Introduction
Models refer to three-dimensions objects that are used to represent natural phenomena or solve existing problems in mathematics, art, science, and geography, among other areas. In mathematics, different types of models are used to solve mathematical situations. They enable learners to simplify and analyze data to arrive at the desired answer. Among the common models used in representing mathematical situations are the analytical and numerical models. Both are applied by learners at different levels to represent mathematical situations. As such, this paper is aimed at explaining how numerical and analytical models can be used to represent mathematical situations. Also, the paper will analyze how commutative, associative, and distribution laws are applied to promote the efficiency and effectiveness of mathematical situations.
Models
Numerical Models
Numerical models refer to mathematical models presented in numerical-time stepping procedure. They appear in the form of tables containing numbers in the form of rows and columns. Most of the numerical models are used to represent mathematical situations that circulate in human lives. For example, they may be used to represent the economic developments of an individual or an organization. Numerical representation offigures in a tabular enables an organization to determine its progress in terms of losses and profits (De Bock, Veracx, & Van Dooren, 2017). These models are usually constructed using computational methods since manual construction can be challenging or can subject the numerals to errors. More so, numerical models are applied in oceans to determine the likelihood of flood occurrences. In this case, a tool is used to measure the extent of sea-level rise such that it records the numerical values representing the extent of the rise in the sea level. Another application is weather forecasting, in which the numerical model is used to determine weather changes by numerical changes in temperature and humidity, among other aspects. Lastly, the model is applied in the measurement of electric energy being consumed by power appliances. A metered device that is usually availed to display the numerals representing the amount of electricity remaining to be used.
Analytical Models
Analytical models refer to mathematical models that have closed-form solutions. It mostly applied in solving mathematical equations that show specific changes in a system, such as reliability and performance (Yao, Huang, Jiang, &Krisp, 2019). The models are divided into simple and complex, depending on how they apply in mathematical situations. First of all, simple analytical models are used to calculate the flow of voltage in fuel cells. They are represented in the form of simple and short equations representing the flow of reversible voltage in cells. On the contrary, complex analytical models are used to solve mathematical equations that determine how different chemical reactions occur in the fuel cell. Also, analytical models are used by business organizations or individuals to track their progress in the saving process. The model is appropriate for the task because it involves equations that lead to closed-form solutions. In short, analytical models solve complex mathematical situations compared to numerical models.
Commutative, Associative, and Distribution Laws
Commutative Laws
Commutative law states that swapping numbers in mathematical situations does not affect the final answer i.e. a + b = b + a, and 3 + 1 = 1 + 3. The law has been can be used to explain the effectiveness of mathematical concepts in situations encountered by individuals at the personal or professional level. For example, in personal situations, putting on the left and right socks is commutative, as any of them can be worn first. Also, the act of purchasing commodities and handing over the money id commutative as any of them can come first as long as the exchange happens at the end (Atangana, & Gómez-Aguilar, 2018). Thus, commutative law can be applied to enhance efficiency and effectiveness in people’s lives.
Associative Laws
In associative law, the laws of numbers in addition and multiplication apply similarly even when the numbers are interchanged i.e. (2 + 3) + 4 = 2 + (3 + 4) = 9. The same law can apply while multiplying numbers i.e. 2 x (3 x 4) = (2 x 3) x 4 = 24. Associative property can as well be applied to enhance effectiveness and efficiency in mathematical situations. For example, it can be used by business individuals to group similar products before adding or multiplying the whole stock (Atangana& Gómez-Aguilar, 2018).
Distributive Laws
According to distributive law, multiplying a certain number by a group of numbers that have been added together is similar to performing each multiplication separately i.e., 2 x (3 + 4) = 2 x 3 + 2 x 4 = 14. The law can also apply in normal mathematic situations. For example, it can be applied in the calculation of bills while buying items whereby similar items are added together before calculating their total cost (Zhang, Guo,&Keigher, 2018). Doing it makes purchasing easier and faster.
Conclusion
In brief, numerical and analytical models can be applied in solving mathematical situations encountered in daily lives. They are as ell applied professionally, such as weather forecasting stations, to calculate daily readings in different aspects of weather. The paper also showed that commutative, associative, and distributive laws enhance effectiveness and efficiency in daily mathematical situations.
References
Atangana, A., & Gómez-Aguilar, J. F. (2018). Decolonization of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. The European Physical Journal Plus, 133(4), 166.
De Bock, D., Veracx, N., & Van Dooren, W. (2017). How students connect mathematical models to descriptions of real-world situations. In Mathematical Modelling and Applications(pp. 233-242). Springer, Cham.
Yao, X. A., Huang, H., Jiang, B., &Krisp, J. M. (2019). Representation and analytical models for location-based big data.
Zhang, S., Guo, L., &Keigher, W. (2018). Extensions of Operators, Liftings of Monads, and Distributive Laws. Applied Categorical Structures, 26(4), 747-765.
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