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Tag Archives: numbers

There are many science-minded authors and working physicists who write popular science books. While there aren’t as many math-minded authors or working mathematicians writing popular math books, it’s not a null set. I’ve explored two such authors recently: mathematician **Steven Strogatz** and author **David Berlinski**.

Strogatz wrote *The Joy of X* (2012), which was based on his New York Times columns popularizing mathematics. I would call that a must-read for anyone with a general interest in mathematics. I just finished his most recent, *Infinite Powers* (2019), and liked it even more.

Berlinski, on the other hand, I wouldn’t grant space on my bookshelf.

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11 Comments | tags: calculus, David Berlinski, derivatives, fun with numbers, integrals, numbers, Steven Strogatz, The Joy of X | posted in Books, Math

Five years ago today I posted, *Beautiful Math*, which is about **Euler’s Identity**. In the first part of that post I explored why the Identity is so exquisitely beautiful (to mathematicians, anyway). In the second part, I showed that the Identity is a special case of **Euler’s Formula**, which relates trigonometry to the complex plane.

Since then I’ve learned how naive that post was! It wasn’t wrong, but the relationship expressed in Euler’s Formula is fundamental and ubiquitous in science and engineering. It’s particularly important in quantum physics with regard to the infamous Schrödinger equation, but it shows up in many wave-based contexts.

It all hinges on the **complex unit circle** and the **exp**(i×π×a) function.

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10 Comments | tags: 3Blue1Brown, complex numbers, complex plane, Euler's Formula, Euler's Identity, Fourier transform, numbers, transcendental numbers | posted in Math

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.

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2 Comments | tags: Matt Parker, numbers, Parker Square | posted in Books, 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.

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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.

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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.

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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

In the *Rational vs Real* post I mentioned that real numbers were each *“an infinitely tiny island separated from direct contact with all other numbers.”* The metaphor of each real number as an island comes from how, given any real number, it’s not possible to name the next (or previous) real number.

It’s easy enough to name a *particular* real number. For instance **1.0** are **3.14159…** real numbers. There are infinitely many more we can name, but given any one of them, there is no way to get to any other number other than by explicitly naming it, too.

This applies to a variety of numeric spaces.

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6 Comments | tags: complex numbers, continuum, integers, math theory, mathematics, natural numbers, numbers, rational numbers, real numbers, Theory of Mathematics | posted in Math

One of the great philosophical conundrums involves the origin of numbers and mathematics. I first learned of it as Platonic vs Aristotelian views, but these days it’s generally called **Platonism** vs **Nominalism**. I usually think of it as the question of whether numbers are *invented* or *discovered*.

Whatever it’s called, there is something transcendental about numbers and math. It’s hard *not* to discover (or invent) the natural numbers. Even from a theory standpoint, the natural numbers are very simply defined. Yet they directly invoke infinity — which doesn’t exist in the physical world.

There is also the “unreasonable effectiveness” of numbers in describing our world.

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6 Comments | tags: math theory, mathematics, natural numbers, nominalism, numbers, Plato, Platonic, Platonism, rational numbers, real numbers, Theory of Mathematics | posted in Math, Philosophy

**1000000**_{2}

**1000**_{4}

**100**_{8}

**40**_{16}

**10**_{64}

**@**_{char}

Comments Off on Round Numbers | tags: base 16, base 2, base 8, fun with numbers, numbers, sun sign | posted in Life

One of my earliest posts was *Analog vs Digital*. A few years later, I wrote about it in more detail (twice). Since then I’ve touched on it here and there. In all cases, I wrote from the perspective that *of course* they’re a Yin-Yang pair.

Recently I’ve encountered arguments challenging that “night and day” distinction (usually in the context of computationalism), so here I’d like to approach the topic with the intent of justifying the difference.

I do agree the grooves on a record, and the pits on a CD, are both just physical representations of information, but the *nature* of that information is what is night and day different.

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17 Comments | tags: analog, continuous, digital, discrete, frequency response, numbers, transfer function | posted in Basics