Summary:
Count + + Numbering system
+ Addition and Subtraction Multiplication
+
+ Division
+ + Square Root Exponentiation
- Logarithms John Napier
Finding a Base
Properties of logarithms
logarithms or Briggs
Esempi di calcolo con i logaritmi
I logaritmi "Naturali"
Usi "quotidiani" dei logaritmi (deciBel, pH...)
Conclusioni, e una "perla" letteraria
+ Il Regolo Calcolatore
Logarithms are not part of arithmetic, but they are a direct consequence of the elevation to power that I mentioned in the previous chapter.
John Napier
The protagonist of this story is the Scotsman John Napier (Napier Italianate, 1550-1617). It was not a mathematician by profession, but a wealthy landowner of noble family, with broad interests ranging from astrology to alchemy, theology (he was a staunch anti-papist) in mathematics. Gossips suggest she gave also to attend the devil and black magic ... but this accusation was quite common in those days!
felt particularly the need to find a way to speed up arithmetic. In this regard, he wrote:
Calculate operation is difficult and slow, and the boredom that often comes out is the main cause of the disaffection that the majority of people feel towards mathematics ...About the calculations, was printed in 1617 (posthumously) in which the book describes the use of so-called sticks or bones Napier (English Napier's Bones), with whom I have already described the how to calculate multiplication , divisions and square roots . The commitment of the Napier in mathematics But do not stop there: in fact provide the world something truly revolutionary!
from the properties of powers (of which I speak here ), and in particular the fact that
The product of two powers that have the same base is equal to a single power with exponent equal to the sum of the exponents of the two Start-up powerbegan studying the possibility of transforming products (powers) in amounts (of exponents). Over the years that dealt with this topic, by Scottish physicist John Craig friend, he came into contact with the scientific environment of Tycho Brahe and learned the new procedure for calculating the multiplication con il metodo di prostaferesi (ne parlo qui ). Questi contatti gli dettero ulteriori stimoli a proseguire con le sue ricerche... per cui cominciamo a vedere di cosa si trattava! ▲
Ricerca di una Base
Il lavoro di Nepero si basa una semplice intuizione. Se devo moltiplicare diecimila per un milione posso fare:
passando dai numeri alle loro potenze di dieci, sommando gli esponenti, e alla fine calcolando la potenza di dieci con il nuovo esponente.
L'uso di soli esponenti interi is not of much use. Napier then tried to use decimal exponents, making the simple proportions: if 100 is at 1000 (factors to be multiplied) is like 2 to 3 (members), it is that the number 550 (the arithmetic mean between 100 and 1000) could match the 'exponent 2.5 (arithmetic mean between 2 and 3)? If so you could do:
but evidently the result of the power (red) and the product calculated in the normal way (green) do not match. By making several attempts, Napier realized that things considerably improved by using bases closer and closer to unity. At the end opted for this basic
Vediamo cosa succede. Ammettiamo di conoscere i valori delle seguenti potenze (in rosso alcuni numeri che serviranno fra poco):
Se proviamo a calcolare il valore per l'esponente 3 come media di quelli del 2 e del 4 otteniamo
che è praticamente identico al valore reale:
Facciamo una riprova con la moltiplicazione 0,9999998 x 0,9999997 utilizzando questi numeri, che indico con gli stessi colori rosso e verde che ho usato sopra:
Il valore trovato è praticamente esatto, infatti calcolando il prodotto and the power of the sum of the exponents, is almost equal numbers:
Napier then discovered to be on track just did not like all those decimal places: then multiply all its members to 10000000. The numbers calculated above then become:
For now it seems that the result is "linear," meaning that the growth exponent decreases to the same extent also the power, but it is not always the case. We see higher values:
is that the intervals between the exponents are no longer proportional alle differenze fra i numeri sulla sinistra.
