No, a bajillion is not a real number in the mathematical sense. It is often used informally to represent an extremely large, but unspecified quantity.
The term ‘bajillion’ belongs to a family of invented terms (like ‘zillion’ or ‘gazillion’) that people use when they want to express the idea of a large number without giving a precise value. These terms are often humorous or illustrative, helping to convey the idea of ‘a whole lot’ rather than a specific figure.
In mathematical terms, real numbers include all the rational numbers (like integers and fractions) and irrational numbers (like π or √2). Since ‘bajillion’ doesn’t correspond to any defined value within these categories, it does not qualify as a real number.