1 1 number theory

The document discusses properties of real numbers. It defines real numbers and distinguishes between rational and irrational numbers. Rational numbers can be expressed as ratios of integers, while irrational numbers have non-terminating, non-repeating decimals. It also outlines relationships between subsets of real numbers and properties that real number operations satisfy, such as commutativity, associativity, identities, inverses, and distribution.

The document introduces key concepts in algebra including variables, constants, types of numbers (counting, integers, rational, irrational, real), graphs, averages, and positive and negative numbers. It provides examples and guidelines for understanding these concepts. Variables represent quantities that can vary, while constants represent fixed values. Different number sets are explained and visualized on a number line. Averages are calculated by adding values and dividing by the total count. Positive numbers are greater than zero, while negative numbers are less than zero.

The document discusses the distributive property in algebra. It provides definitions of key terms like term, coefficient, and like terms. It gives examples of using the distributive property to simplify expressions and solve problems involving perimeter. The distributive property allows multiplying a number by the sum of two other numbers, distributing the number factor across the addition.

This document reviews representations of different types of numbers on the number line. It discusses natural numbers, integers, rational numbers like terminating and repeating decimals. Irrational numbers like √2 that are non-terminating and non-repeating are also reviewed. Key properties of real numbers are listed, including that every point on the number line corresponds to a unique real number, and real numbers satisfy closure, commutative, associative, identity and inverse properties.

This document discusses the history and key concepts of real numbers. It provides background on how real numbers developed from ancient civilizations working with simple fractions to the formal acceptance of irrational numbers. Key figures discussed include Euclid, Hippasus, and developments in ancient Egypt, India, Greece, the Middle Ages, and Alexandria. Fundamental ideas covered include Euclid's lemma, the fundamental theorem of arithmetic, prime factorisation, and the distinction between rational and irrational numbers.

1) The document discusses the real number system, including identifying integers, rational numbers, and irrational numbers. Real numbers have important properties like closure under addition and multiplication. 2) Intervals of real numbers can be described using inequality notation like x ≥ 1 or -4 < x < 1, or interval notation like [1, +∞) or (-4, 1). 3) Absolute value of a real number a, written |a|, represents the distance of the number from the origin on the number line and is always non-negative.

Rational numbers can be represented as fractions p/q where p and q are integers and q is not equal to 0. Rational numbers are closed under addition, subtraction, and multiplication, meaning these operations on rational numbers will always result in another rational number. Division of rational numbers is not closed as division by 0 is undefined. Rational numbers exhibit properties like commutativity for addition and multiplication but not for subtraction or division. They also demonstrate the associative and distributive properties. Every rational number has an additive inverse and a multiplicative inverse or reciprocal. Zero is the additive identity and one is the multiplicative identity.

* Use inductive reasoning to identify patterns and make conjectures * Use properties of equality and deductive reasoning to create algebraic proofs

This document discusses rational and irrational numbers. It begins by recalling rational numbers as numbers that can be written as p/q where p and q are integers and q is not equal to 0. Irrational numbers cannot be written in this form and include numbers like √2, √3, and π. Together, rational and irrational numbers make up the set of real numbers. The document then explores properties of rational numbers, such as when their decimal expansions terminate or repeat periodically. It introduces the Fundamental Theorem of Arithmetic and uses it to prove results about rational numbers and their representations as decimals.

The document discusses inequalities and their solutions. It defines absolute and conditional inequalities and explains how to represent solutions using interval, set, and graphical notation. Methods are presented for solving linear, polynomial, rational, and absolute value inequalities. Key steps include determining intervals where an expression is positive or negative, identifying valid intervals based on inequality signs, and expressing the solution in interval notation. Examples are provided throughout to demonstrate these techniques.

The document discusses graphing and solving linear inequalities. It explains that when graphing inequalities, an open dot is used for < or > and a solid dot for ≤ or ≥. An example problem finds the average speed of a runner faster than Sue by representing the faster runner's speed as a variable x and setting up the inequality x > 2/10. It also outlines the rules for solving inequalities, which are the same as equations except the inequality sign must be flipped when multiplying or dividing by a negative number.

The properties of rational numbers, Square and square roots, cube and cube roots- A rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.Since q may be equal to 1, every integer is a rational number.

This document summarizes Chapter 3 of a textbook on C++ programming. It discusses various flow control structures in C++ like boolean expressions, if/else statements, switch statements, loops, and more. Key topics covered include boolean logic, precedence rules for expressions, short-circuit evaluation, nested if/else statements, the switch statement syntax, break and default in switch, scope rules for blocks, and while vs do-while loops.

The document discusses sets and operations involving sets of real numbers. It defines key sets such as the real numbers, rational numbers, integers, natural numbers, whole numbers, and irrational numbers. It also covers properties of real numbers like closure, commutativity, associativity, distributivity, identity, and inverse properties as they relate to addition and multiplication. Examples are provided to illustrate set notation and properties. The document contains exercises asking to identify which properties justify certain statements involving real number operations.

