10 Questions
What is the set of numbers that includes all rational and irrational numbers?
Real numbers
What is the characteristic of an irrational number when expressed as a decimal?
It is non-terminating and non-repeating
What is the purpose of Euclid's Division Lemma?
To find the HCF of two numbers
What does the Fundamental Theorem of Arithmetic state?
Every composite number can be expressed as a product of prime numbers in a unique way
Which property of rational numbers states that the order of numbers does not change the result of addition and multiplication?
Commutative Property
What is the result of the sum, difference, product, and quotient of two rational numbers?
Always a rational number
What is the result of the product of two irrational numbers?
Always a real number
What is the use of the Fundamental Theorem of Arithmetic?
To find the prime factorization of a number
What is the characteristic of a rational number when expressed as a decimal?
It terminates or repeats
Which property of rational numbers states that the order in which numbers are added or multiplied does not change the result?
Associative Property
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.
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.
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