MATHEMAT

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INDEX

WHAT IS MATHEMATICS

POLYNOMIALS

NUMBER SYSTEM

HERON’S FORMULA

WHAT IS MATHEMATICS

o Mathematics is the abstract study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.

o Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects.

POLYNOMIALS

In mathematics, polynomials are the simplest class of mathematical expressions (apart from the numbers and expressions representing numbers). A polynomial is an expression constructed from variables (also called indeterminates) and constants (usually numbers, but not always), using only the operations of addition, subtraction, multiplication, and non-negative integer exponents (which are abbreviations for several multiplications by the same value). However, the division by a constant is allowed, because the multiplicative inverse of a non-zero constant is also a constant. For example, x2 − x/4 + 7 is a polynomial, but x2 − 4/x + 7x3/2 is an algebraic expression that is not a polynomial, because its second term involves a division by the variable x (the term 4/x), and also because its third term contains an exponent that is not a non-negative integer (3/2).

A polynomial function is a function which is defined by a polynomial. Sometimes, the term polynomial is reserved for the polynomials that are explicitly written as a sum (or difference) of terms involving only multiplications and exponentiation by non negative integer exponents.

POLYNOMIALS IN ONE VARIABLE

A polynomial P in one variable x is formally defined as a follows

P(x) = p0 + p1x + ... + pnxn

where the pi are constants. If n = 0 we identify the polynomial with the constant p0. If pn  0 then we say the polynomial has degree n. If pn = 0 then we drop the corresponding term unless n = 0; the degree of the constant polynomial 0 is considered undefined.

EXAMPLES :- 2y + 4 is a polynomial in y of degree 1, as the greatest power of the variable y is 1ax2 +bx + c is a polynomial in x of degree 2, as the greatest power of the variable x is 23p4 -10p3 + 2p – 4/3 is a polynomial in p of degree 4, as the greatest power of the variable p is 4100 is also a polynomial (constant polynomial or monomial - that which contains only one term) in any variable, say x, because 100 is same as 100x0, and we know that x0 = 1.

TYPES OF POLYNOMIALS

1. Linear Polynomial: A polynomial of degree 1 is called a linear polynomial. Example : 3x, 5y + 6, 9p + q

2. Quadratic Polynomial: A polynomial of degree 2 is called a quadratic polynomialExample: in ax2 + bx + c, the degree is 2

3. Cubic Polynomial:A polynomial of degree 3 is called a cubic polynomial. Example: a3 +b3 + 3a2b + 3ab2

Standard Form of a Polynomial:If the terms in a polynomial are written in ascending or descending powers of the variable in it, then the polynomial is said to be in Standard Form. Examples:3x3 - 9x2 + 2 is in standard form, as the powers of the variable x are in descending order. -9 + 6x – 4/5 (x3) + x4

is also in standard form, as the powers of the variable x are in ascending order.

ZEROS OF POLYNOMIALS

In the previous section we studied the end-behavior of polynomials. We know that

a polynomial’s end-behavior is identical to the end-behavior of its leading term. Our

focus was concentrated on the far right- and left-ends of the graph and not upon what

happens in-between.

In this section, our focus shifts to the interior. There are two important areas of

concentration: the local maxima and minima of the polynomial, and the location of

the x-intercepts or zeros of the polynomial. In this section we concentrate on finding

the zeros of the polynomial.

REMAINDER THEOREM

• The remainder theorem states that if  is divided by , then the remainder is .For example, when  is divided by , the remainder (if we don't care about the quotient) will be . When  is divided by , the remainder is . However, this theorem is most useful when the remainder is 0 since it will yield a zero of . For example, is divided by , the remainder is , so 1 is a zero of  (by the definition of zero of a polynomial function).

FACTORISATION OF POLYNOMIALS

In mathematics and computer algebra, factorization of polynomials or polynomial factorization refers to factoring a polynomial with coefficients in a given field or in the integers into irreducible factors with coefficients in same domain. Polynomial factorization is one of the fundamental tools of the computer algebra systems.

The specification of the field is fundamental, as, for example, the polynomial x2−2 is irreducible over the integers and the rational numbers (it has no non-constant factors), while it is factorized into (x-\sqrt{2})(x+\sqrt{2}) over the field of real numbers.

Theoretically, there is always a factorization into irreducible polynomials of any polynomials with coefficients in a field: that is, polynomial rings are unique factorization domains. However, one wants an algorithm to perform this factorization in a finite number of steps.

ALGEBRAIC IDENTITIES

An algebraic equation which is true for all values of the variables occurring in the relation is known as analgebraic identity. 

