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Questions and Answers
What is the set of numbers that includes all rational and irrational numbers?
What is the set of numbers that includes all rational and irrational numbers?
What is the characteristic of an irrational number when expressed as a decimal?
What is the characteristic of an irrational number when expressed as a decimal?
What is the purpose of Euclid's Division Lemma?
What is the purpose of Euclid's Division Lemma?
What does the Fundamental Theorem of Arithmetic state?
What does the Fundamental Theorem of Arithmetic state?
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Which property of rational numbers states that the order of numbers does not change the result of addition and multiplication?
Which property of rational numbers states that the order of numbers does not change the result of addition and multiplication?
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What is the result of the sum, difference, product, and quotient of two rational numbers?
What is the result of the sum, difference, product, and quotient of two rational numbers?
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What is the result of the product of two irrational numbers?
What is the result of the product of two irrational numbers?
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What is the use of the Fundamental Theorem of Arithmetic?
What is the use of the Fundamental Theorem of Arithmetic?
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What is the characteristic of a rational number when expressed as a decimal?
What is the characteristic of a rational number when expressed as a decimal?
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Which property of rational numbers states that the order in which numbers are added or multiplied does not change the result?
Which property of rational numbers states that the order in which numbers are added or multiplied does not change the result?
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Study Notes
Real Numbers
- Real numbers are a combination of rational and irrational numbers.
- They can be represented on the number line.
- Real numbers include all rational and irrational numbers.
- Examples: 0, 1, 2/3, π, √2, etc.
Irrational Numbers
- Irrational numbers are non-terminating, non-repeating decimals.
- They cannot be expressed as a finite decimal or fraction.
- Examples: π, e, √2, etc.
- Irrational numbers are not rational, but they are real.
Euclid's Division Lemma
- Euclid's Division Lemma states that for any two positive integers a and b, there exist unique integers q and r such that: a = bq + r, where 0 ≤ r < b
- This lemma helps in finding the HCF (Highest Common Factor) of two numbers.
- It is used in the proof of the Fundamental Theorem of Arithmetic.
Fundamental Theorem Of Arithmetic
- The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers in a unique way.
- This theorem helps in finding the prime factorization of a number.
- Example: 12 = 2 × 2 × 3 (unique prime factorization)
Properties Of Rational Numbers
- Closure Property: The sum, difference, product, and quotient of two rational numbers is always a rational number.
- Commutative Property: The order of rational numbers does not change the result of addition and multiplication.
- Associative Property: The order in which rational numbers are added or multiplied does not change the result.
- Distributive Property: The multiplication of rational numbers can be distributed over addition.
- Existence of Additive and Multiplicative Identities: 0 and 1 are the additive and multiplicative identities for rational numbers, respectively.
- Existence of Additive and Multiplicative Inverses: Every rational number has an additive and multiplicative inverse.
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Description
Test your understanding of real numbers, including rational and irrational numbers, Euclid's Division Lemma, the Fundamental Theorem of Arithmetic, and properties of rational numbers. Learn about number systems, algebra, and mathematical theorems.