Revision Notes for CPT Chapter: Ratio & Proportion, Indices, Logarithms Subject: Quantitative Aptitude

This document defines and explains key concepts related to real numbers and algebraic expressions. It introduces sets and subsets of real numbers like integers, rational numbers, and irrational numbers. It describes properties of real numbers including addition, multiplication, order, and absolute value. It also covers representing real numbers on a number line, algebraic expressions, and properties of negatives.

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Taylorism or scientific management is an approach to work optimization developed by Frederick W. Taylor in the late 19th/early 20th century based on breaking jobs down into small, repetitive tasks and setting optimal processes and output goals for workers. It focuses on analyzing jobs, enforcing rigid discipline, minimizing interpersonal interactions, and incentivizing productivity through piece-rate pay systems. The approach aims to maximize efficiency using scientific methods, specialized roles for planning and performance, and strict managerial control.

Scientific management, developed by Frederic Winslow Taylor in the 1880s-1890s, aimed to improve economic efficiency, especially labor productivity, through analyzing and optimizing workflows. It involved time and motion studies to eliminate unnecessary movements, specialization of tasks, establishment of performance standards, and use of monetary incentives. While scientific management improved productivity and reduced costs, it also faced criticism for potentially exploiting workers and creating monotony. Many of its principles of efficiency and standardization, however, remain relevant to modern management.

Functionalism views society as a system of interconnected parts that work together to maintain stability. Key functionalist theorists like Emile Durkheim and Talcott Parsons believed that social institutions like the family, education, religion, and the economy serve important functions in socializing individuals and promoting social cohesion. Functionalism interprets social change as slow and gradual to preserve social equilibrium. It also sees some social inequalities as inevitable and functional for motivating social mobility. However, functionalism has been criticized for overlooking social conflicts and providing an overly harmonious view of society.

Functionalism views society as a structure of interconnected parts that work together for the benefit of the whole, similar to the human body or a clock. Key institutions like law, education, and religion serve important functions in maintaining social order and enforcing shared norms and values. Functionalists argue these institutions allow for social stability and change by socializing individuals, dividing labor, and discouraging deviant behavior. Early functionalist theorists like Evans-Pritchard and Firth studied non-Western societies and distinguished between social structure and social organization.

The Hawthorne experiments conducted in the 1920s and 1930s studied the effects of various workplace conditions on productivity. Led by Elton Mayo, the studies found that social and psychological factors strongly influenced worker behavior and output. Specifically, participation in decision-making, attention from managers, good social relationships among coworkers, and feeling valued on the job all increased productivity, regardless of physical working conditions. The experiments concluded that non-financial motivations are important for worker satisfaction and performance.

- Marxism is a political, economic, and sociological theory developed by Karl Marx and Friedrich Engels that focuses on class conflict between the bourgeoisie and proletariat as the driving force behind history. - Under capitalism, the bourgeoisie own the means of production and exploit the proletariat, who must sell their labor for wages. This causes the proletariat to experience alienation and false consciousness until they develop class consciousness and overthrow the bourgeoisie in a communist revolution. - After overthrowing the capitalists, the proletariat would establish a temporary dictatorship before creating a true communist society without social classes where people share according to their abilities and needs.

Henry Ford introduced a new economic model called Fordism in the early 1900s. Fordism involved using assembly line production methods to enable mass production and consumption. It organized workers into a highly efficient production process and also aimed to increase wages to boost mass consumption. Fordism spread widely after World War I as companies adopted assembly line production and countries sought to emulate the US economic model to recover from the Depression. It transformed industries and cities as large factories concentrated production and people migrated to urban areas for jobs.

This presentation shows in a crisp form the 3 Ps viz. Promise, Personality & Perseverance for successful branding of any product or service.

The document discusses the concepts of Fordism, McDonaldization, and Post-Fordism. Fordism refers to Henry Ford's system of mass production using assembly lines and the standardization of tasks. This lowered costs and increased wages, allowing for mass consumption. McDonaldization describes how society has taken on characteristics of fast food restaurants, such as efficiency, predictability, calculability, and control. While these seem beneficial, they can also result in irrational outcomes. Post-Fordism emerged in response to market saturation, focusing on specialized goods and diverse production rather than mass markets and standardization. However, some argue Post-Fordism coexists with an evolving Fordism, and flexible specialization is not widespread.

Marxism is an economic and sociopolitical worldview created by Karl Marx and Friedrich Engels in the 19th century. It views society and history through the lens of class struggle and proposes that capitalist societies will inevitably give way to socialist societies, and in turn to communist ones. Neo-Marxism extends Marxism by incorporating elements from other traditions like critical theory and psychoanalysis. While Marxism focuses on economic determinism, Neo-Marxism considers broader social and intellectual influences. Criticisms of Marxism argue that it takes too materialistic an approach, cannot be falsified, ignores gender roles, and overstates the importance of economics.