ronaldweinland.info Magazines BOOK OF NUMBERS BY SHAKUNTALA DEVI PDF

BOOK OF NUMBERS BY SHAKUNTALA DEVI PDF

The Book of Numbers [Devi Shakuntala] on ronaldweinland.info *FREE* shipping on qualifying offers. We can't live without numbers. We need them in our daily chores. View The Book of ronaldweinland.info from CSC 51 at Sardar Patel University. SHAKUNTALA DEVI'S NUMBEBS Everything you always wanted to know about . Shakuntala Devi - 5 Books DOWNLOAD 5) The Book of Numbers . Astrology for u if someone has it in pdf send to [email protected] Author: MADLYN KREISEL Language: English, Spanish, Arabic Country: South Sudan Genre: Biography Pages: 705 Published (Last): 22.04.2015 ISBN: 719-5-70832-465-3 ePub File Size: 24.47 MB PDF File Size: 12.45 MB Distribution: Free* [*Register to download] Downloads: 35819 Uploaded by: MARINE Originally I wrote Bhagavad-gétä As It Is in the form in which it is presented now. When this book Bhagavad-Git Beginning English Conversation. This Pin was discovered by BankingPDF. Discover (and save) your own Pins on Pinterest. Shakuntala Devi's Book of Numbers Everything You Always Wanted to Know About Numbers, But was Difficult to Understand We can't live without numbers.

A fraction is called an improper fraction, when its numerator is greater than its denominator. For example, or. On the other hand a fraction is called a 'proper' fraction when it has a numerator, smaller than its denominator. Such problems are known as Empirical Problems. To give an example, if you come across a problem which says 'with the ten digits, 9, 8, 7,6, 5, 4, 3, 2, 1, 0, express numbers whose sum is unity: each digit being used only once, and the use of the usual notations for fractions being allowed with the same ten digits express numbers whose sum is '. There is no limit to the making of such questions, but their solutions involve little or no mathematical skill. These are considered Empirical Problems.

The number plate showed the bus number as a perfect square and also if the plate was turned upside down.? I came to know from the bus company they had only five hundred buses numbered from 1 to S From this I was able to deduce the bus number.

Indian "Human Computer" Shakuntala Devi passes away

Can you tell what was the other number? However there are certain occasions when the hands are exactly opposite each other. Can you give a simple formula for calculating the times of these occasions? A man was being accused of having stolen certain valuable jewels and trying to run away with them, when he was caught by a smart police officer who overtook him.

In cross examination the lawyer for accused asked the police officer how he could catch up with the accused who was already seven steps ahead of him, when he started to run after him. So the number of steps 1 required were fewer than his. IS One day while on her rounds she sold i an orange more than half her oranges to the first customer. To the second customer she sold i an orange more than half of the remainder and to the third and the last customer she sold i an orange more than half she now had, leaving her none.

Can you tell the number of oranges she originally had? Oh, by the way, she never had to cut an orange. I picked it up, noted the number and took it home. In the afternoon the plumber called on me to collect his bill. As I had no other money at home, I settled his account with the hundred rupee note I had found. Later I came to know that the plumber paid the note to his milkman to settle his monthly account, who paid it to his tailor for the garments he had had made.

The tailor in turn used the money to buy an old sewing machine, from a woman who lives in my neigh- bourhood. This woman incidentally, had borrowed a hundred rupees from me sometime back to buy a pressure cooker. She, remembering that she owed me a hundred rupees, came and paid the debt. I recognised the note as the one I had found on the footpath, and on careful examination I discovered that the bill was counterfeit.

How much was lost in the whole transaction and by whom? Tinku, Rinku and Jojo, the three brothers, divided the nuts in the following manner: As often as Tinku took four Rinku took three and as often as Tinku took six Jojo took seven. With this data can you find out what were the respec- tive ages of the boys and how many nuts each got?

We began to wonder how old the couple must have been each at the time of their marriage! Can you figure it out? A wholesale merchant came to me one day and posed this problem. Every day in his business he has to weigh amounts from one pound to one hundred and twenty- one pounds, to the nearest pound.

To do this, what is the minimum number of weights he needs and how heavy should each weight be7 I decided to send them through my boy servant Harish. On settlement Harish received Rs 2. How many glasses did Harish break? This num- ber is three times the sum of its digits.

