7

A few recommendations: Fernando Q. Gouvêa's $p$-adic numbers: An Introduction. This text gives a good introduction to the $p$-adic number system and the properties of the space of $p$-adic numbers, vector spaces over $\mathbb{Q}_p$, and the metrically completed algebraic closure of $\mathbb{Q}_p$. There is also some discussion of $p$-adic analysis near ...


5

This is probably better suited to the History of Science and Mathematics SE, but I'll take a semi-informed stab at it. Since we read from left to right, placing the most significant digits in that same order give us the best opportunity to quickly comprehend the magnitude of the number. For instance, the speed of light is 299,792,458 meters per second. I ...


4

I will answer with an example. I seek the Bezout coefficients for 99 and 707. First I execute the Euclidean algorithm: $$ \begin{align*} 707 &= 7 \cdot 99+14\\ 99 &= 7 \cdot 14+ 1 \end{align*} $$ Now, I will recursively "backtrack" $$ \begin{align*} 1 &= 99-7\cdot 14\\ &= 99-7 \cdot (707-7 \cdot 99)\\ &=50 \cdot 99 - 7 \...


3

Which academic subjects examine what the advantages and disadvantages of the various number bases are, e.g. besides base ten: base twelve, base sixteen, base eight, base two and the ways that they can be written and pronounced. I don't know whether this has been investigated academically. You should use Semantic scholar, Google scholar and other academic ...


2

Benjamin Hutz has a recent book that could be appropriate: An Experimental Introduction to Number Theory. This book presents material suitable for an undergraduate course in elementary number theory from a computational perspective. It seeks to not only introduce students to the standard topics in elementary number theory, such as prime factorization and ...


2

Probably you know all of the following, but just to have it on the record: Most seven year olds are not fluent with fractions, but if you have one that is especially skillful in manipulating them, he might be able to learn how to relate the Euclidean algorithm to the continued fraction, learn to manipulate continued fractions and to understand their ...


2

Simplest explanation I've seen is to take $a, b$ integers and consider the set $\{u a + v b\}$ for integer $u, v$. It is a bunch of integers, so it has to contain a smallest positive one, call it $d = u_0 a + v_0 b$. Now divide $a$ by $d$: $a = q d + r$, by the Euclidean "algorithm" $0 \le r < d$. You see that: $\begin{align*} a &= (...


1

While this is not quite an app, I believe it fits within the spirit of the question: Desmos activities. https://teacher.desmos.com/collections/featured While some Desmos activities are very math-forward, others follow Dan Myers' "Three Acts" format, giving students an interesting scenario and asking them to solve it with math, and showing the ...


1

Our numbering system comes from copying the Hindu-Arabic numbering system. They write from right to left, so in their system, they start from the least significant digit. However, taking over the ordering as-is was easier, otherwise operations would need to have been adapted. So we write numerals in the same order even if we generally write text in the ...


1

There's a really nice book by Svetlana Katok called "p-adic analysis compared to Real."


1

Some of my students love Number Theory: Step by Step published in Dec. 2020 by Kuldeep Singh. The author provides solutions online to ALL exercises. And the author uses color.


1

A recent book is Number Theory and Geometry: An Introduction to Arithmetic Geometry by Álvaro Lozano-Robledo, AMS 2019. It's particularly nice for students who want to see how number theory works in the context of curves, especially lines, conics and cubics. It might attract some to studying elliptic curves and arithmetic/diophantine geometry. The author has ...


Only top voted, non community-wiki answers of a minimum length are eligible