Which of the following results in a sum or a product that is rational? Choose all that apply.
- A rational number added to an irrational number results in a sum that is irrational.
- An irrational number added to another irrational number can result in a sum that is rational or irrational, depending on the specific numbers involved.
- A rational number multiplied by any other number (rational or irrational) will result in a product that is rational.
To determine which operations result in a sum or product that is rational, we need to understand what a rational number is.
A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.
Now, let's consider the operations:
1. Addition: The sum of two rational numbers will always be rational. Therefore, addition results in a sum that is rational.
2. Subtraction: The difference between two rational numbers will always be rational. Therefore, subtraction results in a sum that is rational.
3. Multiplication: The product of two rational numbers will always be rational. Therefore, multiplication results in a product that is rational.
4. Division: The quotient of two rational numbers is not always rational. For example, dividing 2 by 3 would result in the irrational number 0.6666... However, if the divisor is not zero, then division can result in a rational number.
Therefore, the operations that result in a sum or product that is rational are addition, subtraction, and multiplication.