#
Category Archives: Math

At the beginning of the month I posted about a neat Japanese visual method for multiplying smallish numbers. Besides its sheer visual attractiveness, it’s interesting in allowing one to multiply numbers without reference to multiplication tables (which, let’s face it, typically require rote memorization).

As I mentioned last time, my interest in multiplication is linked to my interest in generating Mandelbrot plots, which is a multiplication-intensive process. But for those learning math, digging into basic multiplication has some instructive value.

With that in mind, here are some other **multiplication** tricks.

Continue reading

4 Comments | tags: Cartesian product, multiplication | posted in Math

**123 × 321 = 39,483**

My interest in number multiplication goes back to exploring algorithms for generating Mandelbrot plots, which can require billions of multiplication operations on arbitrary precision numbers (numbers with *lots and lots* of digits).

**Multiplying** two numbers — calculating their *product* — is computationally intense because of the intermediate Cartesian product. Multiplying two 12-digit numbers creates a 24-digit result (12+12), but it *also* has an intermediate stage involving **144** (12×12) single digit multiplications.

Recently I learned an intriguing Japanese *visual* multiplication method.

Continue reading

23 Comments | tags: Cartesian product, Japanese multiplication method, Mandelbrot, multiplication | posted in Math

*Expert Logician*

For a little Friday Fun I have a logic puzzle for you. I’ll give you the puzzle at the beginning of the post, detour to some unrelated topics (to act as a spoiler barrier), and then explain the puzzle in the latter part of the post. I would encourage you to stop reading and think about the puzzle first — it’s quite a challenge. (I couldn’t solve it.)

The puzzle involves an island with a population of **100** blue-eyed people, **100** brown-eyed people, and a very strange social practice. The logic involved is downright nefarious, and even after reading the explanation, I had to think about it for a bit to really see it. (I still think it’s twisted.)

To be honest, I’m kinda writing this to make sure **I** understand it!

Continue reading

20 Comments | tags: Daniel Craig, George Lazenby, James Bond, logic, logic puzzle, NASA, Pierce Brosnan, Randall Munroe, Roger Moore, Sean Connery, Terry Tao, Timothy Dalton | posted in Math

Four years ago I started pondering the **tesseract** and four-dimensional space. I first learned about them back in grade school in a science fiction short story I’d read. (A large fraction of my very early science education came from SF books.)

Greg Egan touched on tesseracts in his novel *Diaspora*, which got me thinking about them and inspired the post *Hunting Tesseracti*. That led to a general exploration of multi-dimensional spaces and rotation within those spaces, but I continued to focus on trying to truly understand the tesseract.

Today we’re going to visit the 4D space *inside* a tesseract.

Continue reading

3 Comments | tags: 1D, 2D, 3D, 4D, cube, dimensions, square, tesseract | posted in Math, Sideband

I just finished *Humble Pi* (2019), by **Matt Parker**, and I absolutely loved it. Parker, a former high school maths teacher, now a maths popularizer, has an easy breezy style dotted with wry jokes and good humor. I read three-quarters of the book in one sitting because I couldn’t stop (just one more chapter, *then* I’ll go to bed).

It’s a book about mathematical mistakes, some funny, some literally deadly. It’s also about how we need to be better at numbers and careful how we use them. Most importantly, it’s about how mathematics is so deeply embedded in modern life.

It’s my third maths book in a month and the only one I *thoroughly* enjoyed.

Continue reading

1 Comment | tags: Matt Parker, numbers, Parker Square | posted in Books, Math

Multiplying by *i*…

Recently I did a series of posts about how the complex numbers arise from a natural progression of math realizations. I’ve done posts in the past about how the natural numbers lead through the integers and rationals to the real numbers. (And I’ve done posts about how weird the real numbers are, but that’s another topic.)

I recently came across another way a progression of obvious natural questions directly leads to the necessity of a new type of number, and this progression takes us all the way from the naturals to the complex numbers.

All by asking, *“What do you get when you…”*

Continue reading

5 Comments | tags: complex numbers, complex plane, group theory, groups, integers, irrational numbers, natural numbers, rational numbers, real numbers, sets, town barber paradox | posted in Math

In recent posts I’ve presented the complex numbers and the complex plane. Those were just stepping stones to this post, which involves a basic fact about the Mandelbrot set. It’s something that I stumbled over recently (after tip-toeing around it many times, because math).

This is one of those places where something that seems complicated turns out to have a fairly simple (and kinda cool) way to see it when approached the right way. In this case, it’s the way multiplication *rotates* points on the complex plane. This allow us to actually visualize certain equations.

With that, we’re ready to move on to the “heart” of the matter…

Continue reading

18 Comments | tags: cartioid, complex numbers, fun with numbers, Mandelbrot, Mandelbrot fractal, real numbers | posted in Math

In the first post I explained why the mathematical “imaginary” number *i* is “real” (in more than one sense of the word). That weird number is just a stepping stone to the complex numbers, which are themselves stepping stones to the **complex plane**.

Which, in turn, is a big stepping stone to a fun fact about the Mandelbrot I want to write about. (But we all have to get there, first.) I think it’s a worthwhile journey — understanding the complex plane opens the door to more than just the Mandelbrot. (For instance, Euler’s beautiful “sonnet” also lives on the complex plane.)

As it turns out, the complex numbers cause this plane to “fly” a little bit differently than the regular X-Y plane does.

Continue reading

3 Comments | tags: complex numbers, complex plane, fun with numbers, imaginary unit, numbers, real numbers | posted in Math

Graph of *ax*^{2} for diff *a* values.

(green < 1; blue = 1; red > 1)

This is a little detour before the main event. The first post of this series, which explained why the imaginary unit, *i*, is important to math, was long enough; I didn’t want to make it longer. However there is a simple visual way of illustrating exactly why it seems, at least initially, that the original premise isn’t right.

There is also a visual way to illustrate the solution, but it requires four dimensions to display. Three dimensions can get us there if we use some creative color shading, but we’re still stuck displaying it on a two-dimensional screen, so it’ll take a little imagination on our part.

And while the solution might not be super obvious, the *problem* sure is.

Continue reading

4 Comments | tags: complex numbers, fun with numbers, imaginary unit, numbers, parabola, real numbers, x-squared | posted in Math

Yes, this is a math post, but don’t run off too quickly. I’ll keep it as simple as possible (but no simpler), and I’ll do all the actual math so you can just ride along and watch. What I’m about here is laying the groundwork to explain a fun fact about the Mandelbrot.

This post is kind of an origin story. It seeks to explain why something rather mind-bending — the so-called *“imaginary numbers”* — are actually vital members of the mathematical family despite being based on what seems an impossibility.

The truth is, math would be a bit stuck without them.

Continue reading

16 Comments | tags: complex numbers, complex plane, fun with numbers, imaginary unit, integers, Leopold Kronecker, natural numbers, numbers, rational numbers, real numbers | posted in Math