But decimal floats fall short in subtle ways. Here is the simplest example - sales tax. In Ontario it's 13%. If you buy two items for $0.98 each, the tax on each is $0.1274. There is no legal, interoperable mechanism to charge the customer a fractional number of cents, so you just can't do that. If you are in charge of producing an invoice, you have to decide where to perform the rounding(s). You can round the tax on each item, which is $0.13 each, so the total is ($0.98 + $0.13) × 2 = $2.22. Or you can add up all the pre-tax items ($1.98) and calculate the tax ($0.2548) and round that ($0.25), which brings the total to $0.98×2 + $0.25 = $2.21, a different amount. Not only do you have to decide where to perform rounding(s), you also have to keep track of how many extra decimal places you need. Massachusetts's sales tax is 6.25%, so that's two more decimal places. If you have discounts like "25% off", now you have another phenomenon that can introduce extra decimal places.

If you do any kind of interest calculation, you will necessarily have decimal places exploding. The simplest example is to take $100 at 10% annual interest compounded annually, which will give you $110, $121, $133.1, $146.41, $161.051, $177.1561, etc., and you will need to round eventually. Or another example is, 10% annual interest, but computed daily (so 10%/365 per day) and added to the account at the end of the month - not only is 10%/365 inexact in decimal arithmetic, but also many decimal places will be generated in the tiny interest calculations per day.

If you do anything that philosophically uses "real numbers", then decimal FP has zero advantages compared to binary FP. If you use pow(), exp(), cos(), sin(), etc. for engineering calculations, continuous interest, physics modeling, describing objects in 3D scene, etc., there will necessarily be all sorts of rational, irrational, and transcendental numbers flying around, and they will have to be approximated in one way or another.

Many technologies have been doing that for centuries. For example: The printing press making books available to the masses, standardized spelling in English (it was a mess before!), radio and TV broadcasting speech thus creating more uniform accents nationwide, the Internet spreading all kinds of information globally instantly, even memes (literally thinking and writing alike).

And then she writes about logic cells, hardware pipeline architectures, RTL (register-transfer level), ASIC floorplans, and tapeout. Building hardware is hard, as iteration cycles are long and fabricating stuff costs real money. That was quite a journey.

Her "about" page adds more context that helps everything make sense:

> Professionally, I have a Masters in Electrical and Computer Engineering, have previously worked as a CPU designer at ARM and an FPGA engineer at Optiver. -- https://essenceia.github.io/about/

Furthermore, it locks down the web browser's settings so that you cannot use a proxy server to bypass Zscaler's MITM. I saw this behavior in Mozilla Firefox, where the proxy option is set to "No proxy" and all other options are disabled and grayed out; I imagine that it does the same to Google Chrome. If you try to modify the browser's .ini(?) file for proxy settings, Zscaler immediately overwrites it against your will. Zscaler worked very hard to enforce its settings in order to spy on the computer user.

And as you'd expect, if you open up the Zscaler GUI in the system tray, you are presented with the option to disable the software if you have the IT password. Which of course, you don't have. Then again, that might be an epsilon better than the Cybereason antivirus software, which just has a system tray icon with no exit option, and cannot be killed in Task Manager even if you are a local administrator, and imposes a heavy slowdown if you're open hundreds of small text files per second.

My article isn't written as a step-by-step tutorial and doesn't come with example numbers. But mine fills in certain things that xnacly doesn't cover: random prime generation, efficiently calculating the decryption exponent d from (n, e) by using a modular inverse, using modular exponentiation instead of power-then-modulo.