IMPORTANT IDENTITIES:

There are 8 important algebraic identities which are given below:

POLYNOMIALS OF DIFFERENT DEGREES:

Identity I:

(x + y)2 = x2 + 2xy + y2

Identity II:

(x - y)2 = x2 - 2xy + y2

Identity III:

x2 - y2 = (x+ y)(x - y)

Identity IV:

(x + a)(x + b) = x2 + (a + b)x + ab

CONTINUED…….

Identity V:

(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx

Proof: 

Let x + y = k then, 

(x + y + z)2 = (k + z)2 

      = k2 + 2kz + z2        (Using identity I) 

      = (x + y)2 + 2(x + y)z + z2 

      = x2 + 2xy + y2 + 2 xz + 2yz + z2 

      = x2 + y2 + z2 + 2xy + 2yz + 2zx (proved)

Identity VI:

(x + y)3 = x3 + y3 + 3xy(x + y)

CONTINUED………..

Identity VII:

(x - y)3 = x3 - y3 - 3xy(x - y)

Identity VIII:

x3 + y3 + z3 - 3xyz = (x + y + z)(x2 + y2 + z2 - xy - yz - zx)

Proof: 

R.H.S.        = (x + y + z)(x2 + y2 + z2 - xy - yz - zx) 

       = x(x2 + y2 + z2 - xy - yz - zx) + y(x2 + y2 + z2 - xy - yz - zx)                                                 + z(x2 + y2 + z2 - xy - yz - zx) 

       = x3 + xy2 + xz2 - x2y - xyz - zx2 + yx2 + y3 + yz2- xy2 - y2z - xyz                                               + zx2 + zy2 + z3 - xyz - yz2 - xz2

       = x3 + y3 + z3 - 3xyz = L.H.S.        (proved)

NUMBER SYSTEM

A number is a mathematical object used to count, label, and measure. In mathematics, the definition of number has been extended over the years to include such numbers as 0, negative numbers, rational numbers, irrational numbers, and complex numbers.

Mathematical operations are certain procedures that take one or more numbers as input and produce a number as output. Unary operations take a single input number and produce a single output number. For example, the successor operation adds 1 to an integer, thus the successor of 4 is 5. Binary operations take two input numbers and produce a single output number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic.

A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (telephone numbers), for ordering (serial numbers), and for codes (e.g., ISBNs).

In common usage, the word number can mean the abstract object, the symbol, or the word for the number.

CLASSIFICATION OF NUMBERS

Natural 0, 1, 2, 3, 4, ... or 1, 2, 3, 4, ...

Integers ..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...

Rational a⁄b where a and b are integers and b is not 0

Real The limit of a convergent sequence of rational numbers

Complex a + bi or a + ib where a and b are real numbers and iis the square root of −1

Important number systems           

NATURAL NUMBER The most familiar numbers are the natural numbers or

counting numbers: 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including 0 (cardinality of the empty set, i.e. 0 elements, where 0 is thus the smallest cardinal number) in the set of natural numbers.[citation needed] Today, different mathematicians use the term to describe both sets, including 0 or not. The mathematical symbol for the set of all natural numbers is N, also written \mathbb{N}, and sometimes \mathbb{N}_0 or \mathbb{N}_1 when it is necessary to indicate whether the set should start with 0 or 1, respectively.

In the base 10 numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base 10 system, the rightmost digit of a natural number has a place value of 1, and every other digit has a place value ten times that of the place value of the digit to its right.

INTEGERS

The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a minus sign). As an example, the negative of 7 is written −7, and 7 + (−7) = 0. When the set of negative numbers is combined with the set of natural numbers (which includes 0), the result is defined as the set of integer numbers, also called integers, Z also written \mathbb{Z}. Here the letter Z comes from German Zahl, meaning "number". The set of integers forms a ring with operations addition and multiplication.

RATIONAL NUMBERS

o A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero Integer number denominator. Fractions are written as two numbers, the numerator and the denominator, with a dividing bar between them. In the fraction written m⁄n or m \over n \,

o m represents equal parts, where n equal parts of that size make up m wholes. Two different fractions may correspond to the same rational number; for example 1⁄2 and 2⁄4 are equal, that is: {1 \over 2} = {2 \over 4}.\,

o If the absolute value of m is greater than n, then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator 1. For example −7 can be written −7⁄1. The symbol for the rational numbers is Q (for quotient), also written \mathbb{Q}.

REAL NUMBER The real numbers include all of the measuring numbers.

