You can often hear that mathematics is the language of nature. This is true. Any natural system (physical, chemical, biological, social, political, etc.) can be described with mathematical symbols. In other words, we describe nature through some mathematical sentences. With such a description, we can notice that some mathematical sentences logically imply other mathematical sentences, interpreted as new information about our system. In other words, we manage to predict the behavior of the system. This process is called mathematical modeling.
For example, look at organic chemistry (it's okay if you haven't studied it yet). As you know, chemistry deals with molecules consisting of atoms, while in organic chemistry, the main atoms are carbon (C), hydrogen (H), oxygen (O), and nitrogen (N). Living organisms are made up of these atoms, which prompted the use of the word "organic.”
We can describe an organic molecule by saying how many C, H, O, and N atoms are in it. Thus, the notation CH4 means that our molecule has one C atom and four H atoms. Similarly, the formula C2H6 indicates a molecule with two C atoms and six H atoms. The CH4 and C2H6 molecules are called methane and ethane, respectively. These gaseous compounds burn well in air (that is why they are used as fuel).
The formulas we have used are called molecular formulas. Molecular formulas are a mathematical way of storing information about organic compounds. These formulas do not describe organic compounds best since one molecular formula can correspond to several molecules that differ in how their atoms are connected. Here, we have to say that the C atom always combines with four other atoms: the N atom with three, the O atom with two, and the H atom with one. We say that C, N, O, and H have a valence of 4, 3, 2, and 1, respectively. If we are talking about CH4 or C2H6 molecules, then there are no problems. There is precisely one way to represent them, in which the atoms have their prescribed valences:
However, for example, there are two molecules with the molecular formula C2H6O:
The images above are called structural formulas. Structural formulas provide an accurate description of organic molecules. However, it is worth noting that from a mathematical point of view, a structural formula is nothing more than a graph. Graph theory is a part of discrete math (like combinatorics, mathematical logic, etc.) We have encountered it before (for example, in the article What is symmetry).
As often happens in mathematical modeling, purely mathematical questions are highly relevant in a practical sense. In the example above, such an open mathematical question counts the number of structural formulas corresponding to a given molecular formula. For example, we already know that two structural formulas correspond to C2H6O. Try to determine the number of structural formulas corresponding to C6H14.
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