By the way for Python, modular exponentiation is pow(x, y, m) (since 3.0), and modular inverse is pow(x, -1, m) (since 3.8, Oct 2019). https://docs.python.org/3/library/functions.html#pow

It really feels like we're slowly marching back to the era of mainframe computers and dumb terminals. Maybe the democratization of hardware was a temporary aberration.

    int crc32(byte[] data) {
        int crc = ~0;
        for (byte b : data) {
            crc ^= b & 0xFF;
            for (int i = 0; i < 8; i++)
                crc = (crc >>> 1) ^ ((crc & 1) * 0xEDB88320);
        }
        return ~crc;
    }

    int crc32(byte[] data) {
        int crc = ~0;
        for (int i = 0; i < data.length * 8; i++) {
            crc ^= (data[i / 8] >> (i % 8)) & 1;
            crc = (crc >>> 1) ^ ((crc & 1) * 0xEDB88320);
        }
        return ~crc;
    }

* https://www.nayuki.io/page/deflate-specification-v1-3-html

* https://www.nayuki.io/page/simple-deflate-decompressor

* https://github.com/nayuki/Simple-DEFLATE-decompressor

Year 2019? Oh no, not by a longshot. I've been using Microsoft-branded PowerToys since Windows XP, around the year 2001. Wikipedia says:

> PowerToys are available for Windows 95, Windows XP, Windows 10, and Windows 11 (and explicitly not compatible with Windows Vista, 7, 8, or 8.1).

https://en.wikipedia.org/wiki/Microsoft_PowerToys

> Death is a popular escape from deferred taxes. When you die, your obligations to the government vanish. Your heirs inherit assets/property at market value. Their assets depreciate from new cost bases.

The article talks about taxes in the USA, and I think the treatment of taxes at death is unfair by giving a significant tax advantage to people who hold assets till death, especially with the step-up basis. The way Canada handles it seems more reasonable to me:

> Capital property generally includes real estate, such as homes and cottages, investments like stocks, mutual funds or crypto-assets, and personal belongings like artwork, collections or jewelry. When a person dies, they are considered to have sold all their property just prior to death, even though there is no actual disposition or sale. This is called a deemed disposition and may result in a capital gain or capital loss

-- https://www.canada.ca/en/revenue-agency/services/tax/individ...

In exchange, Canada does not have an inheritance tax. All taxation is resolved in the estate of the deceased person before the money or assets are passed on without further taxation.

(Yonge, [jʌŋ], [jɑndʒ]); (Strachan, [sdʒɹɑn], [ˈsdʒɹa.tʃæn]); (Tecumseth, [tə.ˈkʌm.zi], [ˈti.kəm.sɛθ]); (Markham, [ˈmɑr.kʌm], [ˈmɑrk.hæm]), (Etobicoke, [ɛ.ˈtoʊ.bɪ.koʊ], [ɛ.ˈtoʊ.bɪ.koʊk]).

See also: https://www.youtube.com/watch?v=i2cyg6bFeRc , https://www.youtube.com/watch?v=1PmeDWvwD8M

> 10 microns

"Micron(s)" is a deprecated word since 1967 and "micrometre(s)" must be used instead. The reason is that it is a non-standard word; if "micron" is accepted, then we should also accept the nonsensical words "millin", "nanon", "kilon", etc. The metric system is supposed to be easy to learn with consistent rules and as few special cases as possible.

> 4.8mm ... 0.01mm ... "0.002mm tolerance"

These numbers are correct, but it's harder to quickly skim the text and make comparisons because the number of decimal places vary. It would be better to write 4.800 mm, 0.010 mm, 0.002 mm to make the reader's job easier. Or convert everything to whole micrometres, like 4800 μm, 10 μm, 2 μm.

> withstand over 4,000 Newtons

Almost correct, but the unit must be decapitalized to "newtons". This is similar to how other name-based units are decapitalized - like "100-watt light bulb", "12 amps", "3 gigahertz".

> 2-3 Newton insertion force

It must be written as "2–3 newtons". When the unit name is written out in full, it follows normal English pluralization rules (e.g. metres, seconds, volts, pascals, kelvins, ohms, teslas). The only exceptions are hertz and siemens, because they already end with -s or -z.

DVDs and BDs are digital.