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Mathematical formulae are useful rules expressed using symbols or letters. In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a chemical formula. The informal use of the term formula in science refers to the general construct of relationship between given quantities. The plural of formula can be spelled either as formulas (from the most common English plural noun form) or, under the influence of scientific Latin, formulae (from the original Latin).

In mathematics, a formula generally refers to an identity which equates one mathematical expression to another with the most important ones being mathematical theorems. Syntactically, a formula is an entity which is constructed using the symbols and formation rules of a given logical language. For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion. However, having done this once in terms of some parameter (the radius for example), mathematicians have produced a formula to describe the volume of a sphere in terms of its radius: V = 4/3nr3

Having obtained this result, the volume of any sphere can be computed as long as its radius is known. Here, notice that the volume V and the radius rare expressed as single letters instead of words or phrases. This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate formulas which are larger and more complex. Mathematical formulas are often algebraic, analytical or in closed form.

In modern chemistry, a chemical formula is a way of expressing information about the proportions of atoms that constitute a particular chemical compound, using a single line of chemical element symbols, numbers, and sometimes other symbols, such as parentheses, brackets, and plus (+) and minus (?) signs. For example, H2O is the chemical formula for water, specifying that each molecule consists of two hydrogen (H) atoms and one oxygen (O) atom. Similarly, O?

denotes an ozone molecule consisting of three oxygen atoms and a net negative charge.

In a general context, formulas are a manifestation of mathematical model to real world phenomena, and as such can be used to provide solution (or approximated solution) to real world problems, with some being more general than others. For example, the formula F = ma is an expression of Newton’s second law, and is applicable to a wide range of physical situations. Other formulas, such as the use of the equation of a sine curve to model the movement of the tides in a bay, may be created to solve a particular problem. In all cases, however, formulas form the basis for calculations.

Expressions are distinct from formulas in that they cannot contain an equal’s sign (=). Expressions can be liken to phrases the same way formulas can be liken to grammatical sentences.

Decimal numbers use 10 digits, which are combined to make numbers of any size. The position of the digit determines what it means in any number. For example, the 2 in the number 200 is ten times the size of the 2 in the number 20. Each position of a number gives a value ten times higher than the position to its right. So 9867 means 7 units, plus 6 x 10, plus 8 x 10 x 10, plus 9 x 10 x 10 x 10. As decimal numbers are based on the number 10, we say that this is a base -10 number system.

We have learnt that the decimals are an extension of our number system. We also know that decimals can be considered as fractions whose denominators are 10, 100, 1000, etc. The numbers expressed in the decimal form are called decimal numbers or decimals.

For example: 5.1, 4.09, 13.83, etc.

A decimal has two parts:

(a) Whole number part

(b) Decimal part

These parts are separated by a dot (.) called the decimal point.

• The digits lying to the left of the decimal point form the whole number part. The places begin with ones, then tens, then hundreds, then thousands and so on.


• The decimal point together with the digits lying on the right of decimal point form the decimal part. The places begin with tenths, then hundredths, then thousandths and so on…

Example:

(i) In the decimal number 211.35; the whole number part is 211 and the decimal part is .35

(ii) In the decimal number 57.031; the whole number part is 57 and the decimal part is .031

(iii) In the decimal number 197.73; the whole number part is 197 and the decimal part is .73

The binary system is another way of counting. Instead of being a base-10 system, it is a base-2 system, using only two digits: 0 and 1. Again, the position of a digit gives it a particular value. 1010101 means 1 unit, plus 0 x 2, plus 1 x 2 x 2, plus 0 x 2 x 2 x2, plus1 x 2 x 2 x 2 x 2, plus 0 x 2x 2 x 2 x 2x 2,plus 1 x 2 x 2 x 2 x 2 x 2 x 2. 1010101 is the same as 85 in decimal numbers.

When you learn math at school, you use a base-10 number system. That means your number system consists of 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. When you add one to nine, you move the 1 one spot to the left into the tens place and put a 0 in the ones place: 10. The binary system, on the other hand, is a base-2 number system. That means it only uses two numbers: 0 and 1. When you add one to one, you move the 1 one spot to the left into the twos place and put a 0 in the ones place: 10. So, in a base-10 system, 10 equal ten. In a base-2 system, 10 equal two.

In the base-10 system you're familiar with, the place values start with ones and move to tens, hundreds, and thousands as you move to the left. That's because the system is based upon powers of 10. Likewise, in a base-2 system, the place values start with ones and move to twos, fours, and eights as you move to the left. That's because the base-2 system is based upon powers of two. Each binary digit is known as a bit.

Don't worry if the binary system seems confusing right now. It's fairly easy to pick up once you work with it a while. It just seems confusing at first because all numbers are made up of only 0s and 1s. The familiar base-10 system is as easy as 1-2-3, while the base-2 binary system is as easy as 1-10-11.

You may WONDER why computers use the binary system. Computers and other electronic systems work faster and more efficiently using the binary system, because the system's use of only two numbers is easy to duplicate with an on/off system. Electricity is either on or off, so devices can use an on/off switch within electric circuits to process binary information easily. For example, off can equal 0 and on can equal 1.