Real numbers are usually written using decimal numerals, in which a decimal point is placed to the right of the digit with place value 1. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. Thus

123.456\,

represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6 thousandths. In saying the number, the decimal is read "point", thus: "one two three point four five six". In the US and UK and a number of other countries, the decimal point is represented by a period, whereas in continental Europe and certain other countries the decimal point is represented by a comma. Zero is often written as 0.0 when it must be treated as a real number rather than an integer. In the US and UK a number between −1 and 1 is always written with a leading 0 to emphasize the decimal. Negative real numbers are written with a preceding minus sign:

-123.456.\,

COMPLEX NUMBER

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose, historically, from trying to find closed formulas for the roots of cubic and quartic polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: the square root of −1, denoted by i, a symbol assigned by Leonhard Euler, and called the imaginary unit. The complex numbers consist of all numbers of the form

\,a + b i or

\,a + i b

where a and b are real numbers. In the expression a + bi, the real number a is called the real part and b is called the imaginary part. If the real part of a complex number is 0, then the number is called an imaginary number or is referred to as purely imaginary; if the imaginary part is 0, then the number is a real number. Thus the real numbers are a subset of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a Gaussian integer. The symbol for the complex numbers is C or \mathbb{C}.

COMPUTABLE NUMBER

• Moving to problems of computation, the computable numbers are determined in the set of the real numbers. The computable numbers, also known as the recursive numbers or the computable real's, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. Equivalent definitions can be given using μ-recursive functions, Turing machines or λ-calculus as the formal representation of algorithms. The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.

HISTORY OF NUMBER SYSTEM :- FIRST USE OF NUMBERS• Bones and other artifacts have been discovered with marks

cut into them that many believe are tally marks. These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.

• A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. Nonetheless tallying systems are considered the first kind of abstract numeral system.

• The first known system with place value was the Mesopotamian base 60 system (ca. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt

ZERO

• The use of 0 as a number should be distinguished from its use as a placeholder numeral in place-value systems. Many ancient texts used 0. Babylonian (Modern Iraq) and Egyptian texts used it. Egyptians used the word nfr to denote zero balance in double entry accounting entries. Indian texts used a Sanskrit word Shunye or shunya to refer to the concept of void. In mathematics texts this word often refers to the number zero.[6]

• Records show that the Ancient Greeks seemed unsure about the status of 0 as a number: they asked themselves "how can 'nothing' be something?" leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of 0 and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of 0. (The ancient Greeks even questioned whether 1 was a number.)

NEGATIVE NUMBER

• The abstract concept of negative numbers was recognized as early as 100 BC – 50 BC. The Chinese Nine Chapters on the Mathematical Art (Chinese: Jiu-zhang Suanshu) contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative.[7] This is the earliest known mention of negative numbers in the East; the first reference in a Western work was in the 3rd century in Greece. Diophantus referred to the equation equivalent to 4x + 20 = 0 (the solution is negative) in Arithmetica, saying that the equation gave an absurd result.

• During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brāhmasphuṭasiddhānta 628, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots."

IRRATIONAL NUMBER

• The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800 and 500 BC.[9] The first existence proofs of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but he could not accept irrational numbers, so he sentenced Hippasus to death by drowning.

TRANSCENDENTAL NUMBERS AND REAL'S 

• The first results concerning transcendental numbers were Lambert's 1761 proof that π cannot be rational, and also that en is irrational if n is rational (unless n = 0). (The constant e was first referred to in Napier's 1618 work on logarithms.) Legendre extended this proof to show that π is not the square root of a rational number. The search for roots of quintic and higher degree equations was an important development, the Abel–Ruffini theorem (Ruffini 1799, Abel 1824) showed that they could not be solved by radicals (formulas involving only arithmetical operations and roots). Hence it was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations). Galois (1832) linked polynomial equations to group theory giving rise to the field of Galois theory.

• The existence of transcendental numbers[10] was first established by Liouville (1844, 1851). Hermite proved in 1873 that e is transcendental and Lindemann proved in 1882 that π is transcendental. Finally Cantor shows that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite, so there is an uncountably infinite number of transcendental numbers.

INFINITY AND INFINITESIMALS

• The earliest known conception of mathematical infinity appears in the Yajur Veda, an ancient Indian script, which at one point states, "If you remove a part from infinity or add a part to infinity, still what remains is infinity." Infinity was a popular topic of philosophical study among the Jain mathematicians c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.

PRIME NUMBER

• Prime numbers have been studied throughout recorded history. Euclid devoted one book of the Elements to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic, and presented the Euclidean algorithm for finding the greatest common divisor of two numbers.

• In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the Renaissance and later eras.

HERON’S FORMULA

• In geometry, Heron's (or Hero's) formula, named after Heron of Alexandria,[1] states that the area T of a triangle whose sides have lengths a, b, and c is

• T = \sqrt{s(s-a)(s-b)(s-c)}

• where s is the semiperimeter of the triangle:

• s=\frac{a+b+c}{2}.

