Real Numbers - Proofs based on irrational numbers for Class 10th maths.
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Números reales
This document discusses several mathematical concepts including sets, real numbers, inequalities, and absolute value. It provides examples and properties for each concept. Sets can be defined by listing elements or with a common characteristic. Real numbers include natural numbers, integers, rationals, and irrationals. Properties of real numbers include closure under addition and multiplication. Inequalities can be solved using the same methods as equations while maintaining the inequality sign. Absolute value gives the distance of a number from zero and has properties related to products and sums.
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This document provides examples of solving inequalities by multiplying or dividing each side by positive and negative numbers.
It begins with 6 examples of solving inequalities such as x > 18, g ≤ -23, and t ≤ 19. Then it discusses the properties of inequalities when multiplying or dividing each side by positive and negative numbers. Specifically, it states that the inequality sign remains the same when multiplying by a positive number but must be reversed when multiplying by a negative number.
Finally, it provides step-by-step examples of solving inequalities by multiplying and dividing each side by positive and negative numbers. This includes writing the inequality, simplifying by multiplying or dividing, and determining the solution set.
Chapter 2.6
This document discusses division of real numbers. It defines division as multiplying by the reciprocal. It provides examples of dividing numbers and simplifying expressions using division. It also discusses properties of division including that the quotient of numbers with the same sign is positive and opposite signs is negative. It cannot divide by zero since zero has no reciprocal. An example problem calculates velocity by dividing distance by time.
Lesson 1 1 properties of real numbers
1) The document provides an overview of properties and operations of real numbers including identifying different types of real numbers like integers, rational numbers, and irrational numbers.
2) It discusses ordering real numbers and using symbols like <, >, ≤, ≥ to compare them. Properties of addition, multiplication and other operations are also covered.
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Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed 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, but not division. They are commutative for addition and multiplication, but not for subtraction or division. Addition is associative for rational numbers, but subtraction is not.
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Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q ≠ 0. Rational numbers are closed under addition, subtraction, and multiplication but not division. Addition and multiplication of rational numbers are commutative, but subtraction and division are not. Addition is associative for rational numbers, but subtraction is not.
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This document provides information about solving and graphing inequalities using the properties of inequality. It defines inequalities and their solution sets. It introduces the addition, subtraction, multiplication, and division properties of inequality, explaining how these properties can be used to solve inequalities and determine if resulting inequalities are equivalent. It provides examples of solving and graphing various inequalities using these properties.
Aditya Class 8th
Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q ≠ 0.
Rational numbers are closed under addition, subtraction, and multiplication, but not division. Addition, subtraction and multiplication of rational numbers are commutative, but division is not. Addition of rational numbers is associative, but subtraction is not.
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Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q ≠ 0. Some key properties of rational numbers are:
1) Rational numbers are closed under addition, subtraction, and multiplication but not division. For any two rational numbers a and b, a + b, a - b, and a × b are rational, but a ÷ b may not be rational if b = 0.
2) Addition and multiplication of rational numbers are commutative but subtraction and division are not.
3) Addition and multiplication of rational numbers are associative but subtraction and division are not.
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Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q ≠ 0. Some key properties of rational numbers are:
1) Rational numbers are closed under addition, subtraction, and multiplication but not division. For any two rational numbers a and b, a + b, a - b, and a × b are rational, but a ÷ b may not be rational if b = 0.
2) Addition and multiplication of rational numbers are commutative but subtraction and division are not.
3) Addition and multiplication of rational numbers are associative but subtraction and division are not.
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The document provides examples of applying these different techniques to solve specific inequalities.
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This document provides learning objectives and content about rational and irrational numbers for a Class 9 mathematics lesson. It begins by defining different types of numbers - natural, whole, integers, rational, and irrational - and provides examples. It then explains rational numbers as those that can be written as fractions p/q, and irrational numbers as those that cannot be expressed as fractions. Various methods are provided for representing and finding rational numbers between two given rational numbers, as well as representing irrational numbers on the number line. Finally, the document discusses operations involving rational and irrational numbers.
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The Real 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).
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This document provides an overview of topics related to algebra, including quadratic equations, inequalities, absolute value, and modulus. It discusses how to form quadratic equations given different conditions on the roots. Methods for finding roots such as splitting the middle term and using the quadratic formula are presented. Properties and techniques for solving different types of inequalities and absolute value equations are outlined. Maximum and minimum values of functions involving modulus are also addressed.
College algebra -REAL Numbers
This document provides an overview of real numbers and their properties. It defines natural numbers, integers, rational numbers, and irrational numbers. It describes operations on real numbers like addition, subtraction, multiplication, and division. Key properties discussed include the commutative, associative, and distributive properties. It also covers fractions, negatives, and finding a common denominator to add fractions.
Numeros reales y planos numericos
Definición de Conjuntos.
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Real Numbers - Proofs based on irrational numbers for Class 10th maths.
- 1. Decimal Expansion Of Rational numbers Euclid’s Division Algorithm Fundamental Theorem Of Arithmetic
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- 3. Q) Prove that √7 is irrational. Let’s say that √7 is actually a rational number. Since √7 is a rational number we can express √7 as √7 = 𝐚 𝐛 where a & b belong to integers and have no common factor other than 1 and b ≠ 0. So , since √7 = 𝐚 𝐛 Squaring on both the sides we get, 7 = 𝐚 𝟐 𝐛 𝟐 Or, 7b2 = a2 ------------------------------ (1) Chapter : Real Numbers Website: www.letstute.com
- 4. ⇒ a2 is divisible by 7. (∵7b2 is divisible by 7) ⇒a is divisible by 7. (∵7 is prime & divides a2, ∴7 divides a) ⇒ Let, a = 7 c ( where c is some integer) Substituting a = 7c in (1), we get 7b2 = (7c)2 7b2 = 49 c2 b2 = 7 c2 ⇒ b2 is divisible by 7. (∵ 7 c2 is divisible by 7) ⇒ b is divisible by 7. (∵7 is prime & divides b2, 7 divides b) a and b are divisible by 7. Chapter : Real Numbers Website: www.letstute.com
- 5. ⇒ 7 is a common factor of a and b, but this contradicts the fact that a and b have no common factor other than 1. This contradiction has arisen because of our incorrect assumption that √7 is rational. Hence, √7 is an irrational number. Chapter : Real Numbers Website: www.letstute.com
- 6. Chapter : Real Numbers Website: www.letstute.com IF ‘p’ (PRIME NUMBER) DIVIDES ‘𝐚 𝟐 ’ THEN ‘p’ DIVIDES ‘a’ WHERE ‘a’ IS POSITIVE INTEGER. RATIONAL NUMBER + RATIONAL NUMBER = RATIONAL NUMBER. RATIONAL NUMBER − RATIONAL NUMBER = RATIONAL NUMBER. 𝐑𝐀𝐓𝐈𝐎𝐍𝐀𝐋 𝐍𝐔𝐌𝐁𝐄𝐑 𝐑𝐀𝐓𝐈𝐎𝐍𝐀𝐋 𝐍𝐔𝐌𝐁𝐄𝐑 = MAY OR MAY NOT BE A RATIONAL NUMBER.
- 7. ALWAYS TRY TO PROVE THESE TYPES OF PROBLEMS BY THE CONTRADICTION METHOD. APPLY THE DEFINATION OF RATIONAL NUMBERS.
- 8. Please visit www.letstute.com to take a test. Chapter : Real Numbers Website: www.letstute.com