Every letter, number, and symbol on a keyboard is represented by an eight-bit binary number. For example, the letter A is actually 01000001 as far as your computer is concerned! To help you develop a better understanding of the binary system and how it relates to the decimal system you're familiar with, here's how the decimal numbers 1-10 look in binary:

Picture Credit : Google

The romans had a number system with a base of 10, as we do, but they used different numerals to write it down. For the numbers one to nine, instead of using nine different numerals, they used only three different letters, combining them to make the numbers. This made it very difficult for them to do even simple calculations, so their advances in mathematics and related fields were not as great as might have been expected from such a far-reaching civilization.

Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet. Modern usage employs seven symbols, each with a fixed integer value:

The use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced in most contexts by the more convenient Arabic numerals; however, this process was gradual, and the use of Roman numerals persists in some minor applications to this day.

One place they are often seen is on clock faces. For instance, on the clock of Big Ben (designed in 1852), the hours from 1 to 12 are written as:

I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII

The notations IV and IX can be read as "one less than five" (4) and "one less than ten" (9), although there is a tradition favoring representation of "4" as "IIII" on Roman numeral clocks.

Other common uses include year numbers on monuments and buildings and copyright dates on the title screens of movies and television programs. MCM, signifying "a thousand, and a hundred less than another thousand", means 1900, so 1912 is written MCMXII. For the years of this century, MM indicates 2000; so that the current year is MMXX (2020).

Picture Credit : Google

  

               Infinity is a number, or value, which is so huge that it cannot be measured. For instance, the distance to the end of the Universe is called infinity, because if there is an end, it is so far away that it could never be measured.

               Sometimes infinity can be the result of a mathematical calculation. For example, the formula for calculating the distance around the outside of a circle is pi times the diameter. Pi is a Greek letter, and it represents a value of approximately 3.14159. Pi can never be fully calculated, because you would finish up with a string of numbers extending to infinity.

Picture credit: google

 

                    Geometry is a part of mathematics that deals with the shape, position and size of things of geometric forms such as squares, triangles, cubes and cones. Its name comes from the Greek words meaning ‘earth measuring’, because it was probably originally invented as a means of surveying and measuring land.

                   The ancient Egyptians also used geometry when constructing buildings and tombs. Nowadays, geometry is important to engineers and architects. It is also essential in navigation because geometry is used to follow charts and maps.

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               Our numbers are called Arabic numerals, although they probably appeared first in India around AD600. By the 800s the Arab numbering system was use throughout Europe because it was much easier to use than the old Roman system. At first the numbers varied in the way they were written, but with the invention of printing the numbers became standardized. The basic Arab numbers are 0 to 9, and can be used to write any combination of numbers.

Picture credit: google

   

                 In some ways, Roman numbering worked like the modern Arabic numeral system where, starting from the left, there are thousands, hundreds, tens and individual units. However, Roman numerals are quite different. One thousand is written as M, five hundred as D, one hundred as C, fifty as L, ten as X and five as V.

               The decimal system describes a numbering system for calculation based on multiples of ten. Multiplying or dividing a number by ten is very easy because only the decimal point needs to be moved. In the decimal system, each number has a value ten times that of the next number to the right. For instance, 5,283 means five thousands, two hundreds, eight tens and three ones. The decimal point simply separates the main number from numbers less than one. The number ten has always been important in mathematics — you can easily count it on your fingers.

Picture credit: google

               The metric system is a group of various units of measurement. Its name comes originally from the metre. It is a decimal system, which means that each unit is ten times bigger, or smaller, than the next unit. Previously, measurements were difficult to calculate; in measurements of length, for example, there were 12 inches in one foot and three feet in one yard, while weight was calculated using ounces and pounds (16 ounces in one pound).

               The metric system was devised in the 1790s in France, and is now used in most countries in the world for all scientific and technical measurements.

Picture credit: google

               Numbers are used to describe the amount of things. We can express numbers in words, by hand gestures or in writing, using symbols or numerals. When we talk about a number we use words (‘five’) rather than the numeral (5), but when we write we use both words and numerals.

               Numbers can describe how many objects there are or their position among lots of objects, such as 1st or 5th for example. Other types of numbers describe how many units of something there are, for example how many kilograms (weight) or metres (length).Numbers are just a convenient way to describe ideas.

Picture credit: google

               All numbers are made from ten figures. These are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. All decimal numbers are also made from the above ten figures.

               Now the question arises, why are there only ten numbers with single figure? Although it cannot be precisely explained, yet some had concluded that this is because ancient people began counting by using their ten fingers. This was easier for them to learn how to handle numbers, as they could use their ten fingers to count.

               It is an established fact that right from ancient times, people tried various systems to write numbers. They showed numbers by marking separate indications. For a number like 10 for instance, they showed 10 lines or 10 drawings of birds or animals. This system was used by Egyptians which are still to be found in ancient monuments or structures. In fact, all civilizations have had their own way of writing and using numbers. Romans invented special signs or alphabets for counting. Such signs are called numerals. For centuries, people used Roman numerals. But they are rather clumsy. They did not have zero.