• Heron's formula can also be written as:

• T=\frac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a+b+c)(a+b+c)}

• T=\frac{1}{4}\sqrt{2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)}

• T=\frac{1}{4}\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}

• T=\frac{1}{4}\sqrt{4a^2b^2-(a^2+b^2-c^2)^2}

• Heron's formula is distinguished from other formulas for the area of a triangle, such as half the base times the height or half the modulus of a cross product of two sides, by requiring no arbitrary choice of side as base or vertex as origin.

HISTORY

• The formula is credited to Heron (or Hero) of Alexandria, and a proof can be found in his book, Metrica, written c. A.D. 60. It has been suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work.[2]

• A formula equivalent to Heron's namely:

• T=\frac1{2}\sqrt{a^2c^2-\left(\frac{a^2+c^2-b^2}{2}\right)^2}, where a \ge b \ge c

• was discovered by the Chinese independently of the Greeks. It was published in Shushu Jiuzhang (“Mathematical Treatise in Nine Sections”), written by Qin Jiushao and published in A.D. 1247.

PROOF

• A modern proof, which uses algebra and is quite unlike the one provided by Heron (in his book Metrica), follows. Let a, b, c be the sides of the triangle and A, B, C the angles opposite those sides. We have

• \cos \widehat C = \frac{a^2+b^2-c^2}{2ab}

• by the law of cosines. From this proof get the algebraic statement:

• \sin \widehat C = \sqrt{1-\cos^2 \widehat C} = \frac{\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2 }}{2ab}.

• The altitude of the triangle on base a has length b·sin(C), and it follows

• \begin{align}

CONTINUED……

• T & = \frac{1}{2} (\mbox{base}) (\mbox{altitude}) \\

• & = \frac{1}{2} ab\sin \widehat C \\

• & = \frac{1}{4}\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2} \\

• & = \frac{1}{4}\sqrt{(2a b -(a^2 +b^2 -c^2))(2a b +(a^2 +b^2 -c^2))} \\

• & = \frac{1}{4}\sqrt{(c^2 -(a -b)^2)((a +b)^2 -c^2)} \\

• & = \sqrt{\frac{(c -(a -b))(c +(a -b))((a +b) -c)((a +b) +c)}{16}} \\

• & = \sqrt{\frac{(b + c - a)}{2}\frac{(a + c - b)}{2}\frac{(a + b - c)}{2}\frac{(a + b + c)}{2}} \\

• & = \sqrt{\frac{(a + b + c)}{2}\frac{(b + c - a)}{2}\frac{(a + c - b)}{2}\frac{(a + b - c)}{2}} \\

• & = \sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}.

• \end{align}

• The difference of two squares factorization was used in two different steps.

PROOF USING THE PYTHAGOREAN THEOREM

• Heron's original proof made use of cyclic quadrilaterals, while other arguments appeal to trigonometry as above, or to the incenter and one excircle of the triangle [2]. The following argument reduces Heron's formula directly to the Pythagorean theorem using only elementary means.

• We wish to prove 4T^2=4s(s-a)(s-b)(s-c). The left-hand side equals

• 4 T^2 = (c h)^2 = c^2(b^2-d^2) = (c b)^2 - (c d)^2

• while the right-hand side equals

• 4s(s-a)(s-b)(s-c) = [s(s-a)+(s-b)(s-c)]^2 - [s(s-a)-(s-b)(s-c)]^2

• via the identity (p+q)^2-(p-q)^2=4pq. It therefore suffices to show

• cb=s(s-a)+(s-b)(s-c)

CONTINUED…….

• and

• cd=s(s-a)-(s-b)(s-c).

• Substituting 2s=(a+b+c) into the former,

• s(s-a)+(s-b)(s-c)=\frac{1}{4}(a+b+c)(-a+b+c) + \frac{1}{4}(a+b+c)(a+b+c) = \frac{1}{4}[(b+c)^2-a^2] + \frac{1}{4}[a^2-(b-c)^2] = \frac{1}{4}[(b+c)^2 - (b-c)^2] = cb

• as desired. Similarly, the latter expression becomes

• s(s-a)-(s-b)(s-c)=\frac{1}{4}[(b+c)^2-a^2] - \frac{1}{4}[a^2-(b-c)^2] = \frac{1}{2}(b^2+c^2-a^2).

• Using the Pythagorean theorem twice, b^2=d^2+h^2 and a^2=(c-d)^2+h^2, allows us to simplify the expression to

• \frac{1}{2}(b^2+c^2-a^2) = \frac{1}{2}[d^2+c^2-(c-d)^2] = cd.

THANKYOU

MADE BY :- MEGHANSH GAUTAM & MOHIB KHAN

OF CLASS IX –A

ROLL NO. 19,20