Finite math

Finite Math is a catch-all title for a broad collection of mathematical topics that are anything but calculus. Its purpose is to give a survey of mathematical analysis techniques used in the working world, though it also gives valuable experience at organizing information and then analyzing it.

In a broader sense, this field of mathematics implements another way we use math to give people experience at analytical thinking. It may introduce concrete evidence of the role of mathematics plays in such disciplines as business, social science, and biological science, politics, education, sociology, philosophy, programming, finance, economics, zoology, urbanism, strategy, architecture, planning, diagnosis, land development, communication networks, elections, energy efficiency, individual retirement accounts, gross national product and the deficit, games, food web, income taxes, inflation, road construction, transportation, toxic waste disposal, personal decision making, lotteries, profit maximization, population growth, quality control, production, psychological scaling, carbon dating, consumer price index, genetics, advertising, agriculture, annuities, tournaments, survey sampling, spread of information and disease, sinking funds, dominance relations, diet determination, sensitivity analysis, investments, etc.

Here is a short list of the main topics it covered:

–Mathematical model building. Math modeling is the act of creating functions or equations that describe a given application or situation. In this course we mainly concentrate on business-oriented ideas such as break-even analysis or depreciation.

–Matrix algebra. Matrices are collections of numbers organized in rectangular arrays. These can effectively represent certain kinds of data or systems of equations. In Finite Math you only get a brief glimpse into how they are used and manipulated, but matrix ideas can arise in both accounting and business analysis, and computer programmers use them as array variables.

–Linear programming. This topic has nothing to do with computer programming, but it is a method for optimizing situations when constraints are in place. For example, if you produce several lines of products but have budgetary constraints on labor and materials, and have production contracts in place that must be filled, then what is the most efficient, profitable way to determine how much of each line to produce, that is, how can you maximize the profit potential? Linear programming is ideally suited to problems of this nature. Linear programming can also be an application of both math modeling and matrix algebra.

–Combinatorics. This is the art of advanced counting. For example, if a room has twenty chairs and twelve people, in how many ways can these people occupy the chairs? And are you accounting for differences in who sits in particular chairs, or does it only matter if a chair has a body in it? These kinds of counting problems are the basis for solving diverse strategical issues in very real life.

–Probability. In order to calculate the chance of a particular event happening you must be able to count all the possible outcomes. Once you understood how to find probabilities then you can begin to understand cycles, outcomes, etc.

–Statistics. Statistics uses probability in order to analyze data and make decisions. In Finite Math you will only get a brief introduction and overview of statistics.

–Logic. Logic is the symbolic, algebraic way of representing and analyzing statements and sentences. You will only get a brief introduction to logic in this course, but the mathematics used in logic are found at the heart of computer programming and in designing electrical circuits.

More advanced courses in Finite Math topics are sometimes called Discrete Mathematics. The word discrete helps explain where Finite Math gets its name. Discrete means broken up or separated. For example, integers are discrete objects because there are non-integer numbers in between them, but real numbers are continuous numbers because there is no identifiable separation between them.

For a maddening exercise in continuity try finding the largest real (i.e., decimal) number less than one. No, it is not 0.999999 . . . (the nines repeating forever), because it can be demonstrated that 0.999999 . . . is equal to 1. Whatever this number is it is impossible to represent it in any other than the most abstract way.

Continuity is in some ways associated with infinity and infinitesimal. Since calculus is concerned with continuous numbers and continuous functions, the subject must confront the ideas of infinity and infinitesimal. Finite Math is a subject that avoids the issues of continuity encountered in calculus, so those topics are lumped into the category of finite mathematics.

Fibonacci Numbers

The Fibonacci numbers Fn are defined recursively by the relation Fn = Fn–1 + Fn–2, where F1 = F2 = 1. The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …

They describe, among other things, the number of ways to tile a 2 x (n–1) checkerboard with 2 x 1 dominoes.

There’s only one way to tile a 2 x 1 board:






Two ways to tile a 2 x 2 board:






Three ways to tile a 2 x 3 board:






Five ways to tile a 2 x 4 board:







Eight ways to tile a 2 x 5 board:








Thirteen ways to tile a 2 x 6 board:










Twenty-one ways to tile a 2 x 7 board: