Month: December 2020

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| < …

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The Radian

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 …

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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 …

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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 …

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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 …

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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 …

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