval (0, ). Likewise, the range of f is the entire real line, ( - , ).
As with exponential functions, the most useful bases turn out to be 2, 10, and e. We usually abbreviate log 10x by log x. Similarly, log ex is usually abbreviated ln x (for natural logarithm). It is most likely unclear what is so natural (or even interesting) about a logarithm with an irrational base (after all, why wouldn't log 10x be the natural logarithm?), but we clear this up in Chapter 6.
6.4
Evaluating Logarithms
Without using your calculator, determine log (1/10), and ln e3.
Since 1/10 = 10 - 1, Similarly, since 0.001 = 10 - 3 , we have that log (0.001) = - 3. Since ln e = log ee1, Similarly, ln e3 = 3.
Observe that one consequence of Definition 6.2 is thateln x = x, for any x > 0.
(6.2)
We demonstrate this as follows. Let
y = ln x = log ex.
By Definition 6.2, we have that
x = ey = eln x.
We can use this relationship between natural logarithms and exponentials to solve equations involving logarithms, as in the following example.
6.5
Solving a Logarithmic Equation
Solve the equation ln (x + 5) = 3 for x.
Taking the exponential of both sides of the equation and writing things backward (for convenience), we have
e3 = eln ( x + 5) = x + 5,
from (6.2). Subtracting 5 from both sides gives us
e3 -5 = x.
As always, graphs provide excellent visual summaries of the important properties of functions.
6.6
Sketching Graphs of Logarithms
Sketch graphs of y = log x and y = ln x and briefly discuss the properties of each.
From a calculator or computer, you should obtain graphs resembling those in Figures 0.58a and 0.58b. Notice that both graphs appear to have a vertical asymptote at x = 0 (why would that be?), cross the x - axis at x = 1 and very gradually increase as x increases. Neither graph has any points to the left of the y - axis, since log x and ln x are defined only for x > 0. The two graphs are very similar, although not identical.
Figure 0.58a
y = log x Figure 0.58b
y = ln x
The properties described above graphically are summarized in the following result.
6.1
For any positive base b 1 , (i) log bx is defined only for x > 0 ,
(ii) log b1 = 0 and
(iii) if b > 1, then log bx < 0 for 0 < x < 1 and logb x > 0 for x > 1
.
(i) Note that since b > 0, for any y. So, if log bx = y, then x = by > 0.
(ii) Since b0 = 1 for any number b 0, log b1 = 0 (i.e., the exponent to which you raise the base b to get the number 1 is 0).
(iii) We leave this as an exercise.
All logarithms share a set of defining properties, as stated in the following theorem.
6.2
For any positive base b 1 and positive numbers x and y, we have (i) log b(xy) = log bx + log by ,
(ii) log b(x/y) = log bx - log by
(iii) log b(xy) = ylog bx.
As with most algebraic rules, each of these properties can dramatically simplify calculations when they apply.
6.7
Simplifying Logarithmic Expressions
Write each as a single logarithm: (a) log 227x - log 23x and (b) ln 8-3ln (1/2).
First, note that there is more than one order in which to work each problem. For part (a), we have 27 = 33 and so, 27x = (33)x = 33x. This gives us
log 227x - log 23x = log 233x - log 23x
= 3xlog 23 - xlog 23 = 2xlog 23 = log 232x.
For part (b), note that 8 = 23 and 1/2 = 2 - 1. Then,
ln 8-3ln (1/2) = 3ln 2-3( - ln 2)
= 3ln 2 + 3ln 2 = 6ln 2 = ln 26 = ln 64.
Notice that from Theorem 6.2 and the definition of the natural logarithm, we have that for any real number x ln (ex) = xln e = x.
(6.3)
We can use this relationship between exponentials and logarithms to solve equations involving exponentials, as in the following example.
6.8
Solving an Exponential Equation
Solve the equation ex + 4 = 7 for x.
Taking the natural logarithm of both sides and writing things backward (for simplicity), we have from (6.3) that
ln 7 = ln ( ex + 4) = x + 4.
Subtracting 4 from both sides yields
ln 7-4 = x.
Using the rules of exponents and logarithms, notice that we can rewrite any exponential as an exponential with base e, as follows. For any base a > 0, we have
(6.4)
This follows from Theorem 6.2 (iii) and the fact that eln y = y, for all y > 0.
