8 Questions
What is a rational number?
A real number that can be expressed as the ratio of two integers.
Which of the following is an example of a rational number?
22/7
What is the result of subtracting 2/3 from 3/4?
1/12
What is an irrational number?
A real number that cannot be expressed as the ratio of two integers.
Which of the following is an example of an irrational number?
π
What is a characteristic of irrational numbers?
They have infinite non-repeating decimals.
What is the main difference between rational and irrational numbers?
Rational numbers can be expressed as a ratio of integers, while irrational numbers cannot.
What is true about the decimal expansion of an irrational number?
It is infinite and non-repeating.
Study Notes
Rational Numbers
- A rational number is a real number that can be expressed as the ratio of two integers, i.e., p/q, where p and q are integers and q ≠ 0.
- Examples: 3/4, 22/7, 1/2, etc.
- Rational numbers can be expressed as finite decimals or recurring decimals.
- Operations on rational numbers:
- Addition: a/b + c/d = (ad + bc)/bd
- Subtraction: a/b - c/d = (ad - bc)/bd
- Multiplication: a/b × c/d = ac/bd
- Division: a/b ÷ c/d = ad/bc
Irrational Numbers
- An irrational number is a real number that cannot be expressed as the ratio of two integers, i.e., it cannot be written in the form p/q, where p and q are integers and q ≠ 0.
- Examples: π, e, √2, etc.
- Irrational numbers have infinite non-repeating decimals.
- Properties of irrational numbers:
- Irrational numbers are non-terminating and non-repeating.
- The decimal expansion of an irrational number is infinite and never repeats.
- Irrational numbers cannot be expressed as a finite decimal or a ratio of integers.
Difference between Rational and Irrational Numbers
- Rational numbers can be expressed as finite or recurring decimals, while irrational numbers have infinite non-repeating decimals.
- Rational numbers can be expressed as a ratio of integers, while irrational numbers cannot be expressed as a ratio of integers.
- Rational numbers are countable, while irrational numbers are uncountable.
Rational Numbers
- A rational number is a real number that can be expressed as a ratio of two integers (p/q), where p and q are integers and q ≠ 0.
- Examples of rational numbers include 3/4, 22/7, and 1/2.
- Rational numbers can be expressed as finite decimals or recurring decimals.
- Operations on rational numbers include:
- Addition: combine numerators (ad + bc) and denominators (bd) to get the result.
- Subtraction: combine numerators (ad - bc) and denominators (bd) to get the result.
- Multiplication: multiply numerators (ac) and denominators (bd) to get the result.
- Division: multiply numerator (ad) by denominator (bc) to get the result.
Irrational Numbers
- An irrational number is a real number that cannot be expressed as a ratio of two integers (p/q).
- Examples of irrational numbers include π, e, and √2.
- Irrational numbers have infinite non-repeating decimals.
- Properties of irrational numbers include:
- Non-terminating and non-repeating decimals.
- Infinite decimal expansion that never repeats.
- Cannot be expressed as a finite decimal or a ratio of integers.
Key Differences
- Rational numbers have finite or recurring decimals, while irrational numbers have infinite non-repeating decimals.
- Rational numbers can be expressed as a ratio of integers, while irrational numbers cannot.
- Rational numbers are countable, while irrational numbers are uncountable.
Test your understanding of rational and irrational numbers, their definitions, examples, and operations such as addition, subtraction, multiplication, and division.
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