Mathematics Assignment

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B i ss Mathematics” (PROFESSOR FAHAD AMDANI)

STUDENT INFORMATION: NAME : SYED OWAIS ALI ID : SP07-BB-0135 COURSE: “BBA IN BANKING AND FINANCE”

: ASSIGNMENT FOR : Search about the following Topics; 1) Logarithm, 2) Antilogarithm, 3) Rational Numbers, 4) Irrational Numbers, 5) Exponential Expression

Logarithm and Antilogarithm In mathematics, a logarithm of a number x in base b is a number n such that x = bn, where the value b must be neither 0 nor a root of 1. It is usually written as

A good way of remembering is by asking: "b to what power (n) equals x?” In other words, it is the exponent or power to which a base must be raised to yield a given number. When x and b are further restricted to positive real numbers, the logarithm is a unique real number. For example, since

We conclude that

Or, in words, the base-3 logarithm of 81 is 4, or the log base-3 of 81 is 4.

The logarithm as a function The function logb(x) depends on both b and x, but the term logarithm function (or logarithmic function) in standard usage refers to a function of the form logb(x) in which the base b is fixed and so the only argument is x. Thus there is one logarithm function for each value of the base b (which must be positive and must differ from 1). Viewed in this way, the base-b logarithm function is the inverse function of the exponential function bx. The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.

Bases The most widely used bases for logarithms are 10, the mathematical constant e ≈ 2.71828... and 2. When "log" is written without a base (b missing from logb), the intent can usually be determined from context:

• • • •

natural logarithm (loge, ln, log, or Ln) in mathematical analysis common logarithm (log10 or simply log) in engineering and when logarithm tables are used to simplify hand calculations binary logarithm (log2) in information theory and musical intervals indefinite logarithm when the base is irrelevant, e.g. in complexity theory when describing the asymptotic behavior of algorithms in big O notation.

Other notations The notation "ln(x)" invariably means loge(x), i.e., the natural logarithm of x, but the implied base for "log(x)" varies by discipline: •

Mathematicians generally understand both "ln(x)" and "log(x)" to mean loge(x) and write "log10(x)" when the base-10 logarithm of x is intended. Calculus textbooks will occasionally write "lg(x)" to represent "log10(x)".

•

Many engineers, biologists, astronomers, and some others write only "ln(x)" or "loge(x)" when they mean the natural logarithm of x, and take "log(x)" to mean log10(x) or, sometimes in the context of computing, log2(x).

•

On most calculators, the LOG button is log10(x) and LN is loge(x).

•

In most commonly used computer programming languages, including C, C++, Java, Fortran, Ruby, and BASIC, the "log" function returns the natural logarithm. The base-10 function, if it is available, is generally "log10."

•

Some people use Log(x) (capital L) to mean log10(x), and use log(x) with a lowercase l to mean loge(x).

•

The notation Log(x) is also used by mathematicians to denote the principal branch of the (natural) logarithm function.

•

A notation frequently used in some European countries is the notation blog(x) instead of logb(x).[citation needed]

This chaos, historically, originates from the fact that the natural logarithm has nice mathematical properties (such as its derivative being 1/x, and having a simple definition), while the base 10 logarithms, or decimal logarithms, were more convenient for speeding calculations (back when they were used for that purpose). Thus natural logarithms were only extensively used in fields like calculus while decimal logarithms were widely used elsewhere. The inverse function of the logarithm, defined such that The antilogarithm in base of is therefore .

Rational Numbers In mathematics, a rational number is a number which can be expressed as a ratio of two integers. Non-integer rational numbers (commonly called fractions) are usually written as the vulgar fraction

, where b is not zero.

Each rational number can be written in infinitely many forms, such as , but is said to be in simplest form is when a and b have no common divisors except 1 (i.e., they are coprime). Every non-zero rational number has exactly one simplest form of this type with a positive denominator. A fraction in this simplest form is said to be an irreducible fraction, or a fraction in reduced form. The decimal expansion of a rational number is eventually periodic (in the case of a finite expansion the zeroes which implicitly follow it form the periodic part). The same is true for any other integral base above one, and is also true when rational numbers are considered to be p-adic numbers rather than real numbers. Conversely, if the expansion of a number for one base is periodic, it is periodic for all bases and the number is rational. A real number that is not a rational number is called an irrational number. The set of all rational numbers, which constitutes a field, is denoted builder notation, is defined as

where

. Using the set-

denotes the set of integers.

The term rational In the mathematical world, the adjective rational often means that the underlying field considered is the field of rational numbers. For example, a rational integer is an algebraic integer which is also a rational number, which is to say, an ordinary integer, and a rational matrix is a matrix whose coefficients are rational numbers. Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a "polynomial over the rationals". However, rational function does not mean the underlying field is the rational numbers, and a rational algebraic curve is not an algebraic curve with rational coefficients.

