EXTENDING THE LESSON If you add any two rational numbers, the sum is always a rational number. So, the set of rational numbers is closed under addition. Is the set of rational numbers closed under subtraction. multiplication,and division also? Explain.

Can you think of any two numbers that, when subtracted, multiplied, or divided, yield a number that is not rational? For example, take the rational numbers 2 and 3.

2 - 3 = -1
2 * 3 = 6
2 / 3 = 2/3

Are any of the results irrational? And are there any other rational numbers that yield different results?

(I assume a formal proof is not requested).

Can you think of any two numbers that, when subtracted, multiplied, or divided, yield a number that is not rational? For example, take the rational numbers 2 and 3.

2 - 3 = -1
2 * 3 = 6
2 / 3 = 2/3

Are any of the results irrational? And are there any other rational numbers that yield different results?

yes this one 2 * 3 = 6

The set of rational numbers is closed under addition, subtraction, multiplication, and division. Let's explore each operation individually:

1. Addition: When you add two rational numbers, the sum will always be a rational number. This is because for any rational numbers a/b and c/d, where a, b, c, d are integers and b, d are nonzero, the sum (a/b) + (c/d) can be written as (ad + bc) / (bd), where ad + bc and bd are both integers. Therefore, the sum is a fraction with integers in its numerator and denominator, satisfying the definition of a rational number.

2. Subtraction: When you subtract two rational numbers, the difference will always be a rational number. We can express the subtraction (a/b) - (c/d) as (ad - bc) / (bd), which again has integers in the numerator and denominator. As a result, the difference is a rational number.

3. Multiplication: When you multiply two rational numbers, the product will always be a rational number. For any rational numbers a/b and c/d, their product (a/b) * (c/d) can be simplified as (ac) / (bd). Both ac and bd are integers, satisfying the condition for a rational number.

4. Division: When you divide one rational number by another, the quotient will always be a rational number, except when dividing by zero. For any rational numbers a/b and c/d, where b and d are nonzero, the division (a/b) ÷ (c/d) can be rewritten as (a/b) * (d/c) = (ad) / (bc). Like previous operations, the numerator and denominator of the quotient are integers, making it a rational number.

In summary, the set of rational numbers is closed under addition, subtraction, multiplication, and division because each operation maintains the property of being a rational number.

To determine if the set of rational numbers is closed under subtraction, multiplication, and division, we need to understand what it means for a set to be closed under an operation.

A set is said to be closed under an operation if performing that operation on any two elements from the set results in another element that is also in the set.

1. Subtraction:
To check whether the set of rational numbers is closed under subtraction, we need to consider the following scenario. Let's take any two rational numbers, a/b and c/d, where a, b, c, and d are integers, and b and d are not equal to 0. Now, let's subtract these two rational numbers: (a/b) - (c/d).

The subtraction of two rational numbers can be calculated as follows:
(a/b) - (c/d) = (ad - bc) / (bd).

Since ad - bc and bd are integers, the result of the subtraction is also a rational number. Hence, we can conclude that the set of rational numbers is closed under subtraction.

2. Multiplication:
To determine if the set of rational numbers is closed under multiplication, we need to consider two rational numbers, a/b and c/d, and find their product: (a/b) * (c/d).

The multiplication of two rational numbers can be calculated as follows:
(a/b) * (c/d) = (ac) / (bd).

Since ac and bd are integers, the result of the multiplication is also a rational number. Therefore, we can conclude that the set of rational numbers is closed under multiplication.

3. Division:
To check if the set of rational numbers is closed under division, we consider two rational numbers, a/b and c/d, and calculate their division: (a/b) ÷ (c/d).

The division of two rational numbers can be calculated as follows:
(a/b) ÷ (c/d) = (a/b) * (d/c) = (ad)/(bc).

Since ad and bc are integers, the result of the division is also a rational number. Thus, we can conclude that the set of rational numbers is closed under division.

In summary, the set of rational numbers is closed under subtraction, multiplication, and division, along with addition. This means that performing any of these operations on two rational numbers will always result in another rational number.