6.9
Rewriting Exponentials as Exponentials with Base e
Rewrite the exponentials 2x, 5x and (2/5)x as exponentials with base e.
From (6.4), we have
and
6.10
Simplifying Logarithmic Expressions
Use the rules of logarithms to simplify the expression 5ln x + 3ln y - 4ln z.
From Theorem 6.2, we have
In some circumstances, it is beneficial to use the rules of logarithms to expand a given expression, as in the following example.
6.11
Expanding a Logarithmic Expression
Use the rules of logarithms to expand the expression ln
From Theorem 6.2, we have that
= 3ln x + 4ln y - 5ln z.
Just as we were able to use the relationship between the natural logarithm and exponentials to rewrite an exponential with any positive base in terms of an exponential with base e, we can use these same properties to rewrite any logarithm in terms of natural logarithms, as follows. For any positive base b (b 1) , we will show that
(6.5)
Let y = lnbx. Then by Definition 6.2, we have that x = bI. Taking the natural logarithm of both sides of this equation, we get by Theorem 6.2 (iii) that
ln x = ln ( by ) = y ln b.
Dividing both sides by ln b (since b 1 , ln b 0 gives us
establishing (6.5).
One use you will find for equation (6.5) is for computing logarithms with bases other than e or 10. More than likely, your calculator has keys only for ln n 0 and log x 0. We illustrate this idea in the following example.
6.12
Approximating the Value of Logarithms
Approximate the value of log712.
From (6.5), we have
6.13
Matching Data to an Exponential Curve
Find the exponential function of the form f ( x) = aebx that passes through the points (0, 5) and (3, 9).
Notice that you can't simply solve this problem by inspection. That is, you can't just read off appropriate values for a and b. Instead, we solve for a and b, using the properties of logarithms and exponentials. First, for the graph to pass through the point (0, 5), this means that
5 = f (0) = aeb 0 = a,
so that a = 5. Next, for the graph to pass through the point (3, 9), we must have
9 = f (3) = ae3b = 5e3b.
To solve for the b in the exponent, we divide both sides of the last equation by 5 and take the natural logarithm of both sides, which yields
from (6.3). Finally, dividing by 3 gives us the value for b:
Thus,
Historically, logarithms were useful for handling large numbers. Given the wide availability of calculators and computers, this particular usage is now only a minor convenience (see the exercises on decibels and the Richter scale). Today, logarithms serve many other important purposes, including manipulating exponential functions.
Fitting a Curve to Data
Consider the population of the United States from 1790 to 1860, found in the accompanying table. A plot of these data points can be seen in Figure 0.59 (where the vertical scale represents the population in millions).Year U. S. Population
1790 3,929,214
1800 5,308,483
1810 7,239,881
1820 9,638,453
1830 12,866,020
1840 17,069,453
1850 23,191,876
1860 31,443,321
Figure 0.59
U. S. Population 1790-1860
This shows that the population was increasing, with larger and larger increases each decade. If you sketch an imaginary curve through these points, you will probably get the impression of a parabola or perhaps the right half of a cubic or exponential. And that's the question: is this data best modeled by a quadratic function, a cubic function, a fourth-order polynomial, an exponential function, or what?
We can use the properties of logarithms from Theorem 6.2 to help determine whether a given set of data is modeled better by a polynomial or an exponential function, as follows. Suppose that the data actually comes from an exponential, say y = aebx (i.e., the data points lie on the graph of this exponential). Then,ln y = ln (aebx) = ln a + ln ebx = ln a + bx.
If you draw a new graph, where the horizontal axis shows values of x and the vertical axis corresponds to values of ln y , then the graph will be the line ln y = bx + c (where the constant c = ln a ). On the other hand, suppose the data actually came from a polynomial. If y = bxn (for any n), then observe thatln y = ln (bxn) = ln b + ln xn = ln b + nln x.
In this case, a graph with horizontal and vertical axes corresponding to x and ln y, respectively, will look like the graph of a logarithm, ln y = nln x + c. Such a semi-log graph (i.e., graphing ln y versus x) lets us distinguish the graph of an exponential from that of a polynomial: graphs of exponentials become straight lines, while graphs of polynomials (of degree 1) become logarithmic curves. Scientists and engineers frequently use semi-log graphs to help them gain an understanding of physical phenomena represented by some collection of data.
