Ah, in math writing, it's easy enough to say more and be not boring and at times be at least interesting, inviting, even exciting!<p>Let's have some examples!!<p>(1) Dimension.<p>So, suppose we are in the first class in linear algebra:<p>"Maybe you have heard that the <i>real line</i> has 1 dimension, is <i>1 dimensional</i>, the <i>plane</i> is 2 dimensional, and the space we live in is 3 dimensional. Well, that's all true enough, but in linear algebra we do better and have more: For one, we get to say clearly what is meant by <i>dimension</i>, that, in particular, why the line, plane, and space are 1, 2, 3 dimensional. For much more, for any positive integer <i>n</i> we have <i>n</i>-dimensional space.<p>Next, in linear algebra <i>n</i>-dimensional space is a relatively easy generalization of what we already know well in dimensions 1, 2, 3.<p>Why might we care? For example, we know well what <i>distance</i> is in dimensions 1, 2, 3, and <i>distance</i> in <i>n</i> dimensions is a straight forward generalization. In dimensions 2 and 3, we understand <i>angle</i>, and also that carries over to <i>n</i> dimensions. For more, with computing it is common to have a list of, say, 15 numbers. Well, for just one benefit, with linear algebra we get to regard that list as a point in <i>n</i> = 15 dimensional space, and doing so lets us do some powerful things with <i>representing</i> and <i>approximating</i> that list."<p>So, we get some sense of <i>previews of coming attractions</i> and some <i>invitation to higher dimensions</i>.<p>(2) Optimization.<p>"There is a subject, with a lot of development just after WWII, called <i>linear programming</i> (LP). The <i>programming</i> is in the English sense of <i>operational planning</i> as in war logistics and planning as was crucial in WWII. The <i>linear</i> is the same as in linear algebra.<p>The main goal, point of LP is to find how to exploit the freedom we have in doing the operations, the work to be done, to get the work done as fast or cheaply as possible, that is, to find an <i>optimal</i> way to do the work.<p>So, the subject LP is part of <i>optimization</i>. There have been some Nobel prizes from applications of LP and other math of optimization to economics. There have been applications of LP to feed mixing, oil refinery operation, management of large projects, and parts of transportation."<p>(3) The Simplex Algorithm.<p>"Maybe in high school algebra you saw the topic of systems of linear equations. Well, it is fair to say that the <i>standard</i> way to solve such a system is Gauss elimination due to C. F. Gauss.<p>The idea is simple: Multiplying one of the equations by some non-zero number and adding the resulting equation to another of the equations does not change the set of solutions. So, doing that in a slightly clever way results in the system of equations with a lot of zeros, about half all zeros, so that the set of solutions is obvious just by inspection.<p>Then for linear programming, in practice the main solution technique is the simplex algorithm, and it is just but done with <i>optimization</i> in mind."<p>(4) Completeness.<p>A <i>rational</i> number can be written as <i>p/q</i> for integers <i>p</i> and <i>q</i>. We will see, easily, that the rational numbers are not up to <i>carrying the load</i>, are not up to doing the work we need done. So we need a more powerful system of numbers -- we need the <i>real</i> numbers.<p>Here is a really simple place the rational numbers fail to do what we want: At times we consider square roots. E.g., the square root of 9 is 3. Well, what is the square root of 2? Suppose that square root were a rational number, i.e., so that<p><i>(p/q)^2 = 2</i><p>Then we have<p><i>p^2 = 2q^2</i><p>so that the left side has an even number of factors of 2 while the right side has an odd number. Tilt. Bummer! That can't be. That's a contradiction.<p>So, there is no rational number that is the square root of 2. So, for something really simple, just finding a square root, the rational numbers fail us, can't carry the load or do the work.<p>The real numbers will let us find the square root of 2 and much more. With the real numbers we get what we call <i>completeness</i>. A joke, basically correct, is that calculus is the elementary consequences of the completeness property of the real numbers. Then we generalize: Banach space is a complete normed linear space. Hilbert space is a complete inner product space. The Fourier transform works because of completeness. So, we move on and see how the real numbers are <i>complete</i> ...."