Can you find the number. Ten-of the ways have two figures in the integ- ral part of the number, but the eleventh expression has only one figure there. Can you find all the eleven expressions?

One day a man walked in and slamming seventy- five paise on the counter requested, 'Please give me some 2 paise stamps, six times as many one paisa stamps, and for the rest of the amount make up some 5 paise stamps. How would you have handled the situation? One day both of them were obliged to return home when each had thirty marbles unsold.

They put together the two lots of marbles and handing them over to a friend asked her to sell them at five for 2 paise. According to their calculation, after all, 3 for one paise and 2 for one paise was exactly the same as 5 for 2 paise.

But when the takings were handed over to them, they were both most surprised, because the entire lot together had fetched only 24 paisel If however, they had sold their marbles separately they would have fetched 25 paise.

Now where did the one paise go? Can you explain the mystery? His wife drives him over to Howrah Station every morning and in the evening exactly at 6 P. One day he was let off at work an hour earlier, and so he arrived at the Howrah Station at 5 P. He started walking home. They drove home arriving 10 minutes earlier than usual. How long did the man have to walk, before he was picked up by his wife?

At each of the 25 stations the passengers can get tickets for any of the others 24 stations. How many different kinds of tickets do you think the booking clerk has to keep?

What is my aunt's share? Ws also have some beautiful bone-china saucers that I recently brought from Japan. Our table top is fifteen times the diameter of our saucers which are also circular.

We would like to place the saucers on the table so that they neither over lap each other nor the edge of the table. How many can we place in this manner? I did some quick cal- culations in my mind. I found that if I walk down twenty-six steps, I require thirty seconds to reach the bottom. However, if I am able to step down thirty-four stairs I would only require eighteen seconds to get to the bottom. If the time is measured from the moment the top step begins to descend to the time I step off the last step at the bottom, can you tell the height of the stairway in steps?

This can be completely covered by 32 cardboard rectangles, each cardboard covering just 2 squares. If it can be done how can we do it? And if it cannot be done, prove it impossible. Every cat killed an equal number of mice. How many cats do you think there were? Ob, by the way let me clarify just two points—it is not one cat killed the lot, because I have said 'Cats' and it is not cats each killed one mouse, because I have used the word 'mice'. I can give you just one clue—each cat killed more mice than there were cats. It was found that the fore wheels of the carriage make four more revolutions than the hind wheel in going 96 feet.

Can you find the circumference of each wheel? Can you find this temperature? Two gentlemen by the name of Sr. Guittierez and Sr. Ibanez decided to have a Llama race over the mile course on the beach sands. They requested me and some of my other friends whom I had met at the resort to act as the judges.

We stationed ourselves at different points on the course, which was marked off in quarter miles. But, the two Llamas, being good friends decided not to part company, and ran together the whole way.

How- ever, we the judges, noted with interest the following results: The Llamas ran the first three quarters in six and three quarters minutes. They took the same time to run the first half mile as the second half. And they ran the third quarter in exactly the same time as the last quarter. From these results I became very much interested in finding out just how long it took those two Llamas to run the whole mile. Can you find out the answer?

The numerals in each part in every case summed to a total of Can you show how the four parts of the clock face was broken? I decided to get only half the area of the window painted. Even after the painting I found that the clear part of the window still remained a square and still measured 4 feet from top to bottom and 4 feet from side to side.

How is it possible? Anupam Patra October 2, at Anonymous October 20, at Biranchi Narayan Nayak October 21, at 8: Goondla Suresh October 24, at 3: Deepak Yadav November 7, at Beben Habibi May 16, at Unknown August 20, at Stacey Jones September 28, at Purva Khare October 17, at 2: Anonymous October 22, at Stiff Roy November 29, at 4: Joseph March 6, at 3: Unknown April 2, at Engineer's Chaupal April 16, at 1: Engineer's Chaupal April 30, at 5: Unknown June 6, at 4: Priya Gupta August 3, at 5: They are marks or signs x meaning multiplied by or -r meaning divided by.

Deduce actually means to infer something from information obtained. In other words, to derive or draw a conclusion by reasoning from given principles. Deduction is a noun derived from the verb 'Deduct'. If derived from Deduce it means to infer, to draw conclusions by reasoning from given principles. To sieve out all the unnecessary information from a problem and draw out the plain facts. However, when this word is derived from Deduct, it means subtracting or taking away.