6.14
Using a Semi-Log Graph to Identify a Type of Function
Determine if the population of the United States from 1790-1860 was increasing exponentially or as a polynomial.
As indicated above, the trick is to draw a semi-log graph. That is, instead of plotting (1, 3.9) as the first data point, plot (1, ln 3.9) and so on. A semi-log plot of this data set is seen in Figure 0.60. Although the points are not exactly colinear (how would you prove this?), the plot is very close to a straight line with ln y - intercept of 1 and slope 0.35. You should conclude that the population is well-modeled by an exponential function. (Keep in mind that here, as with most real problems, the data is somewhat imprecise and so, the points in the semi-log plot need not be perfectly colinear for you to conclude that the data is modeled quite well by an exponential.) The exponential model would be y = P(t ) = aebt, where t represents the number of decades since 1780. Here b is the slope and ln a is the ln y - intercept of the line in the semi-log graph. That is, b = 0.35 and ln a = 1 (why?), so that a = e. The population is then modeled by
P(t ) = e e0.35t million.
Figure 0.60
Semi-log plot of U. S. population.
Greek, logistikh meant arithmetic, or calculating with numbers. In schools, it is dignified as mathematics, of which it is a small part. Now, geometry is real mathematics, but it is not taught any more in American schools. The Greek word appears in logarithms, "things for computing with numbers," another subject no longer making American schools unnecessarily hard. In any case, logistics or arithmetic is very useful; indeed, it is essential to survival in society, and so a proper subject for elementary education. Logistics is probably more familiar in general as the art of moving and quartering troops, the word derived from the French loger, to lodge.
These days, we have the wonderful advantage of the electronic digital computer, notably in the form of the pocket calculator. This powerful device is available at an unbelievably low cost. At the very least, it will peform the four arithmetic functions and take square roots. At only a modest increase in price, trigonometric and exponential functions are added. At other places on this site, the capabilities of the HP-48, an advanced scientific calculator, are discussed in detail. We won't speak of electronic computers in this article, except to note their existence and compare them with other computing resources.
Calculating machines were invented by Blaise Pascal in 1642, by Samuel Morland in 1666, and by Gottfried Liebniz in 1694. These were beyond the constructional skill of the time, and never worked well or became common. Only after about 1820 were calculating machines made practical. Typically, they could add and subtract, and sometimes multiply. They were used chiefly in business; scientific calculations depended on logarithms until recently. In the 20th century, excellent mechanical machines that could perform all four arithmetic operations became available, but were always expensive and heavy. They replaced logarithms to some degree until driven out by the electronic calculator, which not only could be made more capable (giving trigonometric and exponential functions), but, more importantly, were very much cheaper.
Perhaps we should mention first the common method of computing with pen and paper and mental calculations, called algorism, that had its roots in ninth century Baghdad, at the court of Harun-ar-Rashid. This method requires very little equipment, only the educated person's pen and ink, but learning it consumes many years of elementary education, including the memorization of addition and multiplication tables. It is very slow and error-prone, but is still hardly questioned as a fundamental of elementary education. Any really large problem cannot be done without error, unless laboriously checked, so prone are humans to mistakes. In spite of this, it retains a wide currency, and most pupils are brought to a rudimentary level of skill in it. The level is so rudimentary that most people today are capable of only a little addition and subtraction, with multiplication far too much trouble, and long division a mystery. Not many years ago, even the extraction of square roots was taught in American schools; possibly no one is now left there who can perform this wonder. See Square Roots by Hand if you would like to review the procedure. There is an amazing amount of effort expended on algorism in schools with very little result, and it falls almost completely under the despised classification of rote learning. A much smaller amount of effort would produce experts on the pocket calculator. It must, however, be emphasized that mental arithmetic is a very useful skill indeed.
Experience again how cumbersome algorism actually is by reviewing the method of adding a column of multidigit numbers. First, we have to write all the numbers down. Then, we begin at the right-hand column and add the digits mentally, hoping not to forget any of the carries. At the bottom of the column, we write down a digit of the sum, and write the carry at the top of the next column. The addition of single digits in the next column proceeds as before. Finally, we sum the furthest column on the left and have what we hope is the answer. To have any confidence in the result, we then repeat this operation from bottom to top instead, and see if we get the same answer. If some of the numbers in the column are to be subtracted, we must usually add the positive and negative numbers separately, and then do the subtraction with the sums, to avoid insurmountable difficulties in mixing addition and subtraction in the same column (though, of course, it can be done). With subtraction, there is the additional problem of borrows, which have to be explicitly recorded. The order of working is enforced by the requirement that when we write down a number, it cannot be changed. All of this is really very inconvenient and tedious. A good way to check algorism (casting out nines) is explained below.