Ora riporto altri tre numeri, giusto per fare un calcolo di prova:
Proviamo a calcolare il seguente prodotto:
Il calcolo reale sarebbe:
Mancano 7 cifre all'appello! Ma questo è naturale: siccome i numeri di Nepero sono stati tutti moltiplicati per 10000000, c'è proprio questa discrepanza nel risultato. Allora il calcolo di Nepero va aggiornato:
il risultato finale differisce pochissimo dal risultato effettivo, calcolato a mano! ▲
Properties of logarithms
After twenty years of hard work, published in 1614 Napier Mirifica Logarithmorum Canonis Descriptio explaining the operation (57 pages) and provides tables (90 pages!) of these its Logarithms (the term derives from the Greek words logos in the sense of proportion, and arithmos , number). Now let's see the practical aspects of this system.
The speech is relatively simple: once you choose a fixed number to be used as the basis calculating all the powers of the basis, I can build a table to which every member result is a number equal to the exponentiation of the base to that exponent. It is defined thus:
logarithm to base a number xto see some 'better than what it is we start with two numbers (arg) and their logarithms (log):
the exponent is to be given to a for
x (x is called "subject" of the logarithm).
If using a table I can know the values \u200b\u200bof the exponents, ie the logarithms of two arguments, then to do the multiplication of these two arguments (factors) can proceed as follows:
The process is extremely easy to multiplication: it is enough to see twice the tables, do a simple sum and then see " backward. "
Not only that, unlike the method prostaferesi, with which you can only multiply, with the logarithms can be made even divisions:
extractions of the roots of every degree:
and many other beautiful works! ▲
logarithms or Briggs
No wonder that this new method of calculation had immediate recognition in the scientific era. Among the most enthusiastic admirers was Henry Briggs (1561-1630), who in 1615 went to Napier to discuss possible changes to its tables. They agreed on a new approach (which I shall refer very soon), that would have to recalculate all the numerical tables! Given the advanced age now of Napier, the task fell on Briggs, who made a lot of work: more than 30000 calculated logarithms to fourteenth decimal places after the decimal point ... and this was just the beginning, as also published tables of logarithms of sines, cosines, tangents, in short, did a tremendous job!
Briggs The new system of logarithms (or vulgar, or Briggs) is based on the base 10, so:
Intermediate values \u200b\u200bare calculated in various ways. For example, we know that among the properties of powers, there is one for which he raised to the 1 / 2 means the square root of the base. Then calculated by hand the square root of 10 is obtained:
With complicated methods, but also ingenious, and exploiting all possible short cuts and simplifications, in 1617 Briggs published the first table of logarithms of numbers from 1 to 1000. The next results, which I mentioned above, in 1624. ▲
Sample calculations with logarithms
When I attended high school, around 1975, electronic calculators were just beginning to spread, so the curriculum provided for the teaching of calculation with tables of logarithms, which appeared on special books of numerical tables. Now it is obvious that we have a table in two columns, in which the left I have a right number and its logarithm, the same table lends itself to obtaining the antilog just search the log to the right column and find the corresponding number in the left column.
Logarithms are technically divided into two parts: characteristic and mantissa . The characteristic indicates the magnitude of the number : a single-digit whole numbers with a corresponding characteristic 0, 3-digit whole characteristic 2; 4 zeros after the decimal point to -5. Fact:
We have said that the logarithms are exponents, we know that the product of two powers is the sum of the exponents. Each number can always espresso dal prodotto di due numeri, di cui uno esprime l'ordine di grandezza (1 10 100... oppure 0,1 0,01 0,001...) e l'altro un numero con una sola cifra intera e tutti gli altri decimali. Ciascuno di questi due numeri avrà un suo logaritmo, di cui il primo sarà sempre un numero intero (come mostrato nella tabella sopra), mentre l'altro sarà sempre uno zero virgola qualcosa (avendo questo secondo numero una sola cifra intera): il logaritmo di questo secondo numero si chiama mantissa .
