Question

Yes or No? If No, give a reason. Assume that$a$ s and$b$are non-zero real numbers.

(a) Is the sum of two rational numbers always a rational number?

(b) Is the sum of two irrational numbers always an irrational number?

Short answer

(a) Yes, the sum of two rational numbers is always a rational number.

(b) Yes, the sum of two irrational numbers is always an irrational number.

Step-by-step solution

Part a. Step 1. State the concept.

Any rational number can be expressed as a fraction where the numerator and denominator are integers.

So, the sum of two rational numbers is the sum of two such fractions which gives another such fraction where the numerator and denominator are integers and thus, another rational number.

Part a. Step 2. List the objective.

The objective is to determine if the sum of two rational numbers is always a rational number.

Part a. Step 3. Determine the given objective.

According to the given concept, the sum of two rational numbers is always a rational number.

Part b. Step 1. State the concept.

Any irrational number cannot be expressed as a fraction where the numerator and denominator are integers.

So, the sum of two irrational numbers also cannot be expressed as a fraction where the numerator and denominator are integers and thus is another irrational number.

Part b. Step 2. List the objective.

The objective is to determine if the sum of two irrational numbers is always an irrational number.

Part b. Step 3. Determine the given objective.

According to the given concept, the sum of two irrational numbers is always an irrational number.