Binary means consisting of two parts or things. The Binary number system is actually a system of notation which uses the base 2 combined with place value of notation. Since the scale is 2, there are only two digits or 'bigits' 0 and 1. The main use of the Binary System, is its usefulness in electronic computing machines. When an electric circuit is open, in other words, when there is n o current, the situation may be regarded as signifying 0.

A closed circuit signifies 1. The common scale of notation we use in our dayto-day life is the Denary System — base It means that addition and subtraction may be performed in any order. There is no relationship really. This means altogether another thing.

Distributive Law of Multiplications means that the multiplication of a compound expression by a factor is the sum of the partial product of each terms of the expression by the factor. In Arithmetic Eliminate means to get rid of one or more unknown quantities from two or more simultaneous equations. The 'Austrian Method' of subtraction is generally used by shopkeepers in Europe while giving change to a customer.

They give the customer the following money units for his change: A graph is the pictorial representation of Statistics. For example, if we want to show the comparative rainfall in a certain district from month to month, we draw a graph depicting a row of vertical glass tubes, one for each month, each of them shaded, to a depth which represents the number or inches of rainfall for that month.

Many other statistics can be reduced to simple graphs. There are two other ways. Graphs may be expressed in curved lines and horizontal bars also. When statistics are reduced to a simple group of figures, the correct form of graph will present itself without any trouble. Evaluate means to work out a sum to its simplest form.

In other words, to get the exact vlaue of a sum. It is based on the idea of finding the sum of the digits in a number and then adding the digits in the resulting sum, etc. This addition of the digits in a number is further simplified by first discarding or casting away any digits whose sum is 9. The remainder is set down, in each case as the check number. For example, if you want to verify the following multiplication: So the multiplication is correct.

They are just numbers or a series of numbers formed 42 from an arithmetical progression of which the first term is unity and the difference a whole number, by taking the first two, first three, first four, etc. The numbers of each sequence, when represented by points, can be systematically arranged in various geometrical figures such as triangles, pentagons, tetrahedrons, etc.

Synonym of Polygonal Numbers. Cryptograms are mathematical puzzles which are concerned with the association of the letters of the alphabet with the digits.

In simple cryptogram each letter is replaced by one of the digits 0; 1,2, 3, 4, 5, 6 , 7 , 8 , 9 and n digit is represented by more than one letter. Of course, the cryptogram Should make mathematical sense.

The use of the fingers on both hands in counting led to the use of 'ten' as the base of the number system. Latter on symbols were used to represent numbers. In a certain tribe the word meaning 'three' was the word used for 'middle finger'. Zero is used as a place holder when there is no frequency to record in a place in a number. For example, in the 0 holds ten's place, showing that there were no tens to record in the number. Zero is the symbol which makes it possible to show values in our number system without the use of an artificial means to identify place value.

The word 'digit' is derived from the Latin word 'digitus' which means 'finger'. A simple form of Abacus consists of a series of parallel wires or rods fastened in a wooden frame.

The Book of Numbers.pdf

There are counters or beads which are moved along each rod. The position of each rod represents a certain place value. A number system is not complete unless it may be used to express fractional parts of a unit by means of place values.

The decimal point identifies the one's place and therefore it serves as a separatrix between a unit and a part of a unit. Magic Squares consist of a number of integers arranged in the form of a square in such, a way that the sum of the numbers in every row, in every column and each diagonal is the same. Magic Squares originated as an amusement in old times when mystical ideas were associated with particular numbers.

Then introduction into Europe appears to have been in the early part of the fifteenth century. A Magic Square engraved on a silver plate was sometimes prescribed as a charm against the plague.

The Diabolic Magic Square also, known as the Pandiagonal Magic Square, is the one that satisfies the condition that the square should be magic along the broken diagonals as well as along the two ordinary diagonals.

In other words, if a Diabolic Magic Square is cut into two pieces along a line between any two rows or any two columns, and the two 15 10 3 6 4 5 16 9 14 11 2 7 1 8 13 12 46 pieces are interchanged, the real square so formed will also be pandiagonally magic.

