Month: January 2023

A New Home

An essential WordPress feature that I use on this blog is the ability to format mathematical expressions. At the old home, I was using the plugin Katex. I just noticed that it was only working in Firefox. Unfortunately, I’m not sure how long it had been broken.

WordPress.com’s customer service wasn’t helpful, although they did point out that Katex hadn’t been updated for over a year. I found that the MathJax-LaTeX plugin was being actively maintained. On the old host’s service MathJax was horribly broken. It didn’t support colors, which I used in the previous post, An Expanded Multiplication Table. It only supported inline expressions and the background of inline expressions was white which didn’t match the blog’s background.

When I combined those failures with the high cost of the required upgrade at wordpress.com, $25/month, I searched for a new home. I’m glad that I found the problem before the blog renewed in a few weeks at $300 for another year!

When I searched for reviews, this host, IONOS was ranked highly. I already have been using IONOS for hosting my other websites, so I moved the blog there too. It appears that I didn’t lose much in the transition. It took me an afternoon to get the content transferred and an hour or two to get the site reconfigured. Most of that time was spent fixing the color references.

It was easy to find help for MathJax-LaTeX on the Mathematics StackExchange. For example, MathJax basic tutorial and quick reference and that is just scratching the surface of resources.

I wonder why I didn’t make this transition sooner. Spending $12 instead of $300 seems like enough incentive to change, but inertia is a powerful demotivator. My other WordPress blogs can follow.

{{Update: It turns out that the other blogs are stuck because I can’t transfer the media library. However, the financial benefit of transferring them only applies to the first year so the incentive isn’t as strong. It appears that transferring the media library is a difficult problem because its configuration is set by the managed hosting provider. Different hosts might use incompatible file naming schemes. In addition, the two cheaper blogs, blog.wwayneb.com and blog.sesquibits.com only have a limited export tool.}}

An Expanded Multiplication Table

In the Dune stories, mentats are trained to do complex calculations without any mechanical aids. For normal people, instantly knowing the multiplication up to 10 is a useful skill. Extending the multiplication table to twenty seems to require the skills of a mathematical savant.

However, calculating products of up to 20 actually only requires the single digit multiplication table and the ability to add small numbers mentally. My goal is to be able to do these calculations without paper. How?

If one of the numbers is single digit and the other is between 10 and 20, the calculation works like this:

A number between 10 and twenty can be written as \(10 + a\) or \(1a\). For the results you have one single digit multiplication an addition of a number times ten.

$$1a * b = (a*b) + (10 * b)$$

Since \(10 * b\) is just \(b\) shifted left by one, you can get the result by identifying \(a*b\) and then adding \(b\) to the second digit.

$$1\color{blue}{5} * \color{red}{7} = (\color{magenta}{35} + \color{red}{7}0) = 105$$

The mental steps are “calculate \(\color{blue}{5} * \color{red}{7}\)” and then “add \(\color{red}{7}\) to the second digit.”

For the product of a single digit times and a number under twenty, you always use the normal multiplication table and single digit addition.

If both numbers are between 11 and 19, the calculation works out this way.

$$a * 1b = (10 +a) * (10 + b) = a * b + 10 * (a + b) + 100$$

To start, multiply the singles place values. For example \(1\color{blue}{5} * 1\color{red}{7}\) starts as \(\color{magenta}{35}\).

Then add \(\color{blue}{5} +\color{red}{7}\), giving \(\color{orange}{12}\).

Add that in the 10s place \(\color{magenta}{35} + 10 * \color{orange}{12}\) giving \(\color{brown}{155}\) and finally, increment the hundreds digit, giving \(\color{green}{255}\)

This involves multiplying single digit numbers, adding a pair of single digits, adding a digit and a number less than twenty and incrementing a digit. It works out to needing having six or so digits in mind at once which can be less than one’s estimated working memory of 7 digits.

If one of the numbers is 10, add a zero to the right of the other value. For twenty, double the other number and add a zero. Now there’s a simple way to multiply two numbers where both are 20 or less.

With a little practice, I can do this in my head. More importantly, I’m confident of my result so that I’m getting so that I don’t need to double check on a calculator.