# Square numbers (or squared numbers)

Many of us are familiar with the concept of the square of a number. Surely, you heard it once upon a time in math classes in high school. However, what if this concept is only a small part of a vast and truly interesting topic? Let's reveal some details.

## What is a full square

A perfect square is an integer that is the square of an integer. In other words, a perfect square is the product of two equal integers. Based on the definition, the square root of a full square is taken completely, so the geometric embodiment of a full square is the area of a square with a side expressed as an integer equal to the square root of the original full square.

For a more precise disclosure of the topic, let us recall the definition of integers. Integers are called all natural (used to count objects) and their opposite numbers and zero. Accordingly, the set of integers does not include finite or infinite fractions and complex numbers.

Examples of perfect squares are, for example, the following numbers: 9 (square of the number 3), 49 (square of the number 7), 676 (square of the number 26). But the number 15 cannot be represented as a product of two equal integers, so it is not a perfect square.

It is interesting that the concept of a perfect square can be extended to include, for example, rational numbers. In this case, a full square is a fraction, which is the ratio of two square integers.

## About curly numbers

A full square is the most common example of a classic figurative number, that is, a number that can be graphically expressed using geometric shapes. The concept of figurative numbers arose, according to researchers, as early as the 6th-4th centuries BC and is directly related to the Pythagoreans. Ancient Greek philosophers learned algebra, largely relying on geometric foundations, so natural numbers were associated with a set of points on the plane and in space. Actually, the very name "full square" owes its appearance to this particular approach to the study of mathematics.

Figured numbers are traditionally generalized to multidimensional spaces. For example, on a plane, curly numbers are associated with polygons according to certain rules, and in three-dimensional space they are associated with various polyhedra.

The Pythagoreans attached great importance and greatness to the concept of curly numbers, so such well-known ancient mathematicians as, for example, Diophantus of Alexandria, Hypsicles of Alexandria and Eratosthenes of Cyrene were engaged in their study. Entire scientific papers and studies were devoted to the comprehension and structuring of the theory of curly numbers. So, fragments of the book of Diophantus of Alexandria "On Polygonal Numbers", written, according to some estimates, in the 3rd century BC, have survived to our time.

By the way, curly numbers were of interest not only to ancient mathematicians. Many mathematicians of the Middle Ages were also engaged in them: Gerolamo Cardano, Fibonacci, and even the great scientists of modern times - Leonard Euler, Joseph Louis Lagrange, Pierre de Fermat, Carl Friedrich Gauss.

Thus, the topic of curly numbers, including their brightest representatives - full squares, has attracted the attention of mathematicians since ancient times.