In the magic square given here, the sum of the numbers in each row, column and in the two diagonals is The Geometrical Mean of any number or numbers is the root of their product which is represented by their number. For example, the Geometrical Mean of two numbers is the square root of the product of the two numbers. The Geometrical Mean of three numbers is the cube root of the product of the three numbers. The measure had its origin in something concrete 47 unlike numbers, which are abstract.

To start with some parts of the human body was used as a basis for establishing a suitable standard for a measure of distance, and as each particular standard for a measure was established an effort was made to divide the standard into smaller units of measure.

As such since there is no direct relationship between our system of number and our collection of measures, and as man learned to use different measures, he tried to make an arbitrary relationship between different units. Thus, the ratio of the foot to the yard became 1 to 3, and the ratio of the foot to the mile 1 to That's rather a funny number. King Henry VII of England, in the fifteenth centurychanged it to exactly 8 furlongs, 8 X yards which are equal to ft.

If the numerator of a simple fraction is unity, it is 48 called a unity fraction. Use the one that makes it possible for you to cancel the units you do not need. It gives two 3 ft 3 ft 1 Yd ' Both are, of course, equal to 1.

In this case try one of the two fractions. If it works good. If not, try the other one. The metric is related to our system of numbers, 49 whereas the English system of Linear measures, in reality is not a system but a collection of independent measures.

The units in the metric system are interrelated, and so they constitute a system. The comparison of the English System to Metric System is as follows: Metric System English System 12 inches—1 foot 3 feet —1 yard yards —1 rod rods—1 mile 10 millimetres—1 centimetre 10 centimetres—1 decimetre 10 decimetres —1 metre 10 metres —1 decametre 10 decametres —1 hectametre 10 hectametres—1 kilometre The metric system has been in use less than years, whereas the English System of measures is as old as our number system.

A Committee of Scientists in France, in the early part of the nineteenth century, formulated a system of measures, which is a decimal system—the same base as our number system. In this system the measure is disassociated 50 from any tangible object, such as the length of an arm or a foot. A Meter, the Standard Unit of the Metric System is one-ten millionth of the distance from the equator to the pole along the meridian.

Since Metric System was constructed as a decimal system by scientists who knew that number in general, and that it can be applied anywhere to any situation in which quantitative relationships are present, it is a much more practical system.

On the other hand, the English System is a product of necessity, with its standards originating from tangible concrete things which had no relationship to our number system.

The originators of these measures, perhaps, were unaware of the generality of a decimal system. Therefore, in this system we have unrelated units. The basic practical unit of time is the mean Solar day. We define all other time units in terms of it. A complex fraction has a fraction or mixed number for one or both of its terms, thus: A fraction is called an improper fraction, when its numerator is greater than its denominator. For example, or. On the other hand a fraction is called a 'proper' fraction when it has a numerator, smaller than its denominator. When the Harmonic Progression is inverted and thereby turned into an Arithmetical progression, 52 then after finding the required number of means by the method for Arithmetic Means, and these are inverted, they become the Harmonic Means required. There are problems that are not necessarily supported by any established theory of laws but are based upon immediate experience rather than logical conclusions.

Such problems are known as Empirical Problems. To give an example, if you come across a problem which says 'with the ten digits, 9, 8, 7,6, 5, 4, 3, 2, 1, 0, express numbers whose sum is unity: There is no limit to the making of such questions, but their solutions involve little or no mathematical skill.

These are considered Empirical Problems. Sometimes certain problems are put leading to arithmetical results which are obviously impossible. Such problems are known as Arithmetical fallacies. The class of problems dealing with the reconstruction of arithmetical sums from which various digits have been erased are called Arithmetical Restorations. This kind of exercise has attracted a good deal of attention in recent years.

That every number greater than 4 can be expressed as ths sum of two odd primes. Claus in The problem goes like this: These discs are of different radii, and initially they are placed all on one peg, so that the biggest is at the bottom, and the radii of the successive discs decrease as we ascend: The problem is to shift the discs from one peg to another in such a way that a disc shall never rest on one smaller than itself, and finally to transfer the tower i.

The number of separate transfers of single discs which one must make to effect the transfer of the tower is: A de Polignac has conjenctured that every even 55 number is the difference of two consecutive primes in infinitely many ways.

