Finite math and precalculus both refer to math before calculus. Finite mathematics, however, is a catch-all title representing preparatory math focusing on advanced math outside of calculus, while precalculus is more narrowly defined as the algebra knowledge necessary to perform calculus, often called algebra 3.
TL;DR (Too Long; Didn't Read)
If you're going to study calculus, a precalculus course is immensely helpful in preparing you to make some big conceptual leaps. If you're not going on to study calculus a finite math course might be more immediately useful, depending on your career plans.
The transition from Algebra I and Algebra II to precalculus and finite math can be a drastic change. High school or college algebra students might become accustomed to thinking arithmetically and solving problems with factoring, inequalities, polynomials, complex numbers, and similar techniques. However, precalculus and finite math work to reframe how we think about different aspects of math. They share some new approaches, but there are significant diversions in content and focus as each course progresses.
The Goal of Finite Math
In finite math classes, the goal is to give students enough information to use mathematical analysis in the real world, at jobs or at home. Topics covered include matrix algebra, linear transformations, probability, statistics, logic, graphs, set theory, and discrete mathematics. You learn simple, immediately useful ways to count, calculate, add, subtract, multiply and divide. While success in finite math can be immensely helpful in the real world, it does not necessarily prepare you for a full calculus class.
The Value of Precalculus
Precalc, also called algebra 3, is the highest-level algebra class you can take before going into calculus. In this course, you become comfortable with quantitative literacy and logic, such as algorithms, logic and proofs, functions, geometry, trigonometry, statistics and probability. You learn how to form relationships between numbers in a way that provides you more information about what the numbers mean. For example, that might mean solving for an unknown variable by constructing an equation. You also become more comfortable manipulation those variables that stand in for unknown numbers.
Calculus is the next advanced class after algebra and precalculus, and it introduces students to the great conceptual leaps of differentiation and integration. Differentiation allows you to take apart mathematical functions to understand their behavior by taking derivatives, looking at rates of change across infinitely small intervals. Integration lets you put functions back together, adding together infinitely many values across varying intervals. You must have strong algebra skills to be successful in calculus, and at the college level, calculus will include and introduce many of the topics discussed in finite math.
Advanced Mathematics Courses
Calculus courses are prerequisites for many different different math, physics, and computer science courses later in college. While finite math can provide a good introduction to some very interesting topics, many of them will be covered in more rigorous detail in classes that follow calculus. Linear algebra, differential equations, and a full course in discrete math will cover topics like linear programming, vector spaces, determinants, graph theory, optimization, combinatorics, and number theory. Many of these classes also focus on proof based or higher level understanding of these topics.
These classes will cover some topics that might seem familiar from intermediate algebra and trigonometry – like systems of linear equations, integers, exponentials, and logarithmic functions, but the key to moving beyond the first year understanding often revolves in viewing the same math with a different perspective. Instead of just solving a system of linear equations for the proper variables, we might look at theorems and definitions that can help us look at the dimensionality or generalize the situation.