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Rational numbers are introduced as numbers that can be expressed in the form of a fraction. The section covers various properties such as closure, commutativity, associativity, and the role of zero and one in operations involving rational numbers.
In this section, we define rational numbers as numbers expressible in the form of a fraction where both the numerator and denominator are integers, and the denominator is non-zero. The chapter begins by emphasizing the need for rational numbers in solving various equations that can't be addressed solely using whole numbers or integers.
Understanding these properties is crucial for manipulating rational numbers effectively in various mathematical contexts.
Rational Numbers: Numbers that can be expressed as a fraction of integers.
Closure Property: Rational numbers are closed under addition, subtraction and multiplication.
Commutativity: Addition and multiplication of rational numbers are commutative.
Associativity: Addition and multiplication of rational numbers are associative.
Identity Elements: Zero as the additive identity, and one as the multiplicative identity.
Rational ratio, it's a fraction's friend; with p and q, the rules extend.
Once upon a time, in a land of numbers, there lived fractions p/q making rules about addition and multiplication—commutative and associative; they spread joy and closure!
C.A.S. - Closure, Associativity, and Subtraction (not commutative) are the main traits of rational number operations.
Example 1: The number 3 is a rational number because it can be written as 3/1.
Example 2: The fraction 3/4, -5/2, and 0 are all rational numbers.
Term: Rational Numbers
Definition: Numbers that can be expressed in the form p/q, where p and q are integers and q is not zero.
Numbers that can be expressed in the form p/q, where p and q are integers and q is not zero.
Term: Closure Property
Definition: A property indicating that performing an operation on two numbers of a set will result in a number that is also in the set.
A property indicating that performing an operation on two numbers of a set will result in a number that is also in the set.
Term: Commutative Property
Definition: A property of certain operations where a + b = b + a or a × b = b × a.
A property of certain operations where a + b = b + a or a × b = b × a.
Term: Associative Property
Definition: A property that states that for a group of numbers, the way in which they are grouped does not change the result of the operation.
A property that states that for a group of numbers, the way in which they are grouped does not change the result of the operation.
Term: Identity Element
Definition: A special number that, when used in an operation, does not change the value of the other number.
A special number that, when used in an operation, does not change the value of the other number.
Term: Distributive Property
Definition: A property that allows the multiplication of a number by a sum or difference, e.g., a(b + c) = ab + ac.
A property that allows the multiplication of a number by a sum or difference, e.g., a(b + c) = ab + ac.