Counting Boards and the Abacus
Algorism is a rather late development, not widespread in Europe until the 16th or 17th century, and hardly known elsewhere in the world. It began as a learned, not a popular, practice. Scientific and mathematical workers had used similar methods from antiquity, especially in dealing with sexagesimal numbers, but algorism did not penetrate to everyday life until then. The traditional and universal method of computing among the general public was to use movable counters that represented certain values, and actually to add or subtract counters representing the numbers under consideration. Counting on fingers is an obvious example of this, and one that was widely used and elaborated into complex systems, though now totally forgotten. The word "digit" comes from this, though digital computers do not use fingers. Chinese number sticks were another example. The word calculate even comes from calculus, a "pebble," typically used as a marker. The analogous Greek yhfos (psehphos), a "rounded pebble," gave rise to the verb yhfizw (psehfizo), which refers more commonly to voting, which was also done with pebbles, than to calculating. Parallel lines were drawn on a tabletop or scratched in the clay, representing units, tens, hundreds and so forth, creating a counting board, and even giving us the name of the table on which sales were made in a shop, the counter. When ten pebbles accumulated on the units line, they were removed and replaced by one pebble on the 10's line. The method was based on the decimal system (though it could just as well have been based on other radicals). These calculating boards were universal in every place where there was commerce, money and a developed numerical system. They were more accurate, and faster, than algorism, which probably required some time and effort to be accepted.
The number of markers that had to be handled was reduced by introducing markers with multiple values, especially 5 (Chinese number rods had values 1 or 5 depending on their orientation). A much greater advance was making the markers captive beads in the slots or on the rods of a portable counting board, now called an abacus. Abacus is simply the Latin for a counting board, from the Greek abax for the same thing. It is not known when or where the portable abacus was devised, but it was universal in the classical world, and small portable abaci were carried by anyone who had to do arithmetic. Roman numerals are simply an input-output notation for the abacus; no one even thought of doing algorism with them, and there was never any need to do so. Similarly, we use our number symbols with a pocket calculator, but the calculations are done in binary. The Roman abacus had 5 1's markers, and 2 5's markers in each slot, as well as markers for special purposes, such as halves. The markers were called claviculi, "little nails." Only two examples are known to have survived; they are illustrated in Menninger. No examples from earlier times are known.
The abacus is now associated with China and Japan, but it is a relatively late comer there. For information on Chinese numbers and the history of the abacus, see Chinese Numbers. It could have appeared in the 13th century, perhaps with the Mongols, but there are obscure earlier references. Its similarity to the Roman abacus suggests that that was the ultimate source, and this was very possible, since there were direct trade relations between the classical world and China, and Mongol traders were a bridge between East and West. It could even have been introduced by the Roman soldiers captured by the Persians and sold to the Chinese emperor as engineers. Most were later ransomed, but many found China much to their liking. The suan-p'an, the Chinese abacus, superseded the traditional calculating rods, and was transformed into its present form by the introduction of bamboo for the Roman brass, and beads on rods rather than buttons in slots. It retains to this day two 5's markers and 5 1's markers on each rod. The suan-p'an came to Japan as the soroban, which was streamlined and improved there into a very elegant device.
A typical soroban has 21 rods, on each of which is one 5's marker and 4 1's markers (some soroban apparently had 5 1's markers). The 5's marker is above the horizontal dividing bar, while the 1's markers are below it. The markers are given value by being moved to the bar with the index finger or thumb. While away from the bar, they "do not count." When using the soroban, one concentrates on each rod and then passes to the next, working from left to right, as numbers are read. There is no mental arithmetic as in algorism, but only consideration of 10's or 5's complements. The answers appear as if by magic, and are very seldom in error. The addition or subtraction is already done when the number has been entered.
