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If a is a rational number and b is an irrational number, then the sum a + b is
A) rational.
B) imaginary.
C) irrational.
D) an integer.

Solution:

A rational number is a type of real number, which is in the form of p/q where q is not equal to zero. Any fraction with non-zero denominators is a rational number.

Irrational numbers are real numbers that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio, such as p/q, where p and q are integers, q≠0.

Let us assume that a = p/q and b = m/n are rational numbers and x an irrational number.

To prove: rational + irrational = rational

p/q + x = m/n

x = m/n - p/q

x= (mq-np)/nq

Clearly, the numerator is an integer and the denominator is an integer and the whole x is rational.

However, this contradicts our assumption that x is irrational.

For example, consider two rational numbers

a = 2/3 and b = √5

The sum of two numbers a + b = 2/3 + √5

This cannot be simplified further and thus the sum is an irrational number.

Therefore, the sum a + b is an irrational number.