Which operations are defined for any two real
numbers?
addition
subtraction
multiplication
division
The Correct Answer and Explanation is :
For any two real numbers, the following operations are defined:
- Addition: The sum of two real numbers is always a real number. For two real numbers (a) and (b), the operation (a + b) results in another real number. This is the most fundamental arithmetic operation and is always possible for any pair of real numbers.
- Subtraction: The difference between two real numbers is also always a real number. For real numbers (a) and (b), the operation (a – b) results in another real number. Subtraction is essentially the addition of a negative number, and it works similarly to addition in terms of real number results.
- Multiplication: The product of two real numbers is always a real number. Multiplication of two real numbers (a \times b) results in a real number. This operation is closed over real numbers, meaning any multiplication of two real numbers will not result in a value outside the set of real numbers. The multiplication of positive or negative real numbers follows specific rules regarding the signs of the results.
- Division: Division is defined for any two real numbers except when the divisor is zero. That is, for two real numbers (a) and (b), the operation (a \div b) is defined as long as (b \neq 0). If (b = 0), division by zero is undefined because it leads to indeterminate results. Division of real numbers is the inverse of multiplication, and it maintains the properties of real numbers as long as division by zero is avoided.
Explanation:
Real numbers encompass all rational and irrational numbers, including integers, fractions, and numbers like (\pi) and (\sqrt{2}). The operations of addition, subtraction, and multiplication are universally defined for all pairs of real numbers, which means that no matter the values of (a) and (b), the result of these operations will always be a real number.
Division, however, requires caution. While division is defined for any pair of real numbers, it is only valid if the divisor is nonzero. Dividing by zero does not yield a real number result because there is no real number that satisfies the equation (0 \times x = a) for any nonzero (a).
In summary, all four operations—addition, subtraction, multiplication, and division (except by zero)—are well-defined for any two real numbers. The rules of arithmetic ensure that these operations are consistent and closed within the set of real numbers, except in the case of division by zero.