Table of Contents
What are Rational Numbers?
Rational numbers are numbers that can be expressed as a fraction, where the numerator and denominator are both integers. Rational numbers include every whole number, as well as every fraction.
Closure Property
Closure Property is a mathematical concept that states that a function defined within a certain set of boundaries (domain) will always produce the same result for the same set of inputs.
Addition of Rational Numbers Under the Closure Property
The closure property states that the sum of two rational numbers is another rational number. This is because the sum of two rational numbers is defined as the result of adding the two numbers together and then reducing the result to its lowest terms.
Subtraction of Rational Numbers Under the Closure Property
The closure property states that the sum or difference of two rational numbers is also a rational number.
For example, the sum of 2/3 and 1/4 is 5/12.
Multiplication of Rational Numbers Under the Closure Property
The closure property states that the product of two rational numbers is also a rational number. To illustrate this, consider the following examples:
1/2 × 1/4 = 1/8
3/4 × 2/5 = 3/10
In each case, the product of the two rational numbers is a rational number.
Why is the Mathematical Operation of Division not Under the Closure Property?
The mathematical operation of division is not under the closure property because it is not an associative operation.
Commutative Law
For every number a and number b, a + b = b + a.
This law states that the order of addition does not affect the sum of two numbers.
Commutative Law of Addition
For all real numbers a and b,
a + b = b + a.
Commutative Law of Multiplication
For all real numbers a and b, a × b = b × a.
Associative Law
The associative law states that the product of two or more associative operations is still associative.
Importance of Rational Numbers and Their Properties
Rational numbers are important because they are the foundation of mathematics. Every other type of number can be reduced to a rational number. This is why rational numbers are so important in mathematics – they are the most basic type of number.
Some of their properties include that they are always rational numbers, they can be written as a decimal, and they can be represented by a point on a number line. Rational numbers are also always between two whole numbers.
