Questions and Answers
What is the set of rational numbers denoted by, and what is the condition for q?
The set of rational numbers is denoted by Q, and the condition for q is q ≠ 0.
What is the formula for subtracting two rational numbers, and what is the key step in this process?
The formula is (p/q) - (r/s) = (ps - qr) / qs, and the key step is to subtract the numerators and keep the common denominator.
What is the formula for multiplying two rational numbers, and what is the key step in this process?
The formula is (p/q) × (r/s) = (pr) / (qs), and the key step is to multiply the numerators and denominators separately.
What are the three properties of rational numbers, and what do they describe?
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What is the formula for dividing two rational numbers, and what is the key step in this process?
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How do you simplify the expression (2/3) - (1/4) using the formula for subtracting rational numbers?
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What is the result of multiplying (2/3) and (3/4) using the formula for multiplying rational numbers?
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Study Notes
Rational Numbers
Definition
- A rational number is a number that can be expressed as the quotient or fraction of two integers, i.e., p/q, where p and q are integers and q ≠ 0.
- The set of rational numbers is denoted by Q.
- Rational numbers can be represented in decimal form, where the decimal expansion either terminates or repeats.
Subtraction
- To subtract two rational numbers, subtract the numerators and keep the common denominator.
- Formula: (p/q) - (r/s) = (ps - qr) / qs
Multiplication
- To multiply two rational numbers, multiply the numerators and denominators separately.
- Formula: (p/q) × (r/s) = (pr) / (qs)
Properties
- Commutative Property: The order of the numbers does not change the result of addition and multiplication.
- Associative Property: The order in which we add or multiply numbers does not change the result.
- Distributive Property: a(b + c) = ab + ac, where a, b, and c are rational numbers.
Division
- To divide two rational numbers, invert the second number (i.e., flip the numerator and denominator) and then multiply.
- Formula: (p/q) ÷ (r/s) = (p/q) × (s/r) = (ps) / (qr)
Rational Numbers
Definition
- A rational number is a number that can be expressed as the quotient or fraction of two integers, p/q, where p and q are integers and q ≠ 0.
- The set of rational numbers is denoted by Q.
- Rational numbers can be represented in decimal form, where the decimal expansion either terminates or repeats.
Operations
Subtraction
- To subtract two rational numbers, subtract the numerators and keep the common denominator.
- Formula: (p/q) - (r/s) = (ps - qr) / qs
Multiplication
- To multiply two rational numbers, multiply the numerators and denominators separately.
- Formula: (p/q) × (r/s) = (pr) / (qs)
Properties
Commutative Property
- The order of the numbers does not change the result of addition and multiplication.
Associative Property
- The order in which we add or multiply numbers does not change the result.
Distributive Property
- a(b + c) = ab + ac, where a, b, and c are rational numbers.
Division
- To divide two rational numbers, invert the second number (i.e., flip the numerator and denominator) and then multiply.
- Formula: (p/q) ÷ (r/s) = (p/q) × (s/r) = (ps) / (qr)
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Description
Learn about the definition, representation, and subtraction of rational numbers, including the formula for subtracting two rational numbers.