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. All the numbers inside the rational numbers circle are rational, and all the numbers inside the irrational numbers are irrational. 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). And we know that no such real numbers exist. Therefore, the diagram is incorrect.
So, what is the correct real number system diagram? Below is the correct one. The union of rational numbers and irrational numbers are the real numbers as shown.
To students: Do not believe everything you read on the internet. You must learn how to verify them using other resources such as books and other reputable websites such as Wolfram’s MathWorld.