The sum of a rational number a and an irrational number b is always an irrational number. This can be understood through the properties of rational and irrational numbers.
A rational number is defined as a number that can be expressed as the quotient of two integers, where the denominator is not zero. Examples include 1/2, 3, and -4. On the other hand, an irrational number cannot be expressed as a simple fraction. It has a decimal expansion that neither terminates nor repeats, such as √2, π, or e.
When you add a rational number to an irrational number, the result cannot be expressed as a fraction of two integers. This is because the addition of a rational number does not eliminate the non-repeating nature of the decimal expansion of the irrational number. Thus, the result remains irrational.
For example, if we take a = 2 (a rational number) and b = √3 (an irrational number), their sum is a + b = 2 + √3. This sum cannot be simplified to a fraction, confirming that it is indeed an irrational number.
Therefore, the final answer is that the sum of a rational number and an irrational number is always irrational.