Applications of Number Patterns in Mathematics
Series, sequences, and sequential functions primarily encapsulate concepts from the realm of mathematical analysis, which is a branch of mathematics concerned with the study of limits, continuity, differentiation, integration, and infinite series.
This field of mathematics overlaps primarily with Number Theory among other fields such as topology, algebra, probability, analysis and applied mathematics (which we will be discussing about). When taking series/sequences and graphing them, drawing them, adding structure or form to it, or any other mathematical relation we can connect to these other fields in mathematics.
Before discussing the applications of patterns in mathematics, we need first to understand what sequences and series actually mean and how they are different from each other. Although this basic concept is taught in school it is necessary to understand what they are and what they mean as a concept in mathematics.
Sequences: are ordered lists of numbers indexed by natural numbers. This order usually has a value which helps it continue for however long it is supposed to. This number has different names for different types of sequences. The most common sequences are the:
- Harmonic sequence
- Geometric Sequence
- Arithmetic Sequence
- Quadratic sequence
- Fibonacci sequence
These sequences seem to be the most occurring and common ones throughout our real-life applications and learning and thus, we have defined them this way. The possibilities for a combination of sequences with various ratios and differences have endless possibilities.
The other part of sequences, or as I’d like to call it the sequel to sequences is the series.
A series is essentially the sum of all objects or terms in a sequence. The emphasis on individual terms is much lesser and the order isn’t as important since we are looking at the sum and not just how the order progresses further as we continue to build the sequence.
For example a sequence would look like this: 2, 4, 6, 8,… and series of the same sequence would look somehting like this: 2 + 4 + 6 + 8… = __
Both the sequence and series can go up to infinity. For series, we can usually only obtain a finite value from the above sequences given, if it is a converging geometric sequence.
Although it may not seem as if sequences and series are applicable in the real world, both concepts have various applications in the real world. The fields in which these concepts are used range from engineering and physics to even machine learning and data mining.
In engineering and physics, series and sequences are used extensively to model and analyze various physical phenomena. For example, Fourier series are used to represent periodic functions in signal processing and electrical engineering. Sequences are used in discrete-time modelling of systems, such as in control theory.
Whereas in the field of finance and economics, sequences and series are used in the analysis of financial markets, time series analysis, and modelling economic behaviours. For instance, geometric sequences are used to model compound interest and exponential growth, while arithmetic sequences can model linear trends in financial data. In this example, we can see that we are using topology along with sequences to model linear trends and exponential growth.
Further, sequences and series play a crucial role in computer science and information theory. They are used in the analysis of algorithms, particularly in the study of time and space complexity. Additionally, sequences are fundamental in coding theory, where error-correcting codes are designed to transmit data reliably over noisy channels. There are even multiple instances where we use sequences to analyse patterns and even come to solutions for mathematical-based problems using computer science.
In probability and statistics, sequences and series are used to model random processes and analyze data. For instance, in probability theory, sequences of random variables are studied to understand phenomena like random walks and stochastic processes. In statistics, time series analysis relies heavily on understanding and modelling sequences of data points over time.
As mentioned before, number theory plays a huge role in the application of sequences and series. This concept is essential in number theory and cryptography. Number sequences are studied to understand the properties of integers, prime numbers, and other number sets. In cryptography, sequences and series are used in the design and analysis of cryptographic algorithms, such as RSA encryption, which relies on the properties of sequences of prime numbers.
Another major mathematical concept in which sequences and series are used is calculus. In calculus and mathematical analysis, sequences and series are used to study the convergence and divergence of functions and to approximate functions. For example, Taylor series expansions are used to approximate functions with polynomials, providing a powerful tool for solving differential equations and understanding functions locally.
Lastly, in machine learning and data mining, sequences and series are used in time series analysis, sequential pattern mining, and sequence prediction tasks. Recurrent neural networks (RNNs) and Long Short-Term Memory (LSTM) networks are architectures designed specifically to model sequences of data, making them valuable tools for tasks like natural language processing and speech recognition.
This effectively shows that sequences and series can be used in various applications and the increasing use of AI and ML will continue to make use of basic concepts of mathematics in the real world.
