Also, if you haven't heard about these other, tangentially-related, fun "base" extensions (beyond non-integer bases, etc), be sure to check out Gray Code and Hamming Codes.

This is all so neat!

en.wikipedia.org/wiki/Gray_code

en.wikipedia.org/wiki/Hamming...

Then I went further down the rabbit hole and found:

en.wikipedia.org/wiki/Non-sta...

Aaaaaaaaah, this was such a good present from my student!

I thought it was problematic that some numbers had more than one representation. For example, one-half can be written as either

0.1111111... or
1. -1 -1 -1 -1 -1 -1 ...

...but then I got over it, because even in our normal notation, one-half can also be written as

0.5000000... or
0.4999999...

"Yeah, I see lower-case i all the time in my class, too." -me, without saying anything else about my sections of advanced algebra.

[Five entire eternities later]

The Engineering teacher: "WAIT A SECOND. Were you being CHEEKY?!"

i was.

2² + 3² + 6² does equal 7² -- whoa!

Alright, time to generalize into a generic base.

b² + (b+1)² + (b² + b)² = (b² + b + 1)²

is an identity!

I wonder if there are any other identities like this. The reason this looks like it can be in binary is in how there are no coefficients anywhere. Hmm..

(2/2) I know that mental health has become main-stream for teens to hear and talk about: in music, in fashion, etc. I have seen this hoodie countless times. I wonder if teens will make it fashionable to wear self-harm bandages to take the stigma off of those who wear them out of necessity.

Follow-up.

(Also, I love the "As predicted", as though it were the scientist equivalent of "as per my previous email".)

"The President’s budget request for NASA in fiscal year 2026 totals $18.8B. This represents a significant shift from the $24.8B enacted in 2025..."

In 2024, NASA got 0.37%.
In 2025, NASA got 0.34%
In 2026, NASA is slated to be further reduced to 0.27%.

www.planetary.org/space-policy...

I like how each category notes approximately which "geomagnetic latitude" is likely to experience aurora overhead. This is different from "regular" latitude because the earth's rotational axis isn't aligned with the magnetic field's axis. Because magnet fields cause auroras, we use its lats instead!

At this link can be found a description of the categories in the space weather scales. It's the same idea as the category scale we have for hurricanes, or the EF scale for tornadoes, or the magnitude scale for earthquakes. I like the rightmost column the most. (www.spaceweather.gov/noaa-scales-...)

I think this would go over their heads, and not just because it took place on a starship.

But for real: this is a moment in pop culture for adults to connect with teenagers about something cataclysmic happening in that space. Consider starting the year off with a bid for cross-cultural connection?

..., then counted the third column as five again, and then subtracted one to correct for how the third column wasn't a full five.

I feel really, really gross at Brain right now.