Introducing Omnifinites and the Arithmetic Errorless Infinity Calculator

Hunter Christopher Fred; Ryan Bennett Greenwood; Shane Mathew Palmquist

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Conference: 2023 ASEE Annual Conference & Exposition
Location: Baltimore , Maryland
Publication Date: June 25, 2023
Start Date: June 25, 2023
End Date: June 28, 2023
Conference Session: Mathematics Division (MATH) Technical Session 2
Tagged Division: Mathematics Division (MATH)
Page Count: 18
DOI: 10.18260/1-2--43849
Permanent URL: https://peer.asee.org/43849
Download Count: 296

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Paper Authors

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Hunter Christopher Fred Western Kentucky University

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HUNTER C. FRED is a senior completing his Bachelor of Science degree in civil engineering from the School of Engineering and Applied Sciences at Western Kentucky University. Mr. Fred may be reached at hunter.fred689@topper.wku.edu.

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Ryan Bennett Greenwood Western Kentucky University

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RYAN B. GREENWOOD is a senior completing his Bachelor of Science degree in civil engineering from the School of Engineering and Applied Sciences at Western Kentucky University. Mr. Greenwood may be reached at ryan.greenwood349@topper.wku.edu.

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Shane Mathew Palmquist Western Kentucky University

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SHANE MATHEW PALMQUIST is a full professor of structural engineering in the School of Engineering and Applied Sciences at Western Kentucky University (WKU) and is the Boyce D. Tate Professor in civil engineering. Dr. Palmquist is a licensed professional engineer in Kentucky and Florida. Prior to becoming a faculty member at WKU, Dr. Palmquist was a bridge engineer for Lichtenstein Consulting Engineers in Natick, Massachusetts. He earned his B.S. degree in civil engineering from the University of New Hampshire, 1995; an M.S. in structural engineering from the University of Rhode Island, 1996; and his Ph.D. in structures/materials engineering from Tufts University, 2003. His research interests include applied mathematics, cable supported structures, and high-performance materials used in construction. Dr. Palmquist may be reached at shane.palmquist@wku.edu.

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Abstract

Typically, in applied mathematics, only real and complex numbers are considered and utilized in mathematical computations and analyzes. Nonfinite numbers, such as infinitesimals and infinites, are most often omitted from these studies. The purpose of this paper is to present a new number system called omnifinites that includes all finite and nonfinite numbers, which will help make these types of studies possible. All arithmetic for omnifinite numbers is fully defined, which overcomes the arithmetic challenges of less robust number systems such as the reals or hyperreals, which are plagued with math related errors such as division by zero and indeterminant forms. In addition, unlike the reals and hyperreals which are open systems where there is no largest number, the omnifinite number system is closed. Two (2) new numbers are introduced. These new numbers are absolute infinity and negative absolute infinity and they are the largest positive number and largest negative number, respectively. The new number system also requires changes in definitions of fundamental mathematical objects, specifically, the definition of a point. In classical Euclidian geometry, a point is defined as a sizeless quantity that has a location. In omnifinite geometry, a point is defined as a nonzero dimensional object, meaning in addition to having a location, it also has size.

As an application of this new number system, an omnifinite infinity calculator has been developed and created as a computer-based software tool. To the authors’ knowledge, this is first of its kind arithmetic errorless computational program. The arithmetic errorless calculator was presented for review to a large group of engineering seniors majoring in civil, mechanical, and electrical engineering, who used the calculator hands-on and completed a survey. Results show that a new number system called omnifinites, which is a modification of the hyperreals, may be created and developed that is reasonable and logical based on the perspective of senior engineering students surveyed. The resulting arithmetic errorless calculator allows for inputs and outputs that may be finite, nonfinite, or a combination. This calculator software tool was found by the engineering students surveyed to be intuitive to learn, easy to use, and fun to operate. This software tool will be shown at the presentation and attendants may actually use this calculator in real time.

Citation
Format

Fred, H. C., & Greenwood, R. B., & Palmquist, S. M. (2023, June), Introducing Omnifinites and the Arithmetic Errorless Infinity Calculator Paper presented at 2023 ASEE Annual Conference & Exposition, Baltimore , Maryland. 10.18260/1-2--43849

TY - CPAPER
AB - Typically, in applied mathematics, only real and complex numbers are considered and utilized in mathematical computations and analyzes. Nonfinite numbers, such as infinitesimals and infinites, are most often omitted from these studies. The purpose of this paper is to present a new number system called omnifinites that includes all finite and nonfinite numbers, which will help make these types of studies possible. All arithmetic for omnifinite numbers is fully defined, which overcomes the arithmetic challenges of less robust number systems such as the reals or hyperreals, which are plagued with math related errors such as division by zero and indeterminant forms. In addition, unlike the reals and hyperreals which are open systems where there is no largest number, the omnifinite number system is closed. Two (2) new numbers are introduced. These new numbers are absolute infinity and negative absolute infinity and they are the largest positive number and largest negative number, respectively. The new number system also requires changes in definitions of fundamental mathematical objects, specifically, the definition of a point. In classical Euclidian geometry, a point is defined as a sizeless quantity that has a location. In omnifinite geometry, a point is defined as a nonzero dimensional object, meaning in addition to having a location, it also has size.

As an application of this new number system, an omnifinite infinity calculator has been developed and created as a computer-based software tool. To the authors’ knowledge, this is first of its kind arithmetic errorless computational program. The arithmetic errorless calculator was presented for review to a large group of engineering seniors majoring in civil, mechanical, and electrical engineering, who used the calculator hands-on and completed a survey. Results show that a new number system called omnifinites, which is a modification of the hyperreals, may be created and developed that is reasonable and logical based on the perspective of senior engineering students surveyed. The resulting arithmetic errorless calculator allows for inputs and outputs that may be finite, nonfinite, or a combination. This calculator software tool was found by the engineering students surveyed to be intuitive to learn, easy to use, and fun to operate. This software tool will be shown at the presentation and attendants may actually use this calculator in real time.

AU - Hunter Christopher Fred
AU - Ryan Bennett Greenwood
AU - Shane Mathew Palmquist
CY - Baltimore , Maryland
DA - 2023/06/25
PB - ASEE Conferences
TI - Introducing Omnifinites and the Arithmetic Errorless Infinity Calculator
UR - https://peer.asee.org/43849
DO - 10.18260/1-2--43849
ER -