Se andiamo a vedere le tabelle dei logaritmi, queste non presentano neanche la virgola negli argomenti: i numeri 7, 70, 700 e così vengono implicitamente ricondotti al numero 7,000. Qui sotto indico in verde le cifre significant numbers, and the mantissa of its logarithm:
See you then as you get the full value of the logarithms. We said that the logarithm of the product is equal to the sum of logarithms, then we consider separately the magnitude and significant digits. For example the number 700 has characteristic 2 (in red), mantissa 0.8451 (green):
I am on the bottom line instead of a smaller unit: the procedure is the same, but it has one more complication: since the logarithm is negative, the decimal part becomes the complement of the mantissa, in practice becomes 0.8451 0.1549 (gray, right). To overcome this problem (especially likely to avoid mistakes) the logarithms is less than zero indicate differently:
The mantissa is unchanged, and -5 feature is referred to as a 5 with a dash above . In this way the log is "positive," but we must consider that the feature (the only part of a whole) is actually negative.
Let us now make some calculations with these numbers:
In calculations that follow, numbers in gray are the logarithms; green in the real values \u200b\u200bof the various operations, in red the values \u200b\u200bcalculated using the logarithms:
Considering that I used logarithms with only 4 digits of mantissa, there is really bad! ▲
Logarithms "Natural"
Natural Logarithms or Neperiani are built on a different basis from 10 to Briggs, the base is a number that is identified with the letter and :
which is obtained by calculating the following expression:
This expression tells us that the number and be calculated (semplifico!) raising the infinitesimal power a number that is larger than the infinitesimal! The result is what I wrote above: trust.
And where did that odd number? The "guilty" of Leonhard Euler (Euler said in Italian, 1707-1783), who was one of the greatest mathematicians ever (one of three so-called principles of mathematics together to Archimedes of Syracuse and Gauss, which We've already had to do arithmetic in this story). Euler gave an essential contribution to ' mathematical analysis that is part of mathematics that deals with derivatives, integrals and differential equations. He discovered that this number would jump out of hell from all sides, it is equally important as mathematics in the greek pi, which binds the diameter of the circumference of the circle. About greek pi: Euler was right to "baptize" the following numbers:
- π (pi greek): Do not believe it, but it took the eighteenth century to give the final name at this number known to antiquity. It seems that Euler, attributing this Greek letter, wanted to pay tribute to Archimedes.
- the (the lowercase letter i) the value of the square root of -1, which is the imaginary unit (of which I speak here).
- and (the letter and lower case): the number we're talking about. Here it seems that the magnanimity of Euler was a bit 'less marked, as this "and" very much like the initial of his last name ...
But what does this number and with Napier? Let's review the formula for logarithms:
If we do a multiplication and division for 10000000 (numbers in green) we get the right expression, which is a new formula for calculating logarithms derived directly from that of Napier.
short, the red part of the expression above, which I reproduce below for clarity
looks a lot like this other expression
The change of sign in the brackets to get involved 1 / e instead of and , the rest, Napier did not use a value n infinite, but is "limited" value 10000000, which is already "enough" to give great results.
Napier, unknowingly had invented a base for his logarithms structurally similar to the value and Euler: for this is ascribed to the authorship of the Napier's number, and say Neperiani logarithms calculated with this basis. It is certain that Napier could not even suspect that the importance should have that number in the analysis ... ▲
Use "daily" of logarithms
Maybe not know, but the scales measuring the following phenomena:
- The magnitude of the stars (index of their brightness)
- The index pH of hydrogen ion activity (measure of acidity) ;
- The Richter scale (a measure of the intensity of earthquakes)
- the decibel measuring the intensity of sound waves;
are all logarithmic scale , which measures the logarithm of relationship between what is measured and a standard value taken as reference. We see these steps closely.
The modern magnitude scale was chosen so that (simplified), chosen a very bright star that you give a magnitude 0 (Vega was chosen, in the constellation Lyra), the magnitude of 5 corresponds to a brightness 100 times lower. That formula is: 
where the minus sign means that the magnitude increases when the star is less bright, the magnitude M is calculated, F is the flow of light from the star we are observing, V is the brightness of Vega, and the value 2.5 is used to make a star 100 times fainter than Vega has its magnitude 5.
With this way of indicating the brightness of stars is certainly easier to say that the maximum brightness of Pluto is magnitude 13.65, rather than saying that Pluto has a brightness duecentottantottomilaquattrocento (288 400) times less than Vega!
