If you search images of the real number system on the internet, you will be surprised that there are a lot of incorrect real number system diagrams. One of the common incorrect diagrams is shown in the first figure below. If we interpret the diagram, all the numbers inside the the oval are real numbers. That is, our universal set is the set of real numbers.
Notice that there are numbers that are outside the rational number circle and also outside the irrational number circle that are real numbers (the solid bluish part). But we know that real number is either rational or irrational. Therefore, the numbers on the bluish part of the diagram outside the rational and irrational number circles do not exist. » Read more