Ciro Santilli @cirosantilli 37

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Carbon microphone  Updated 2025-07-01  +Created 1970-01-01

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CC BY 3.0  Updated 2025-07-01  +Created 1970-01-01

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Metabolite  Updated 2025-07-01  +Created 1970-01-01

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Pharmaceutical company  Updated 2025-07-01  +Created 1970-01-01

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Product definition of the exponential function  Updated 2025-07-01  +Created 1970-01-01

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e^{x} = lim_{n \to \infty} (1 + \frac{x}{n})^{n}

(1)

The basic intuition for this is to start from the origin and make small changes to the function based on its known derivative at the origin.

More precisely, we know that for any base b, exponentiation satisfies:

$b^{x + y} = b^{x} b^{y}$ .
$b^{0} = 1$ .

And we also know that for

b = e

in particular that we satisfy the exponential function differential equation and so:

\frac{d e ^{x}}{d x} (0) = 1

(2)

One interesting fact is that the only thing we use from the exponential function differential equation is the value around

x = 0

, which is quite little information! This idea is basically what is behind the importance of the ralationship between Lie group-Lie algebra correspondence via the exponential map. In the more general settings of groups and manifolds, restricting ourselves to be near the origin is a huge advantage.

Now suppose that we want to calculate

e^{1}

. The idea is to start from

e^{0}

and then then to use the first order of the Taylor series to extend the known value of

e^{0}

e^{1}

E.g., if we split into 2 parts, we know that:

e^{1} = e^{1/2} e^{1/2}

(3)

or in three parts:

e^{1} = e^{1/3} e^{1/3} e^{1/3}

(4)

so we can just use arbitrarily many parts

e^{1/ n}

that are arbitrarily close to

x = 0

e^{1} = (e^{1/ n})^{n}

(5)

and more generally for any

x

we have:

e^{x} = (e^{x / n})^{n}

(6)

Let's see what happens with the Taylor series. We have near

y = 0

in little-o notation:

e^{y} = 1 + y + o (y)

(7)

Therefore, for

y = x / n

, which is near

y = 0

for any fixed

x

e^{x / n} = 1 + x / n + o (1/ n)

(8)

and therefore:

e^{x} = (e^{x / n})^{n} = (1 + x / n + o (1/ n))^{n}

(9)

which is basically the formula tha we wanted. We just have to convince ourselves that at

lim_{n \to \infty}

, the

o (1/ n)

disappears, i.e.:

(1 + x / n + o (1/ n))^{n} = (1 + x / n)^{n}

(10)

To do that, let's multiply

e^{y}

by itself once:

e^{y} e^{y} = (1 + y + o (y)) (1 + y + o (y)) = 1 + 2 y + o (y)

(11)

and multiplying a third time:

e^{y} e^{y} e^{y} = (1 + 2 y + o (y)) (1 + y + o (y)) = 1 + 3 y + o (y)

(12)

TODO conclude.

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Radar  Updated 2025-07-01  +Created 1970-01-01

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ScienceClic  Updated 2025-07-01  +Created 1970-01-01

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This is very promising.

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Tetrahedron  Updated 2025-07-01  +Created 1970-01-01

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Emission spectrum  Updated 2025-07-01  +Created 1970-01-01

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Institute of Electrical and Electronics Engineers  Updated 2025-07-01  +Created 1970-01-01

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Isotope of hydrogen  Updated 2025-07-01  +Created 1970-01-01

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Polish letter  Updated 2025-07-01  +Created 1970-01-01

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Binary translation  Updated 2025-07-01  +Created 1970-01-01

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Book of Revelation  Updated 2025-07-01  +Created 1970-01-01

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Magnetic moment  Updated 2025-07-01  +Created 1970-01-01

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Mammal subclade  Updated 2025-07-01  +Created 1970-01-01

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Blog comment hosting service  Updated 2025-07-01  +Created 1970-01-01

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Iteration  Updated 2025-07-01  +Created 1970-01-01

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J. Craig Venter Institute  Updated 2025-07-01  +Created 1970-01-01

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Founded by Craig Venter by joining up other existing institutes.

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Program the Micro Bit in C  Updated 2025-07-01  +Created 1970-01-01

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stackoverflow.com/questions/73877965/how-to-compile-c-c-code-into-a-hex-file-for-the-bbc-microbit

Official support is abysmal, very focused on MicroPython and their graphical UI.

The setup impossible to achieve as it requires setting up the Yotta, just like the impossible to setup Compile MicroPython code for Micro Bit locally on Ubuntu 22.04 with your own firmware setup.

So we just use github.com/lancaster-university/microbit-samples + github.com/carlosperate/docker-microbit-toolchain:

docker pull ghcr.io/carlosperate/microbit-toolchain:latest
git clone https://github.com/lancaster-university/microbit-samples
cd microbit-samples
git checkout 285f9acfb54fce2381339164b6fe5c1a7ebd39d5

# Select a sample, builds one at a time. The default one is the hello world.
cp source/examples/hello-world/* source

# Build and flash.
docker run -v $(pwd):/home --rm ghcr.io/carlosperate/microbit-toolchain:latest yotta build
cp build/bbc-microbit-classic-gcc/source/microbit-samples-combined.hex "/media/$USER/MICROBIT/"

.hex file size for the hello world was 447 kB, much better than the MicroPython hello world downloaded from the website which was about 1.8 MB!

If you try it again for a second time from a clean tree, it fails with:

warning: github rate limit for anonymous requests exceeded: you must log in

presumably because after Yotta died it started using GitHub as a registry... sad. When will people learn. Apparently we were at 5000 API calls per hour. But if you don't clean the tree, you will be just fine.

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