The sine function divided by its angle, and a certain limit

What is $\lim_{x\rightarrow0} \frac{\sin x}{x}$ ? Recall the definition of a limit, repeated here for reference [2]. A function $f(x)$ approaches a limit $A$ as $x$ approaches $a$ if, and only if, for each positive number $\epsilon$ there is another, $\delta$, such that whenever $0 < |x-a| < \delta$ we have $|f(x) – A| < \epsilon$. That is, when $x$ is near $a$ (within a distance $\delta$ from it), $f(x)$ is near $A$ (within a distance $\epsilon$ from it). In symbols we write $\lim_{x \rightarrow a} f(x) = A$. Using this definition, $a$ is $0$ in this case. Now […]

In Chapter 2 of Reference [1], one of the footnotes has a definition of a radian. For a unit circle with a radius of one unit of length called $u$, one radian is the angle corresponding to an arc-length of $u$. In degrees, one radian is approximately 57 degrees, which is perhaps easier to visualize. A diagram of one radian is included here for reference. References [1] Konrad Knopp. Theory and Application of Infinite Series. Dover Publications. 1990.

Exponents

The following rules apply for exponents that are positive integers. The justification is sketched here.   These rules are also said to apply for exponents that are not positive integers–that is, zero and negative integers. If $p$ and $q$ are rational numbers, they are also said to apply. Finally, if $p$ and $q$ are irrational numbers, they are said to hold if $x$ and $y$ are positive–the other cases are not important for my purposes [1]. References [1] Konrad Knopp. Theory and Application of Infinite Series. Dover Publications. 1990.

Irrational Numbers

Are rational numbers the only numbers? The continuity of real numbers is an assumption in the Cantor-Dedekind axiom [1]. The idea of continuity brings up the possibility of a different type of number called an irrational number that is, simply, not a rational number. So, if a number is not a rational number, it is called an irrational number–this definition of an irrational number may need to modified later if new classes of numbers arise. An irrational number is denoted by a Greek letter such as $\sigma$ in contrast to the lowercase italicized letters for rational numbers. The

Definition of a Sequence

The title of Section 2 in book [1] is Sequences of rational numbers. At the beginning of this section, a definition is provided for a sequence, and I quote it here. Definition. If, by means of any suitable process of construction, we can form successively a first, a second, a third, … (rational) number and if to every positive integer $n$ one and only one well-defined (rational) number $x_n$ thus corresponds, then the numbers $x_1, x_2, x_3, …, x_n, …$ (in this order, corresponding to the natural order of the integers $1,2,3,…,n,…$) are said to

Fundamental Laws of Arithmetic

Here is a continuation of this post. Long story short, I am learning number theory from a book [1]. The author of this book [1] begins his own analysis of number theory with the “system of rational numbers, i.e. of numbers integral and fractional, positive and negative, including zero.” There is a brief mention that the rational numbers follow from operations on “the ordered sequence of natural numbers 1,2,3,…” Note that the word “system” is another word for “set,” in the context of number theory. In this reference [1], a variable for a rational number is given by a small italic

Reaching for that Number Theory book (Part I)

I now turn to number theory, because this field seems like a solid foundation for mathematics and physics. I start with Chapter 1 of the esteemed book, Theory and Application of Infinite Series [1]. The title of Chapter 1 is “Principles of the theory of real numbers.” Section 1 is called “The system of rational numbers and its gaps.” I have wondered about numbers and how they relate to the number line encountered in math classes. Hopefully this section clears up some of my confusion, but I don’t want to get my hopes up too high… I do not