A few examples will make it clear how to use the soroban. If you have access to a soroban, it will be much more interesting to do the examples on it. First, you should practice clearing the soroban, and seeing how each of the digits from 0 to 9 can be set. Note that every third rod is marked with a dot. In any calculation, the unit rod should be selected as one of these. This makes it easier to keep track of the digits in a large number, as by commas when the number is written down. Some operations are very easy to do, since enough markers in the right places may be present. For example, set 3 on the units rod. Now add 5 to it, which simply amounts to pushing down the 5 counter. 8 then appears automatically. You do not have to know that 3 + 5 = 8!
5 + 7 is a little more difficult. Set 5 on the units rod. There are not enough counters there to make up 7, so the only thing to do is add 10 on the next rod (do not do it now!) and subtract 3. To subtract 3, we can only subtract 5, by moving the counter up, and add 2, by moving two 1 counters up. In the first case, we had to know that 7 + 3 = 10, and in the second, that 2 + 3 = 5. Well, after this, move a 1's counter up in the next rod to the left. Since we are working from left to right, this changes the number that was there, something we cannot do in algorism, so we are forced to proceed right to left. Now what we have left is 12 on the two rods, which is the answer. At no time did we have to know what 5 + 7 is, only the tens and fives complements so we could manipulate the markers correctly. The 12 appears automatically! We only have to consider how to add and subtract markers from one rod at a time.
Now set the number 100 on the soroban. Let us subtract 1, or actually 001. There is nothing to do until we get to the third rod. We can't subtract 1 from 0, so we must subtract 10 from the rod on the left and add 9 to the rod under consideration. It is easy enough to add 9. Now we have to subtract 1 from the next 0, which again gives us 9. On the next rod, we do have a 1, so all that is necessary is to move it down. The result is 99. Now let's add 1 back. The units rod is 10 - 9, so we replace the 9 by 0. We have to add 1 to the next rod, which gives 0 again, and we have a 1 to add to the third rod. The result is 100. There is never anything to remember, never anything to write down.
Let's do 78 - 33. On the units rod, 3 can be subtracted immediately. On the tens rod, we must play the trick -3 = -5 + 2, and move the 5 and two 1's up. The answer, 45, appears automatically. Now, how about 73 - 38? Starting at the left, we subtract 3 by subtracting 5 and adding 2, giving 4. We can't subtract 8 from 3 in the units column, so we must subtract 10 in the tens column and add 2 in the units column. This means adding 5 and subtracting 3, which we can do. The 1 comes off of the ten's column at once, leaving 3 there. The answer is 35.
Logarithms and the Slide Rule
Until the development of electronic pocket calculators, logarithms were a great convenience in practical calculations, especially in trigonometry. They perform the operations of multiplication, division and raising to a power with great facility. Multiplication and division are reduced to addition and subtraction, and raising to a power to a simple multiplication or division, often by one digit. Logarithms were taught in the high-school trigonometry course, which gave excellent practice in the use of tables. The first chapters of a trigonometry text were devoted to the use of tables of natural (that is, non-logarithmic) trigonometric functions, and to logarithms. Four- or five-place tables of logarithms and trigonometric functions were at the back of the text. Among other things, the student learned how to interpolate in tables. Much of the algebra in trigonometry was devoted to putting relations in a form that could be calculated easily with logarithms (avoiding sums and differences).
The Scots mathematician John Napier, Baron Merchiston (1550-1617), introduced logarithms as an aid for trigonometric calculations in 1614, at the same period in which Indian (Arabic) numerals were being adopted in Western Europe. These digits are very well adapted to numerical tables, though logarithmic tables are possible in any numerical notation, even Chinese numbers, as was indeed done. Napier's logarithms are not those we use now. Napier did not have the advantage of exponential notation, which makes logarithms easy for us. Logarithmic calculations are very different from the digital calculations that can be done by Napier's Bones, another invention of his for performing multiplication that uses Indian numerals to good effect.
