In Arithmetic, we learned all about numbers and the basic ways of combining them (adding, multiplying, subtracting, division, and powering).
In Algebra, we start to learn how to think of objects that take in various numbers into the same formula and what we can do with that.
The focus here is largely formulas involving addition, multiplying, and powering. Subtraction is subsumed by multiplying by negative numbers and does not play a special role. Division by unknown variables is not fully taken up until the Functions book, but we do see some shadows of it in the polynomial chapter.
By the end of this book, you should be comfortable with quite a bit of useful, practical mathematics and well posed to solve:
Here are some highlights of what we will learn how to do:
Model a problem with a linear equation
Solve a linear equation
Model situations with quadratics
Solve a quadratic equation
Find the maximum or minimun of a quadratic
Solve polynomial equations
Recenter polynomials (gateway to calculus)
Graph polynomials
Solve inequalities
Solve absolute value equations
Find out how much one will owe
Amortization
Solving systems of linear equations
Exploring real and complex numbers
Straight lines give us the shortest path between two points. That fundamental insight will allow us to explore deeply not only lines, but all sorts of other geometric notions that come with lines. We will also explore setting up and solving both linear equations linear inequalities.
The natural successor of lines are quadratics. We go from \(x\) to \(x^2\). While the quadratic formula is a way of solving quadratic equations, of more interest with quadratics is finding the high (or low) point of the quadratic. This is what quadratics model for us. We also end up exploring something called imaginary numbers.
With lines firmly understood, the next set of objects to learn about is when we square the unknown variable. We call these quadratics. They are shaped somewhat like a rounded bowl, either upside down (concave down) or rightside up (concave up). A key feature to compute, and easy to do, is the vertex, which is either the highest or lowest point of the quadratic, depending on whether it is concave down or up, respectively. Once we can the vertex, we can then look into solving quadratics. We inevitably end up computing square roots including the seemingly impossible task of computing square roots of negative numbers leading us to imaginary numbers. We conclude with examing a variety of common quadratic problems for solving both quadratic equations and inequalities.
While lines and quadratics form the basis for the most important aspects of some local notions, polynomials are the wide family that can model just about anything over a given limited space. The full notion of that is what calculus does, but we will explore a lot of those techniques using a tool called synthetic division.
What about higher powers of the unknown? We can do that. Those are called polynomials, which includes lines and quadratics. After introducing them, we will learn a crucial technique called synthetic division. Our approach uses that technique in ways rarely seen elsewhere. We will find that not only is synthetic division useful for factoring polynomials (the typical use), but can also be used for essentially doing calculus with polynomials without the magic and craziness of standard calculus. This gives us tremendous insight, both algebraically and visually, early on into fully mastering polynomials with nothing but the basic notions of arithmetic. We conclude with taking powers of polynomials, in particular patterns of expansions.
Algebra is a perfect place to learn all about financial mathematics. While certain techniques will have to wait for logarithms, a great deal of the most useful and insightful stuff can be done with the toolset we already have.
Perhaps the most applicable chapter to everyday life is the one on interest. We explore the questions and concepts of interest compounding and other related topics to personal financial mathematics.
One of the most useful tools in all of mathematics is that of linear systems. These have the complexity to be useful but the simplicity to be solvable.
What if we had a number of variables and linear relations between them? This is the foundation for handling the messiness of real-life with lots of different information. While we delve into it fully in another book later in the series, here we present some commonly useful and applicable approaches. We conclude the chapter with systems involving linear inequalities, a topic called linear programming. It has a rather fun algorithmic solution method called the simplex method.
How do we make sense of a number whose decimal portion goes on forever in new and exciting ways? How can we make sense of imaginary numbers?
Here we investigate in what sense we can say the real and complex numbers are real. How do we define them such that it is clear that they exist? What are their properties? In addition to exploring real numbers, we also delve a little deeper into the relationship of multiplication and rotations in the realm of complex numbers. We conclude this chapter with an investigation into what sense are there different sizes of infinity.