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5.4 Rational Numbers

Set of Rational Numbers Q={x/x =p/q, p, q¹0 are relatively prime integers}.

Properties of Rational Numbers in Addition

Closure Property: For any two rational numbers a/b, c/d their sum a/b+c/d is also a rational number.

Commutative property: For any two rational numbers a/b, c/d we have a/b+c/d = c/d+a/b where a, b, c, d are integers and b, d are not equal to zero.

Associative Property:For any three rational numbers a/b, c/d, e/f we have a/b+(c/d+e/f) = (a/b+c/d)+e/f, where a, b, c, d, e, f are integers and b, d, f are not equal to zero.

Identity: For any rational number a/b where b¹0, we have 0+a/b = a/b+0 = a/b. Here zero is the identity element for addition in rational numbers.

Inverse: For any non-zero rational number p/q there exists a unique rational number -p/q such that p/q+(-p/q) = 0.

p/q, -p/q are additive inverses. Each one is called the additive inverse of the other.

Properties of Rational Numbers in Multiplication

Closure Property:
For any two rational numbers a/b c/d their product a/b*c/d is also a rational number.

Commutative Property:
For any two rational numbers a/b*c/d = c/d*a/b.

Associative Property:
For any three rational numbers (a/b*c/d)*e/f = a/b*(c/d*e/f).

Identity:
If a/b is a rational number then 1*a/b = a/b*1 = a/b. Here 1 is called the multiplicative identity in the set of rational numbers.

Inverse:
If a/b is a rational number then a/b*b/a = b/a*a/b = 1.

If the product of two numbers is one, we say that each number is the multiplicative inverse of the other(Except for zero).

Directions: Answer the following questions. Also write five examples of each property.
Q 1: What is the multiplicative identity in the set of rational number?
1
-1
0

 Q 2: {x/x=p/q, p, q are not equal zero} are called the set of ________.
Rational Numbers
Whole Numbers
Natural Numbers
Integers

Q 3: For any three rational numbers a/b, c/d and e/f, (a/b+c/d)+e/f = a/b+(c/d+e/f), name the property.
Associative Property
Closure Property
Commutative Property
None of these

 Q 4: Set of rational numbers are denoted by ___.
Z
N
Q
W

Q 5: For any two rational numbers a/b, c/d, a/b+c/d = c/d+a/b, name the property.
Inverse
Associative Property
Commutative Property
Identity

 Q 6: What is the additive identity element in the set of rational numbers?
1
0
-1

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