The logarithmic function x = log y is the inverse of the exponential function y = ax, where a is the "base" of the logarithms. Independently of its use in calculations, the logarithmic function is of great importance. If the base a = e = 2.7182818..., then dy/dx = y, a simple and useful result in calculus. Logarithms to the base e are called natural, hyperbolic or Naperian logarithms, often written ln y. An ordinary number can be expressed in "scientific" notation in the form 5.08 x 106, where a number between 1 and 10 is multiplied by a power of 10. If 10 is the base of the logarithms, then log (5 x 106) = log 5.08 + 6. The first part, log 5.08, is the mantissa and the 6 is, of course, the exponent, or characteristic. Then we can find the logarithm of any number using simply a table of mantissas for numbers between 1 and 10 (ranging from 0 to 1), something that is not possible with natural logarithms. These are common, decimal or Briggsian logarithms, often written log y, invented by Lucasian Professor Henry Briggs of Oxford (1561-1631) in 1617. It is clear that the benefits of common logarithms is mainly in the use of a small table and the relation to ordinary decimal numbers. Sometimes the base is written as a subscript to the designation log, especially for other bases than e or 10.
If y = ex and y' = ex', then yy' = exex' = ex + x', or log yy' = log y + log y'. Similarly, log y/y' = log y - log y'. Also, yn = (ex)n = enx, so that log yn = n log y. In particular, log √y = (1/2) log y. These very familiar rules contain all the theory of computations with logarithms. For those who may not have seen logarithmic computations before, let's do 34.5 x 7.48. From 5-figure tables, log 34.5 = 1 + .53782 and log 7.48 = 0 + .87390. The sum of these is 2.41172, or 2 + 0.41172. This is the logarithm of a number between 100 x 2.580 and 100 x 2.581. By interpolation, we get 100 x 2.5806, or 258.06. This happens to be exactly the answer given by a pocket calculator.
It should be said that for numbers smaller than 1, such as 0.0583, or 5.83 x 10-2, the logarithm is written as -2 + 0.76567. Usually, the number was written all together, with a line above the 2. For example, log 1/2 = -log 2 = -0.30103 = -1 + 0.69897, which corresponds to 5.0 x 10-1. Another method, common in trigonometric calculations, was to add 9 and subtract 10 (the -10 might not be explicitly written). The exponents were always handled separately like this, which avoided inconvenience in using the tables, which were always for positive mantissas.
The use of a graduated scale to add and subtract logarithms by taking distances off by dividers (instead of by using numerical tables) was introduced by Edmund Gunter in 1620. (The same that invented the 66-ft chain of iron links for surveying that was much easier to use than poles.) Movable scales that added and subtracted logarithms directly were invented by William Oughtred in 1630. In 1657 these were put into the form of a movable slide in a fixed stock by Seth Partridge. In 1775, the cursor was added by John Robertson. The log-log scale for handling powers was invented by P. M. Roget of France in 1815, but languished until 1900. The modern slide rule is called the Mannheim type, after the French artillery officer who arranged it in convenient form. The history and use of this very important engineer's tool is told in another article, The Slide Rule, to which the reader is referred for further information, and for a picture of a slide rule. The pocket calculator ended the importance of the slide rule, as it did logarithmic tables, in the 1980's. The slide rule is still interesting to look at, calculates rapidly, and does not need batteries.
Nomography
Nomography is graphical computation of a special sort. Instead of solving general problems, as addition can do, very specific problems are solved in such a way that the results can be obtained for different values of the independent variables. A nomographic solution expects the same problem to be solved over and over, and gives the solution practically instantaneously. Nomographs are still as valuable as they always were, especially for complicated relations or finding implicit variables. The word comes from the Greek nomos, for "rule," referring to the functional relationship between the variables that is expressed by the nomograph.
In nomography, quantities are represented by distances in the form of scales. For a linear scale, distances S are proportional to the values of the variable X, or S = xX, where x is the scale factor. This should be quite familiar from maps, where, for example, a scale of 1" to the mile has a dimensionless scale factor x = 1/63,360. We retain this relation even when the quantities are not both distances. A scale in which 40 mm represents 1 atm pressure would have a scale factor of 1/40 atm/mm. Scales may also be nonlinear, as logarithmic scales or square scales. In this case, the scale factor is not constant, but varies with position on the scale. For a logarithmic scale, we may take S = 100 log x mm, so that one cycle corresponds to 100 mm.
The simplest nomograph are stationary adjacent scales. We often see such scales on a thermometer, with Celsius on one side and Fahrenheit on the other. It can be used to read the thermometer on either scale, or just as well to convert between Fahrenheit and Celsius without considering the thermometer. The reader may make adjacent scales of distance from 1" to 10" and commmon logarithms from 0 to 1. Slide rules usually have such scales, which are read by using the cursor to mark corresponding values. Adjacent scales can "solve" the functional relationship f(x,y) = constant, where there are two functionally dependent variables, either of which may be taken as independent.