Suppose we take the even number to be 2, this means that there are infinitely many pairs of primes that are consecutive odd numbers such as 5, 7; II, 13; 17, 19; 29, 31; 41, 43; 59, 61; 71, 73; these are called prime pairs. I n the Troy weight, which is used by jewellers in weighing precious metals and stones, a Carat is equal to 3. The term Karat is a variation of Carat and in form is used in comparing the parts of gold alloys which are in gold.

The comparison is based on the use of 24 Karats to mean pure gold, and therefore 14 Karats means pure gold by weight or 14 parts pure gold and 10 parts alloy. An equation is a symbolic statement that two quantities are equal in value. When an equation is true for all values of the letters, it is an identity, for example: However, when this is not the case, it is known as a Conditional Equation.

Factorial is the product of all positive integers from 1 up to a given number n, inclusive. Factorial is denoted by the symbol n!

There is no connection at all.

The Book of Numbers.pdf

Factoring is another thing altogether. Factoring is actually the process of finding two or more expressions whose product is a given expression. Gills is a term used in Liquid Measure.

Sixteen fluid ounces or 1 pint is equal to 4 Gills. Many practical problems are concerned with the measurements of a circle. And the basic to the measurements is the fact that the ratio of the circumference to its diameter is a constant. No matter what the size of the circle the ratio remains the same. In mathematics, this ratio is represented by the Greek letter 7: However this constant is not an integer and much effort has been spent to find the value of this ratio.

It has been evaluated to a large number of decimal points by electronic calculators. The story of the accuracy to which the value of pi is known is an interesting one.

In the Bible, the value of pi is used as 3 Archimedes had declared the value of pi as less than 3 but greater than 3i2. The value generally used today 3. In , with the use of the Computer Eniac, a group of mathematicians calculated decimals of pi in 70 hours. And recently Daniel. Shanks and John Wrench have published pi to , decimals. It took them 8 hours and 43 minutes on an IBM system to compute this result. In his will, he requested that these 35 numbers be engraved on his tombstone.

This was done. In Germany they still refer to this number as 'Ludolphian Number'. The significant digits of a decimal number are those beginning with the first one, reading from left to right, which is not zero and ending with the last are definitely specified. To 'Round off' a decimal number means to correct it to a specified number off significant digits.

The following rule is followed: The number of non-zero digits are retained and the rest on the right of this are discarded. If the digit to the immediate right of the last digit retained is greater than 5, the last digit retained is increased by 1. Jf the digit to the immediate right of the last digit retained is less than five, the last digit is left unchanged. A Cardinal number of an infinite set is called a Transfinite Cardinal Number. When we reduce a fraction, for example: Since the numerator and the denominator do not have any common prime factors they are said to be 'relatively prime'.

About the middle of the 19th century, this problem known as the 'Four colour' problem related to map making was proposed and remains insolved to this date. The problem involves the colouring of maps using at most four colours.

When two countries have common boundaries, they must have different 61 colours. When two countries have only single points in common they may use the same colour. No one, so far, has been able to produce a map that would require more than four colours. But no one has been able to prove that four colours are sufficient for all maps. However, it has been proved that if a map could be drawn that would require five colours, there would have been at least 36 countries on it. And it has also been proved that five colours are sufficient for all maps, but may not be necessary.

Googol is one of the largest numbers that has ever been named. It is defined as 1 followed by a googol of zeros. It is claimed that there would not even be room between the earth and the moon to write all the zeros in a googolplex! A quantity which may have any value we care tc give it is an Independent Variable. The value of the corresponding variable is given is fixed as soon as the value of the independent variable is known. A number which is not algebraic is called transcendental.

In other words a transcendental number satisfies no algebraic equation whose coefficients are rational numbers. Five suggested marriage, the union of the first even and first genuine odd number.

One was identified with reason. Two with opinion—a wavering fellow is Two, he does not know his own mind. And FOUR with justice, steadfast and square. When people first began to count, little by little they found out how to add, subtract, multiply and divide. And in some countries special devices were invented to make computation easier, especially in dealing with large numbers. The Romans used a counting frame, which is today known as Abacus, in which units, fives, ten and so on were represented by heads, which could be moved in grooves.

Since CALC means lime and marble is a kind of lime-stone, it is clear the heads were of marble. It is a kind of a wedge shaped writing of numerals, which developed in the Mesapotamia valley, and was very much in use in ancient times.

