Understand a weird method of calculus

Question

I see this method of calculus on youtube and my question: is this method valid? How we can understand it? Thanks.

Answer 1

The method is entirely valid! The problem is that it becomes rather impractical for digits greater than say $4$. Also it may not be very practical for numbers with many digits in them.

Let me try to explain the method.

Basic principle: parallel lines crossing

If $m$ parallel lines intersect $n$ parallel lines they will intersect in $m\cdot n$ points. So multiplying the digit $m$ with the digit $n$ will be equivalent to counting the number of intersection points of $m$ parrallel lines crossing $n$ parrallel lines.

This is easiest to see when they intersect at right angles or close to that.

How the 1's places the 10's places etc. come into play

If we write for instance two $2$-digit numbers as $\newcommand{\red}{\color{red}}\newcommand{\blue}{\color{blue}}\red{10a+b}$ and $\blue{10s+t}$ so that $a,b,s$ and $t$ are the digits what we get when multiplying these will be $$ \begin{array}{c:c:c} \mbox{100's}&\mbox{10's}&\mbox{1's}\\ \hdashline & \red{b}\cdot \blue{s} &\\ \red{a}\cdot \blue{s}&& \red{b}\cdot \blue{t}\\ &\red{a}\cdot \blue{t}& \end{array} $$ which can be seen from the basic algebra: $$ (\red{10a+b})\cdot(\blue{10s+t})=100\red{a}\cdot \blue{s}+10(\red{b}\cdot \blue{s}+\red{a}\cdot \blue{t})+1\red{a}\cdot \blue{t} $$ This can easily be extended to a larger scheme.

The rightmost intersecting lines will always be the product of the 1's places of the two numbers. The leftmost intersecting lines will always be the product of the $10^n$'s places for an $(n-1)$-digit number. If we multiply an $(n-1)$-digit number by an $(m-1)$-digit number the pattern is the same. The leftmost intersecting lines will be the $10^n\cdot 10^m=10^{m+n}$ place.

An additional table for the general principle

The most simple examples will be multiplying numbers of the form $11111...$. A table showing which places will be multiplied could be written as follows: $$ \begin{array}{c:c:c:c:c:c} ...&\mbox{10000's}&\mbox{1000's}&\mbox{100's}&\mbox{10's}&\mbox{1's}\\ \hdashline ...&&&&&\\ &\red{1}\cdot \blue{10000}&&&&\\ ...&&\red{1}\cdot \blue{1000}&&&\\ &\red{10}\cdot \blue{1000}&&\red{1}\cdot \blue{100}&&\\ ...&&\red{10}\cdot \blue{100}&&\red{1}\cdot \blue{10}&\\ &\red{100}\cdot \blue{100}&&\red{10}\cdot \blue{10}&&\red{1}\cdot \blue{1}\\ ...&&\red{100}\cdot \blue{10}&&\red{10}\cdot \blue{1}&\\ &\red{1000}\cdot \blue{10}&&\red{100}\cdot \blue{1}&&\\ ...&&\red{1000}\cdot \blue{1}&&&\\ &\red{10000}\cdot \blue{1}&&&&\\ \end{array} $$ where the red numbers show which places from the first number we multiply and the blue numbers show which places from the second number we multiply.