Which operations are defined for any two real numbers?
A. addition
B. subtraction
C. multiplication
D. division
The Correct Answer and Explanation is:
Correct Answer:
A. Addition
B. Subtraction
C. Multiplication
D. Division (with the exception of division by zero)
Explanation:
Operations between real numbers are fundamental in mathematics. The real numbers (( \mathbb{R} )) include all rational numbers (e.g., fractions and integers) and irrational numbers (e.g., (\pi, \sqrt{2})). These operations are defined as follows:
1. Addition
Addition is always defined for any two real numbers. The sum of two real numbers is another real number. For example:
[
3.5 + (-2.1) = 1.4
]
This closure property under addition ensures that no matter which real numbers are added, the result remains within the set of real numbers.
2. Subtraction
Subtraction is also always defined for real numbers. The difference between two real numbers is another real number. For example:
[
5 – 8 = -3
]
Subtraction can also be viewed as the addition of the opposite ((a – b = a + (-b))).
3. Multiplication
Multiplication is universally defined for real numbers, and the product of any two real numbers is also a real number. For instance:
[
(-3.2) \times 4 = -12.8
]
This operation adheres to closure, associativity, and distributive properties.
4. Division
Division is defined for any two real numbers except when the divisor is zero. The result of dividing one real number by another (non-zero) is always a real number. For example:
[
\frac{9}{-3} = -3
]
However, division by zero is undefined because it leads to contradictions in mathematics (e.g., no number satisfies (a \div 0 = b) for any (b)).
Summary
The operations of addition, subtraction, and multiplication are always defined for real numbers. Division is defined only if the divisor is non-zero. Therefore, all four operations are defined in general, with the one exception of division by zero.