Math Misconception: Incorrect Real Number System Diagram

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.

An Incorrect Real Number System Diagram

An Incorrect Real Number System Diagram

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.

correct real number system

Correct Real Number System Diagram

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.

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8 thoughts on “Math Misconception: Incorrect Real Number System Diagram

  1. You are absolute correct. BTW, I believe you used MS PowerPoint to make the correct diagram, or some other program. The diagrams are wonderful way to learn and teach maths,.

    • I haven’t really. Of course, we know that the set of real numbers is a subset of complex numbers. But I was only talking about the set of real numbers.

  2. that is absolutely correct. there are still pictures in the net of the wrong diagram of the set of real numbers.

  3. Hi Guillermo,
    Other important subsets are:
    – algebraic irrational numbers (not integer fractional powers)
    – transcendental irrational numbers (e, Pi, etc…)
    To indicate these subsets just put an horizontal line dividing the irrational numbers set on the right.

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