How Do Mathematicians Name Big Numbers

The question of how mathematicians name big numbers is not as straightforward as one might think. While everyday language often relies on familiar terms like kilo, mega, and giga to quantify large quantities, the field of mathematics has its own methods and conventions for dealing with extremely large numbers.

Scientific Notation and Counting Systems

Mathematicians primarily use scientific notation and counting systems to handle large numbers efficiently. For small numbers, we have terms like micro, nanometer, and milli. As we move towards larger numbers, we use prefix multipliers such as kilo, Mega, and Giga. These prefixes are based on powers of ten.

For example, a kilometer is 1,000 meters, a Megabyte is 1,048,576 bytes, and a Gigabyte is 1,073,741,824 bytes. This system is widely used in various scientific and technical fields to simplify the representation of large numbers.

However, when it comes to extremely large numbers, mathematicians do not create new names for them. The need for such names is minimal, as these numbers are primarily used in theoretical mathematics or theoretical physics, and rarely in everyday applications.

Large Numbers in Nature

Nature itself uses incredibly large numbers. For instance, the frequency of the annihilation photon of an electron is about 1020 Hertz. This number is so large that it challenges our ability to comprehend it fully. The Einstein-Planck relation, E h v, suggests that at the instant of emission, all photons have the same spatial length, and the number of periods in an annihilation photon can be 1020.

Imagine how many periods this could represent! The length of the photon would be almost inconceivable. While we can grasp that this is a large number, the exact dimensions are beyond our current understanding.

Mathematical Notations for Extremely Large Numbers

Mathematicians sometimes do name extremely large numbers for the sake of clarity or historical interest. One such example is the terms googol and googolplex. These terms were suggested by a 9-year-old boy, Milton Sirotta, in 1920. Googol is defined as 10100, and googolplex is defined as 10googol. While these terms are fun and whimsical, they carry no significant mathematical meaning in academic or scientific contexts.

Another example is Skewes’ number and Graham’s number. These numbers are so large that they have their own names but are more of a playful curiosity than a practical necessity. These names come into existence through popular publications and academic discussions.

Constructions for Fast-Growing Functions

To handle extremely large numbers, mathematicians use various useful constructions of fast-growing functions. One such function is the Ackermann function, which grows at an incredibly rapid rate. Another example is the Friedman-Kruskal Tree function, denoted as TREE(n). Studying these functions and their growth rates is both a hobbyist pastime and a serious object of study in areas like proof theory and complexity theory.

The Ackermann function, for instance, is defined as follows:

code
A(m, n)  
  begin{cases} 
   n 1  text{if } m  0 
   A(m-1, 1)  text{if } m  0 text{ and } n  0 
   A(m-1, A(m, n-1))  text{if } m  0 text{ and } n  0 
  end{cases}
/code

Similarly, the TREE(n) function is defined in a similarly complex and rapidly growing manner. These functions serve to demonstrate the bounds of computable and non-computable numbers, pushing the boundaries of mathematical understanding.

In conclusion, while mathematicians have developed methods to handle extremely large numbers through scientific notation and fast-growing functions, the creation of new names for large numbers is not a common practice. These numbers are more often recognized through their properties rather than through common names. The exploration and study of such numbers are essential in understanding the limits of computational and theoretical mathematics.