               But counting became easier after invention of zero. Zero was invented in 6th century in India and was brought to Europe by Arab travellers at a later stage. Having a figure for zero, any number can be made in a simpler way. It may continue in decimal system. Fractions like or  may be written as decimal fractions like 0.50, 0.25 or 0.75. In fact there are many ways of expressing numbers. In everyday terms numbers are used with units like kg; litre or metre and so on.

               It is most interesting to note that computers use a system that use only two digits — 1 and 0. This is called binary system. They play a key role in a modern digital computer. A computer changes numbers and words into codes made up of 0s and 1s. It makes calculation with these codes.

               The earliest known written numbers were those used by Babylonians about 5000 years ago. The symbols used today in English for the ten numbers (0, 1, 2,...) are called Arabic or Hindu-Arabic numerals because they came from India through the Arab countries and reached Europe in the 12th century.

 

          The Greek letter ‘pi’ is a unique number and is defined as the ratio of a circle’s circumference to its diameter. This number is independent of the size of a circle and for all practical purposes its approximate value is taken as 22/7 or 3.1416. In fact, the fraction 22/7 is slightly greater in value than ‘pi’.

          For many centuries mathematicians have been fascinated by its unique characteristic. The strangest thing about this number is that nobody has been able to calculate its exact value. Computer scientists have now computed pi to over one million decimal places.

          At one time the scientists tried to prove that ‘pi’ was a fraction. When any fraction is written in a decimal number, the same digits always appear over and over again in a special pattern. If ‘pi’ were a fraction, there would be a repeating pattern to its digits. But strangely enough a repeating pattern in ‘pi’ could not be found. Finally in 1761, a Swiss mathematician named Johann Heinrich Lambert settled the matter once and for all. He proved that pi is not a fraction.

          Now the question arises what is the significance of ‘pi’ in our daily life? Suppose you have an automobile tyre whose diameter is one metre. If you want to find its circumference, you can find it out by measuring with a tape. Another way of finding the circumference is to multiply the diameter by this strange number ‘pi’. This number is used to calculate the circumference of all the circular objects.

          The mathematicians are still engaged in research in this direction to see if the digits are arranged in any special way. 

          Any positive integer which is greater  than one and divisible by only itself is called a prime number. For example 2,3,5,7,11,13,17,19,23,29, etc. are all prime number – numbers that cannot be split by division by any other number except 1 and the particular number itself.

          The prime numbers lie at the very roots of arithmetic and have always fascinated those dealing with figures. We can take the sequence of the above given series of prime numbers as far as we like, but we will never find a prime number divisible by another. Over the centuries, the world’s greatest mathematicians have tried to do so and always fail, although they have also been unable to prove that no such number exists.

          Every positive integer greater than one can be expressed as the product of only a single set of prime numbers. Despite the fact that prime numbers have been recognized since at least 300 B.C. when they were first studied by the Greek mathematician Euclid and Eratosthenes. Still these numbers have not yet unfolded certain mysteries relating to them.

          There is infinity of prime numbers and in theory anything may happen in infinity. But so far theorists have not been able to even find any particular rule or theory governing the gaps between prime numbers, which still remains a great mathematical mystery.

          However, the highest known prime number was discovered in 1992 by analysts at AEA Technology’s Harwell Laboratory, Oxon. The number contains 227832 digits, enough to fill over 10 fullscap pages. 

                Arithmetic is the study of the addition, subtraction, multiplication and division of numbers. The word ‘arithmetic’ is derived from the Greek word ‘arithmos’ which means numbers. In the beginning of civilization, man used to count his sheep, cows, oxen and other animals on fingers. In fact, the word ‘digit’ which is used to denote numbers from zero to nine, finds its origin in the Latin word ‘digitus’ meaning a finger or toe. Later on, man started counting by putting marks on sticks of wood. But this process ended soon and man started using various signs for each number.

            The Egyptians used straight lines for counting one to ten. The Greeks used the letters of their alphabets for this purpose. Just to make the difference clearer, a small sign used to be affixed to the letters. For example, they would write a’ for one, b’ for two and j’ for ten. The Romans used to write the first five digits as I, II, III, IV, and V. they used to write X for ten, L for fifty, C for hundred, D for five hundred and M for one thousand. In the Roman language even today numbers are written like this.

            The numerals presently in use are called Arabic numerals, because it was from the Arabs that these numerals spread to Europe. Actually, they are Indian by origin and should rightly be called Indian numerals. Zero too is Indian by origin and is called ‘Shoonya’, meaning ‘empty’ or ‘nothing’, which became ‘sifr’ in Arabic, meaning the same. In 1202 an Italian resident prepared the first book of arithmetic based on the Arabic system. The first book on arithmetic in the Latin language was printed in 1478. By that time, the arithmetical methods of addition, subtraction, multiplication and division had fully developed. Mathematicians took centuries to develop the methods now used in arithmetic. Every one who goes to school learns arithmetic. It is a skill necessary in science, business and every day life.