Much more interesting is the case of three variables, for example X + Y = Z or XY = Z. Such relations are expressed by the alignment nomograph, the typical and characteristic tool of nomography. Indeed, the science of nomography is generally largely restricted to such alignment diagrams, in many different forms. In any such nomograph, we have three scales, one for each variable, and they are very often linear. A straight line intersects the three scales at corresponding points. It is best to have a transparent piece on which an accurate and sharp cursor line has been drawn, so that the scales can be read accurately. Using an opaque straightedge covers the scale asymmetrically, so reading is rendered difficult.
The relationship between distances on three parallel lines is a linear one, regardless of the spacing of the lines or their zero points. However, it is easiest to grasp what is going on in the simplest case of three equally-spaced vertical lines whose zeros are on a horizontal line. Consider an arbitrary line cutting the three lines at points A, B and C. The reader can easily prove that the distances from the zero points (which we shall represent by the same letters for brevity) are related by B = (A + C)/2. Suppose that we choose B = zZ, A = xX and C = yY, in terms of scale factors and values of the variables X,Y,Z. Then, 2zZ = xX + yY. If we choose the scale factors to satisfy x = y = 2z, then we have Z = X + Y. The reader can easily make such a nomograph and test its functioning. We are not restricted to linear scales by any means; X, Y and Z can each be related in a complex way to other variables. If B = 50 log Z, A = 100 log X, and C = 100 log Y, then the nomograph will do log Z = log A + log C. When appropriate, the scales can even be extended to negative values and the nomograph will still be valid. The spacing of the lines and the scale factors can be varied to solve the problem in the most convenient way possible.
The simplest nomograph that does multiplication is formed from two parallel lines and a slant line crossing them at points A and B. It is called an N-diagram from its shape. In one useful form, the scales on the parallel lines are linear and have scale factors x and z. Let the Z-scale be measured downwards from a zero at A, and the X-scale measured upwards from a zero at B. If we consider an arbitrary straight line crossing the parallel lines at D and E, and the slant line at C, it is clear that the triangles ACD and BCE are similar, so their sides are proportional. If K is the total distance AB, and S the distance AC, we see that AD/BE = S/(K - S). Since AD = zZ and BE = xX, we have zZ(K - S) = xXS, or Z = X(x/z)S/(K - S). If we want this nomograph to solve Z = XY, it is clear that we must take Y = (x/z)S/(K - S). That is, choosing the scales for z and x determines the scale for Y, and it is not linear. To draw the nomograph, we need S as a function of Y, and this is easily seen to be S = KY/[(x/z) + Y]. Note that this depends only on the ratio of the scale factors for z and X. V = 0 corresponds to S = 0 (point A), while V = ∞ corresponds to S = K (point B). The scales may be extended to negative numbers if appropriate.
As an example, suppose we require a nomograph to solve Francis's weir equation Q = 3.33LH3/2 cfs, where the width L and the head H are in feet. We take Q as Z, L as X, and 3.33H3/2 as Y. Let's put it in a rectangle 8" high and 4" broad. Let 1" = 10 cfs and 1" = 1'. Then, z = 1/10 and x = 1, so that x/z = 10. Then, S = (8.94")Y/(10 + Y). For H = 2', Y = 3.33(2)3/2 = 9.42, and S = 4.34". Mark this point on the slant line with the value of H. The reader may want to draw this nomograph, and plot enough values to calibrate the slant line. We can enter with any two of the variables Q, H and L and find the third one very quickly. This is not a trivial nomograph, and shows clearly how one is constructed.
Draw three lines meeting in a point O, one horizontal, one at 60° and one at 120°. Suppose these lines are intersected by a straight line at points A, B and C. Draw an equilateral triangle OCD on OC. Side OD will be an extension of line OA, while CD will be parallel to OB. From similar triangles, we see that (OA + OD)/DC = OA/OB. OD = OC since OCD is equilateral. Therefore, we find that 1/OA + 1/OC = 1/OB. This nomograph establishes the given relation between distances measured on the three lines. Each of the three lines can be extended to negative values on the other side of O, giving a hexagonal appearance. This is called a hex nomograph, and it finds the reciprocal of the sum of reciprocals.
