9

I don't see anything wrong with what you were trying to say. In fact, I think that is a good way of saying it. Let's draw some pictures with some concrete numbers. Let's take $a=6$ and $b=9$. Then we can draw an $a\times b$ grid as shown below. Now as you said we can show the greatest common divisor $d$ is the side length of the largest square that can ...


8

Clearly there is no historical data that addresses this question I want to know if there are any numerical bases that are notably well-suited for humans to learn and use at an elementary or grade-school level since we have ten fingers and humans have learned only decimal arithmetic for everyday use. I just finished four weekly sessions with fifth ...


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 ...


6

The answer is simple: because it’s trivial. LCD Will Be always 1, even if one number is 0. GCM doesn’t exist, or it’s $\infty$, if you prefer. The first would be a well defined object, but not so useful ;) The second... not even well defined in a Ring or a Field :(


5

I am thinking that, if you had someone who was an expert in this field, they would observe that the immediate concept of counting does not logically imply running out of numerals and having to invent the idea of “ten”.  They would then observe that different values for “ten” result in different artefacts under processes such as addition and multiplication… ...


5

The OP describes a geometrical illustration using the fact that the GCD is the side of the greatest square that tiles a $A \times B$ rectangle. I don't understand why the following is not a "geometrical operation." But perhaps it is slightly different than what the OP had in mind, since he claimed "this multiplication does not represent...any area...," but ...


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

I'll hazard a probably uninteresting answer. Here's an example: Let's run through how this goes in general. Suppose $a,b$ are positive integers such that they share prime-power divisors $p_1^{r_1},\dots, p_m^{r_m}$ (we assume each prime power is as large as is possible). Furthermore, suppose $$ a = p_1^{r_1+s_1}\cdots p_m^{r_m+s_m}a' \qquad \& \qquad ...


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

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

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 ...


1

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


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