# The most famous mathematical paradoxes

By definition, a paradox (in general) is an absurdity, a false conclusion of correct reasoning.

*Example: a law says: “It is forbidden to prohibit.” The content of the law contradicts the law itself. Therefore, it would be forbidden to say that it is forbidden to prohibit and therefore it would be forbidden to say that it is forbidden to say that it is forbidden to prohibit.*

Whether in trigonometry, arithmetic, or statistics, there are many paradoxes in mathematics that have become famous . Here are a few.

### The false paradox of Achilles and the tortoise: The most famous mathematical paradoxes

Without a doubt, one of the most famous mathematical paradoxes is that of Zeno of Elea (490-430 BC).

His statement: with the mathematical knowledge of the time, he said that if a tortoise was left 100 meters ahead, Achilles would never reach it because the tortoise would continue to advance as well.

Of course, it is an absurd claim today, but we have had to wait until modern mathematics (equations, graphs, etc.) to disprove Zeno’s mathematical theory!

### The lost square paradox

We continue with the absurd, but this time in geometry. The missing square is a rational mathematical hypothesis, but ultimately it is based only on a visual illusion that results in an obviously false conclusion.

The problem: based on the *tangram* model (Chinese puzzle), it is about reconstructing a triangle (already formed in advance) with other geometric shapes (squares, rectangles, triangles, etc.).

The solution has a small empty square in the center of the triangle once the reconstruction is done. Therefore, it would have, when rebuilding it, a square lost in the area. Obviously, it is impossible: the small empty space is actually the result of a slight deformation of the perfect triangle, with slightly rounded edges.

**Mathematics teachers like these paradoxes very much** and propose them to their students to open their eyes to some reflections in mathematics (complex theorems, equations, geometric figures, etc.).

Today, there is no need to have a higher education in mathematics or be an engineer to understand the absurdity of these paradoxes. But at that time, there were opinions of all kinds.

Years later, the study of these statements helped **formulate theorems, math exercises, equations, and mathematical concepts** to disprove these hypotheses.

In your next math class, feel free to ask your teacher if they know of other mathematical paradoxes related to other concepts discussed, such as:

- Literal equations,
- The management of a scientific calculator,
- Probability,
- Symmetry,
- Equations (subtraction, addition, division, multiplication)
- Polygons, polynomials,
- Relative numbers, etc. The most famous mathematical paradoxes