On television there is no advertising of cosmetics that does not keep us to know the pH of the various creams ... but raise your hand if you know exactly what the pH! The pH was conceived by the Danish chemist Søren Sørensen in 1909 PL, who was facing some problems with the fermentation of beer. This process requires a very fine control of the acidity of must, which, at that time was expressed through the concentration of hydrogen ions present in solution. These ions are typically in very small, even less than one part per million. Sørensen realized that the calculations would be greatly simplified by referring only to the exponent of the value of concentration, instead of the whole number. Then proposed to call this exponent pH, where p stands for Power (exponent of 10, from the Latin pondus) and H stands for hydrogen, or, to the hydrogen ion (from the Latin Hydrogen).
Today, pH is defined as the opposite of the logarithm of molar concentration of hydrogen ions. Therefore:
concentration that allows to define the degree of acidity or alkalinity of the solution. The pH can assume values \u200b\u200bbelonging to the range 1-14, but since we are talking about logarithms, the value between 1 and the value 14 there are 13 digits or decimal places, of difference!
is therefore much easier to talk in terms of pH of the effective hydrogen ion concentration, and there are acidic if their pH is between 1 and 6, if the neutral value is 7 (such as distilled water) , basic between 8 and 14.
The magnitude of earthquakes, as indicated by the Richter scale, has to do with logarithms. This scale is based on the equivalent power of the quake, measured in pounds of TNT. Then the magnitude of 0 is equivalent to one kilogram of TNT, then two magnitude of each jump is equivalent to multiply the amount of TNT to 1000: 
Again, it is easier to say "Earthquake of the sixth degree of the scale Richter, "rather than" earthquake, the power equivalent to one million tons of TNT!
Bel is the logarithm of the ratio between two powers, the decibel (dB) is the same value, but multiplied by 10. Then: 
where w2 is the measured sound intensity, while w1 è l'intensità di riferimento standard. Quest'ultima è stabilita come il livello minimo di potenza sonora che sia percepibile dall’orecchio umano. Quindi se la potenza da misurare W2 è uguale a quella di riferimento, allora il rapporto W2/W1 è uguale a 1, e il logaritmo è 0. Se la potenza fosse cento volte superiore a quella di riferimento, il logaritmo varrebbe 2, quindi i dB di potenza sarebbero 20.
I decibel sono molto pratici proprio perché una variazione di 10dB comporta un aumento della potenza di 10 volte, mentre a un raddoppio di potenza corrispondono circa 3dB. E tutto questo è molto più pratico che esprimersi in termini lineari (cioè non logaritmici): per esempio a 43dB corrisponderebbe a power 19,953 times greater than the reference, 57 dB to 501,187 times: so many indications damage is certainly less intuitive than the dB (at least to those who have a certain habit). ▲
conclusions, and a "pearl" literary
Napier, creator of the theory of logarithms, he correctly predicted that its results would revolutionize the scientific era. Pierre-Simon de Laplace (important mathematician, physicist and astronomer French, 1749-1827) wrote that Napier had "doubled the life of Astronomers ": logarithms have greatly simplified the performance of the calculations, allowing scientists to lose much less time. Not only that this new calculation technique has also promoted the development of trade and business activities, essentially on the birth of the industrial world since the second part of the seventeenth century.
One last question: in the story "The adventures of three Englishmen and three Russians in Southern Africa" \u200b\u200bby Jules Verne, is a character that Nicolas Palander, who is so absorbed in his thoughts that he realizes that he surrounded by a number of rather hungry crocodiles. His fellow adventures can in some way to drive them out by firing a few shot. On hearing the shots, the Palander recognizes his companions and began to run toward them, waving his notepad in his hand and exclaiming, as the ancient philosopher
- Eureka! I found it!
- Found what? - Ask friends
- An error in the decimal logarithm tables centotreesimo James Wolston!
Verne then explains that this discovery would have collected the prize of a hundred pounds offered as a prize from the publisher of those tables of logarithms ...
... unfortunate that this James Wolston I failed to find any trace anywhere. It is certain that Henry Briggs, the first to have completed comprehensive tables of logarithms, he really put away a sum to be paid to those who had found errors in his tables: a great way to have them corrected by someone else! ▲
Next Chapter: slide rule
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