One application (which I have never seen in an optics text) is to the Gaussian lens formula 1/u + 1/v = 1/f, where u and v are the object and image distances from the centre of a thin lens of focal length f. For a converging lens, f is positive, while it is negative for a diverging lens. Label, for example, line OA as u, OC as v, and OB as f. The behavior of the object and image distances for a thin lens are easily seen. For example, when the object is at an infinite distance, then the image is at the focal point, or v = f.
As a final example, the circle nomograph also multiplies: Z = XY. Divide the circle with a diameter of length 2a that forms the Z axis, while X is measured along the upper semicircle from 0 at the right to ∞ at the left, and Y is measured similarly on the lower semicircle. Let θ be the angle between the ∞ end of the diameter and the point representing X = tan θ. Let φ be the similar angle determining Y = tan φ. The angle at the center to these points is twice the angle at the end of the diameter, of course, which can make them easier to lay out with a protractor. Once you have calibrated the X and Y semicircles, join the necessary points to calibrate the Z axis. The formula for the distance corresponding to Z is S = 2a/(1 + Z), which can also be used. When Z = 0, we see that S = 2a, and when Z = ∞, S = 0. The proof that this works is left to the reader.
Checking
When a pocket calculator is used to make a calculation, it can best be checked by having two people independently perform the calculation. If the two answers agree, then the result is probably correct. Since the calculators are infallible, the only sources of error are in the keying in of numbers and the operation of the calculator. Two operators are unlikely to make the same mistakes. A single operator can do fairly well by taking great care in entering the numbers, checking each entry on the display, and performing the calculation twice, preferably in a different order. Once you have made a mistake, mere repetition will usually produce the same mistake (the mind thinks it knows what to do without wasting time) if only a small interval has elapsed between the calculations.
Algorism is so error-prone that some check should always be made. A very effective classical method is "casting out nines." The key is the reduction of any number to a single digit, the check digit, by first casting out all 9's, then all pairs of numbers adding to 9, and finally adding the digits remaining until only one digit remains. A skeleton computation is then made with the check digits, and the answer should agree with the check digit of the answer. The theory of this check is the expression of numbers as single digits modulo 9, and performing the calculation with these single digits. This is explained in Algebra.
For example, consider the sum 514 + 208 = 722. In 514, cast out the 5 and the 4, leaving the single digit 1. In 208, add 2 + 0 + 8 = 10, 1 + 0 = 1, which is the check digit. The sum of the check digits is 1 + 1 = 2. In 722, cast out the 7 and the 2, leaving the check digit 2, which agrees with the sum of the check digits of the addends. For the product 5048 x 17 = 85816, the check digit of 5048 is 8, as is that of 17. Their product is 64, 6 + 4 = 10, so the product of the check digits is 1. 85816 becomes 5 + 8 + 6 = 19. Casting out the 9, the check digit of the product is 1. The result checks. In division, the product of the check digits of the divisor and the quotient, plus the check digit of the remainder, should equal the check digit of the dividend. As you can see, this is very easy to do, and a valuable check. See if you can find an arithmetic teacher who knows how to do it! (Some few do, and may they be blessed!)
The most common error in transcribing a number is the transposition of neighboring digits. Once made, it becomes almost undetectable by the victim. The check digit from casting out 9's cannot help here (casting out 11's will, however, detect transpositions; see the link above). For example, 154 and 145 both have the same check digit, 1. One way of forming a check digit that will reliably discover such transpositions is used by the Deutsche Bundesbahn in locomotive numbers. For example, the number 103 001 (the first machine in the 03 series of electric locomotives, whose numbers begin with 1) has the check digit (Kennziffer) 4. To find it, write 121 212 beneath the number and multiply each corresponding digit, then sum the result. Here, we find 1+0+3+0+0+2 = 6. The next higher multiple of 10 is 10, from which we subtract the 6, getting 4, which is the Kennziffer. The number would be reported as 103 001 - 4. Now suppose some idiot reported 103 010 - 4 instead. Now, we have 1+0+3+0+1+0 = 5, and 10-5 = 5, but 4 appears instead, so something is wrong. A check digit like this is used in many circumstances today, for example in banking. It is an example of error detection embedded in the number itself. Elaborate error detection schemes of this type are used for digital data, which also must be checked like algorism results. Here the chance of an error is very small, but there is a very large amount of data.
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