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Questions and Answers
What is a necessary condition for a number to be considered rational?
What is a necessary condition for a number to be considered rational?
How does the closure property apply to rational numbers in division?
How does the closure property apply to rational numbers in division?
Which property ensures that the sum of $3/4$ and $5/8$ is the same as the sum of $5/8$ and $3/4$?
Which property ensures that the sum of $3/4$ and $5/8$ is the same as the sum of $5/8$ and $3/4$?
When converting rational numbers into decimal form, what is a possible outcome?
When converting rational numbers into decimal form, what is a possible outcome?
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In the addition operation of two rational numbers, what is the first step?
In the addition operation of two rational numbers, what is the first step?
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What distinguishes a terminating decimal from a repeating decimal?
What distinguishes a terminating decimal from a repeating decimal?
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Which statement is correct regarding the associative property of rational numbers?
Which statement is correct regarding the associative property of rational numbers?
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Which step is compulsory while dividing one rational number by another?
Which step is compulsory while dividing one rational number by another?
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How can rational numbers be used in finance?
How can rational numbers be used in finance?
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Which property is demonstrated when $a/b * (c/d + e/f) = a/b * c/d + a/b * e/f$?
Which property is demonstrated when $a/b * (c/d + e/f) = a/b * c/d + a/b * e/f$?
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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., a/b, where a and b are integers and b ≠ 0.
Properties
- Closure: The result of adding, subtracting, multiplying, or dividing 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: multiplication is distributive over addition.
Representation
- Fraction Form: a/b, where a and b are integers and b ≠ 0.
- Decimal Form: A rational number can be expressed as a terminating or recurring decimal.
Operations
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Addition: To add two rational numbers, follow these steps:
- Find a common denominator.
- Convert both numbers to have the common denominator.
- Add the numerators and keep the denominator.
- Subtraction: Similar to addition, but subtract the numerators.
- Multiplication: Multiply the numerators and multiply the denominators.
- Division: Invert the second rational number and multiply.
Comparing Rational Numbers
- Equality: Two rational numbers are equal if and only if they have the same numerator and denominator.
- Inequality: Compare the numerators and denominators separately to determine the greater or lesser rational number.
Real-World Applications
- Measurement: Rational numbers are used to measure quantities like length, area, and volume.
- Finance: Rational numbers are used to calculate interest rates, investments, and currency exchange rates.
Rational Numbers
Definition
- A rational number is a number that can be expressed as the quotient or fraction of two integers (a/b, where a and b are integers and b ≠ 0).
Properties
- Rational numbers are closed under addition, subtraction, multiplication, and division.
- The commutative property holds for addition and multiplication of rational numbers.
- The associative property holds for addition and multiplication of rational numbers.
- Multiplication is distributive over addition for rational numbers.
Representation
Fraction Form
- Rational numbers can be expressed in fraction form (a/b, where a and b are integers and b ≠ 0).
Decimal Form
- Rational numbers can be expressed as terminating or recurring decimals.
Operations
Addition
- To add two rational numbers, find a common denominator, convert both numbers, add the numerators, and keep the denominator.
Subtraction
- To subtract two rational numbers, follow the same steps as addition, but subtract the numerators.
Multiplication
- To multiply two rational numbers, multiply the numerators and multiply the denominators.
Division
- To divide two rational numbers, invert the second rational number and multiply.
Comparing Rational Numbers
Equality
- Two rational numbers are equal if and only if they have the same numerator and denominator.
Inequality
- Compare the numerators and denominators separately to determine the greater or lesser rational number.
Real-World Applications
Measurement
- Rational numbers are used to measure quantities like length, area, and volume.
Finance
- Rational numbers are used to calculate interest rates, investments, and currency exchange rates.
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Description
Learn about the definition and properties of rational numbers, including closure, commutative, and associative properties.
