In a mathematical sense (ignoring things like nonstandard analysis), 0.999999... is exactly 1, as the absolute difference between them must be smaller than any positive real number, and the only non-negative real number smaller than every positive real number is 0. Since the difference is 0, the two are equal.

Invoking stuff like this
and this, which is gibberish to me, does not help. I'm sorry, I guess I just don't understand.

Thank you for trying though.

Real numbers are what most people think of when they think of numbers. It includes every number from negative infinity to positive infinity, including fractions and things like pi; it does not include the square roots of negative numbers (those numbers are called imaginary numbers). Positive real numbers are those real numbers that are greater than zero.

In other words, for the average layperson, you could just treat the phrase "real number" as if it were just "number".

Some characteristics of the real numbers that are important for the discussion we're having:
* They're ordered. Given two real numbers, if they're not equal, one is greater than the other.
* They're dense. Between any two non-equal real numbers, there's another real number in between them.

It's worth noting that, even if we restrict our attention to the rational numbers (those that can be represented with terminating or repeating decimals), nothing changes in this discussion, as they still have the properties I just mentioned.

(Giving a rigorous definition of "real number" is actually quite tricky, which is probably why the article is hard for you to understand.)

While I have your attention...

In regard to Pi in equations. Is it possible to alter the unitary from 1 to Pi? I mean for example, if the number system is defined as multiples of Pi (1=Pi, 2=2Pi, 3=3Pi etc..). Then wouldn't that make such equations simpler and highlight the other factors that might be more interesting/relevant?

So, the equation for the area of a circle would be 1r^2 or simply r^2 (In Pi units.) Note, there was no need to choose a precision for Pi in this calculation. And rounding errors will not accumulate. When/if it becomes necessary, the units can be converted back to normal notation after all the complicated stuff has been calculated.

Could this provide some sort of advantage with long and complicated calculations that feature many instances of Pi?

YES! Good spotting there.

https://cdn.quotesgram.com/img/88/76/611780195-analyze-this.jpg

Except for the problem that you'd need to change the exponent to (2 / pi) for this to work.

Now, what *would* work is to use not pi, but rather tau = 2 * pi. If you look at many equations, you'll find that 2 * pi appears very frequently, so some say that tau would be a more natural constant to use.

(The only time I naturally see pi working in an exponent is in the imaginary component, as we have the famous equation
e^(i * pi) + 1 = 0
where i is the square root of -1 (or, to be precise, one of the square roots, and it doesn't matter which one we designate as i).)

You've been very informative. Thank you.

You're welcome.

Math really is an interesting and beautiful subject.

Thank you. Excited I am.


Disappointed that the notification bug hasn't been fixed yet.

Why did the lunar lander have four legs? Three would have been lighter and a lot more stable on uneven ground.

Okay, a quick google found this, via Wikipedia. This leads me to believe that rather than providing extra stability, the extra probe found on the fourth leg allowed for a quicker confirmation and greater certainty that the LM was in direct contact with the lunar surface, so that the commander could power down safely. Especially with the decision to remove the probe from the ladder leg; had there been only two such probes available instead of three, the chances would've been greater that one could malfunction, leading to a mission abort instead of a safe landing.

Thanks for trying, but I don't buy it. The precise landing location could not be predicted, and providing stability for an uneven landing location would trump everything else. Taking off would be impossible with too great a tilt, or a wobbly platform. There has to be a better reason.

I can speculate that four legs mean that if one fails, they still have three. Redundancy is an important part of such a dangerous, no-hope-of-rescue mission. But I'm just speculating. I'd love to hear from anyone with any better ideas.

Has the purple dot for the forum replies been fully fixed? I no longer see it linger anymore. Just want to confirm that the problem is permanently fixed and not just temporarily.

I still have it. :(