Arithmetic Two rational numbers and are equal if and only if ad = bc. Two fractions are added as follows

The rule for multiplication is

Additive and multiplicative inverses exist in the rational numbers

It follows that the quotient of two fractions is given by

Properties The set , together with the addition and multiplication operations shown above, forms a field, the field of fractions of the integers . The rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of . The rational numbers are therefore the prime field for characteristic zero. The algebraic closure of numbers.

, i.e. the field of roots of rational polynomials, is the algebraic

The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set. The rationals are a densely ordered set: between any two rationals, there sits another one, in fact infinitely many other ones. Any totally ordered set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers.

Irrational Numbers In mathematics, an irrational number is any real number that is not a rational number, i.e., it is a number not of the form n/m, where n and m are integers. Almost all real numbers are irrational, in a sense which is defined more precisely below. When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common. A measure of a line segment I in this sense is a line segment J that "measures" I in the sense that some whole number of copies of J laid end-to-end occupy the same length as I.

Example proofs The square root of 2 One proof of the irrationality of the square root of 2 is the following reductio ad absurdum. The proposition is proved by assuming the contrary and showing that doing so leads to a contradiction (hence the proposition must be true). 1. Assume that

is a rational number. This would mean that there exist integers a

and b such that a / b =

.

2. Then can be written as an irreducible fraction (the fraction is shortened as much as possible) a / b such that a and b are coprime integers and (a / b)2 = 2. 3. It follows that a2 / b2 = 2 and a2 = 2 b2. 4. Therefore a2 is even because it is equal to 2 b2 which is also even. 5. It follows that a must be even (odd square numbers have odd square roots and even square numbers have even square roots). 6. Because a is even, there exists an integer k that fulfills: a = 2k. 7. We insert the last equation of (3) in (6): (2k)2 = 2b2 is equivalent to 4k2 = 2b2 is equivalent to 2k2 = b2. 8. Because 2k2 is even it follows that b2 is also even which means that b is even because only even numbers have even squares. 9. By (5) and (8) a and b are both even, which contradicts that a / b is irreducible as stated in (2). Since we have found a contradiction, the assumption (1) that must be false; that is to say,

is a rational number

is irrational.

This proof can be generalized to show that any root of any natural number is either a natural number or irrational.

Another proof Another reductio ad absurdum argument showing that known: •

Assume that

is a rational number. This would mean that there exist integers m

and n such that

•

Then

•

Since

is irrational is less well-

.

. , it follows that .

, and it can be shown that

So a fraction in lowest terms is reduced to yet lower terms. That is a contradiction if n and m are positive integers, so the assumption that

is rational must be false.

Similarly, assume an isosceles right triangle whose leg and hypotenuse have respective integer lengths n and m. By the Pythagorean theorem, the ratio m/n equals . It is possible to construct by a classic compass and straightedge construction a smaller isosceles right triangle whose leg and hypotenuse have respective lengths m − n and 2n − m. That construction proves the irrationality of employed by ancient Greek geometers.

by the kind of method that was

Open questions It is not known whether π + e or π − e are irrational or not. In fact, there is no pair of nonzero integers m and n for which it is known whether mπ + ne is irrational or not. Moreover, It is not known whether the set {π, e} is algebraically independent over Q. It is not known whether 2e, πe, π√2, Catalan's constant, or the Euler-Mascheroni gamma constant γ are irrational.

Exponential Expression Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. When n is a whole number, exponentiation corresponds to repeated multiplication:

just as multiplication by a whole number corresponds to repeated addition:

Exponentiation can also be defined for exponents that are not whole numbers. Exponentiation is also known as raising the number a to the power n, or a to the nth power. (Another historical synonym, involution,[1] is now rare and should not be confused with its more common meaning.) The exponent is usually shown as a superscript to the right of the base. Exponentiation is a basic mathematical tool that is used pervasively in many other fields as well, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.

Identities and properties The most important identity satisfied by integer exponentiation is:

This identity has the following consequences:

While addition and multiplication are commutative (for example, 2+3 = 5 = 3+2 and 2·3 = 6 = 3·2), exponentiation is not commutative: 23 = 8, but 32 = 9.

Similarly, while addition and multiplication are associative (for example, (2+3)+4 = 9 = 2+(3+4) and (2·3)·4 = 24 = 2·(3·4), exponentiation is not associative either: 23 to the 4th power is 84 or 4096, but 2 to the 34 power is 281 or 2,417,851,639,229,258,349,412,352.