Lack' of other writing materials led the people to stamp inscriptions on clay bricks with sticks which usually were triangular with sharp edges. The Cuneiform numerals for 1,2 and 3 are V V V vvv. IN- The integer is a whole number as opposed to a fraction such as 3, 6, 8, 15, Whereas integral means consisting of a whole number or an undivided quantity. It is a vero meaning, filling in the intermediate terms of a series of numbers or to insert the intermediate terms of a series. Such fraction that have the same value, but are written differently are equivalent. Yes there is a good deal of difference, though they sound alike.

The word ton originates from the word tub. And a wine maker's tub which is also known as tun traditionally fills One gallon bottles of wine each weighing about 8 pounds.

A Literal Equation is an equation in which letters are used to represent numbers. In mathematics that arise in business and work it is often necessary to find the value of a Literal Expression when the numerical values of the letter variables are given.

This process is known as 'evaluating' the equation. Archimedes wrote, a short treatise in which he made an estimate of the number of grains of Sand in the world. This treatise is known as 'Sand Reckoner'. In this work he hit upon two of the most powerful peculiarities which is a part of the modern number script. He declared that all high numbers should be represented by multiples of simple powers of ten, and he also hit upon the taw 68 which underlies the modern called logarithms.

For example to find the logarithm of 9. At the top of the page there are columns headed by the numbers 0 to 9. We look along the line beginning 98 till we come to the column headed 7. The number given in this place is 9, This is the Mantissa of the logarithm of 9. In the ancient times the Egyptian writer Ahmes 69 wrote down the laws of measuring things. This is known as Rhynd papyrus.

However the plain man could not decipher this writing.

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The method used for obtaining any power for any quantity is known as 'Involution'. The power of a number is obtained by multiplying the logarithm of the number by the index representing the power required. The product thus acquired is the logarithm of the number required. An immense number, innumerable number, is called Myriad. It can also mean ten thousand.

The product of two numbers is equal to the difference of one-fourth the squares of their sum and their difference: This fact has been used 70 to reduce multiplication to addition with the aid of tables of quarter squares. The method is very old. There are many tables of quarter squares and the most extensive table lists quarter squares of integers upto , The method is very useful and economical.

If you do not use the properties of order and the commutative law of multiplication, we obtain an interesting further extension of the complex number system known as Quarternions q. The Theory of Numbers is concerned with the study of the properties of integers or whole numbers. The proofs of many of the Theory use the deepest resources of mathematics.

However, some of the most interesting conjectures are still unproved. Topics of the Theory of the Numbers can be listed as follows? Divisibility and Primality: Elementary Definitions; Factorizations Primes. Residue Classes; Eulers Theorem c. Congruences in one unknown d. Representation by forms: Binary Quadratic forms b. The Numbers Represented by a Quadratic Form: Automorphs and Reduction Binary Quadratic Forms of indefinite e.

Diophantine Equations III. Topics in Analytic Number Theory: Gauss's Class-Number Conjencture b. Additive Theory of Numbers: Partitions b. The waring problem and related problems 72 c. The Goldbach problem V. Diophontine Approximation: Geometry of Numbers b.

Diophontine Approximation VJ. Generalisations of Arithmetic: Algebraic Numbers b. Ideals c. Dyadic Notation is another name for Binary Notation. In the margin of his copy of 'Diophantus' Fermat wrote: I have found a truly wonderful proof which this margin is too small to contain'. Unfortunately, he died soon after, and this theory has never been proved, though almost every great mathematician for the last three centuries has attempted a proof.

A digital computer consists of three parts: Store 2. Executive Unit 3. Control The store is a store of information and memory. The executive unit is a part which carries out the various individual operations involved in a calculation. And it is the duty of the Control that the instructions fed to the machine are obeyed correctly and in the proper order. Associative Law means that the terms of an expression means connected by plus or minus signs can be grouped in any manner.

Here the meaning is that the factors of a product can be grouped in anyway we please. The terms or parts of an expression wmch are connected by plus or minus signs can be written in any order is the Commutative Law.

The factors of any product may be taken in any order—this is the Commutative Law of Multiplication. That addition and subtraction can be performed in any order. That the product of a Compound expression and a factor is the algebraical sum of the partial products of each term of the compound expression and the factor.

That the final index of a letter occuring more than once in a product is the sum of the indices of that letter. It cannot be done. Normal arithmetical operations cannot be carried out with per cent numbers.