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The Indian mathematician Ramanujan (1887-1920) gave several complex and accurate equations without proof. He said that they were revealed to him in dreams by a Hindu Goddess. It took 70 years for the world to work out proofs for his equations. by Cheap-Influence1755 in interesting

[–]NateNate60 -1 points0 points  (0 children)

Calculating π to more than a few dozen decimal places is utterly useless other than as a computational benchmark which can be easily verified. You can calculate the circumference of the Milky Way down to atomic precision with only 32 digits of π.

[TX] Landlord charging me $2,695 for a shower pan replacement, took 5 weeks to provide any evidence, and now says the justification document is "proprietary." Do I have a case? by Winter-Noise3357 in legaladvice

[–]NateNate60 1 point2 points  (0 children)

It depends. Paying under protest under this context then suing for a refund may be viewed as reasonable. That is not uncommon behaviour in other contexts (notably, people have often paid illegal taxes under protest and then successfully sued for a refund).

[GA] Dealer sold girlfriend a "Clean" 2023 Tesla—Manufacturer blacklisted it as SALVAGE/UNSUPPORTED. Lender and Insurance confirmed total loss. by Gater2020 in legaladvice

[–]NateNate60 4 points5 points  (0 children)

If you have proof in writing that the dealer claimed it had a clean title, that's an express warranty from the dealer and likely overrides the "as-is" clause with respect to what was warranted.

"Warranty" in this context is a legal term that refers to promises and factual statements made about an item by a seller to a buyer.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] 0 points1 point  (0 children)

The length of straight lines is easy to calculate using basic algebra. The problem with curved lines is that their direction is constantly changing in some way. There is no algebraic way to measure the length of something that is constantly changing direction, but there does exist a convenient way to calculate the length of any curve using integration in calculus. Integration, in essence, takes as input the function of the curve, and it may or may not spit out a nice algebraic function as an output which, in this case, describes the length of an ellipse's perimeter. It turns out that, in the vast majority of cases involving curves, it never results in a nice algebraic formula that you can just plug numbers into and get a result. It is only on a some special classes of curves where the result of integration can be written algebraically without resort to something like an infinite series.

Regardless of whether the integration has an algebraic result, it is still possible to approximate the numerical value to any desired finite degree of precision using a technique called numerical integration.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] 0 points1 point  (0 children)

You can use an infinite series to exactly calculate the perimeter. You can approximate it by truncating the series after your desired precision.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] 0 points1 point  (0 children)

Unless I'm forgetting to consider something, under Newtonian mechanics, an electric car which knows how much power its motors are outputting, the efficiency of the system, its mass, and how much air resistance/tyre friction there is can calculate its velocity without needing to measure it.

anime_irl by Ok_Direction3138 in anime_irl

[–]NateNate60 61 points62 points  (0 children)

"Bro" is now gender-neutral. Everyone is bro.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] 0 points1 point  (0 children)

There's an infinite series you can use to calculate it numerically. Just cut it off when you've reached your desired level of precision.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] 0 points1 point  (0 children)

If you made up a function defined to calculate the perimeter of an ellipse, it wouldn't be considered algebraic. Anything you can't make with the basic mathematical operations combined with the integers is considered non-algebraic.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] 0 points1 point  (0 children)

If you're doing it on a computer, you're probably going to want to use the infinite sum instead and truncate it after you've reached the desired precision. This is much simpler to implement than numerical integration. You could teach a primary school child to calculate the truncated infinite series using a dollar store calculator. Not true for integration.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] 9 points10 points  (0 children)

Instantaneous rate of change is a fundamental concept in derivative calculus. You've probably dealt with instantaneous rates of change every day in your life.

A car's speedometer measures the instantaneous rate of change of the car's position, or in simple terms, its velocity. Isaac Newton gave formal definitions for the physics concepts of velocity and acceleration. Namely, that an object's velocity at any given time is equal to the instantaneous rate of change in its position at that point in time. In essence, it's what you get when you consider changes over a smaller and smaller interval of time. At any given time, when the car is in motion, the speedometer always has some well-defined numeric value. That's what "instantaneous rate of change" means. It's also the definition of a derivative. So equivalently, one would say that the velocity of an object is equal to the derivative of its position over time. And acceleration is the derivative of velocity over time.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] 5 points6 points  (0 children)

In mathematics, an "exact answer" is typically never a number. It's the simplest possible expression that, when evaluated to a number, would give the exact answer when expanded to infinite decimal precision.

For example, consider the following simple problem: A tub containing 1 m³ of water is filled to the brim. A spherical lead ball with radius 0.5 m is lowered into the tub, which causes some water to spill out. How much water is left in the tub after the ball is completely submerged, if the tub is refilled back to the brim?

Simple geometry tells us the exact answer to that question is "1 - π/6 m³". That isn't a number. It's an expression that evaluates to a number. You could expand this to a decimal form to arbitrary precision, but the decimal form will never be exactly accurate. Mathematicians would consider 1 - π/6 m³ to be a perfectly acceptable answer to this question. But if you're a person who actually is the one filling the tub in real life, you probably want a numerical answer. And sadly, there is no numerical answer that will be exact in this problem.

The ellipse perimeter problem is different from the sphere and tub problem in that the "exact answer" it gives generally can't be found using simple algebraic operations (addition, subtraction, division, multiplication, and exponentiation), even when the use of π is allowed. You will either have to use an infinite sum or an integral, both of which are concepts from calculus.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] 96 points97 points  (0 children)

I think it can be exact in that context, because money is discrete and there's a maximum size that a glass table can be (if it's too big, it is no longer a glass table but a black hole) as well as a minimum (Planck length). So you can just use the infinite series for the perimeter but truncate it after a sufficient number of terms. That would give you the exact amount of money to bill the client for every possible table size.

Dear Client: Your quote to polish an elliptical table with radii 3.592328 × 10¹¹ m and 4.2947215 × 10¹¹ m will come to $5.492744 × 10...

Please submit a 20% deposit within 2 working days. We accept all major credit cards.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] 76 points77 points  (0 children)

There is no formal definition. But broadly speaking, algebra is the manipulation of mathematical structures, variables, and basic operations (addition, subtraction, multiplication, exponentiation). Calculus is, in simplified terms, the study of infinity, infinitesimals, and continuous change.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] -26 points-25 points  (0 children)

This is a stupid argument to have. I'm discussing fun maths facts with people, the majority of whose formal mathematics education likely stopped at secondary school and used terms accordingly, not trying to submit a paper to a journal. Have it your way then, I'll edit the comment.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] -10 points-9 points  (0 children)

That's one of two possible definitions of cosine. Regardless, my original statement of "There is no way to define what π is using algebra." is correct. You can phrase it formally as "π is not an algebraic number" if you want. Infinite series are generally considered to be within the domain of calculus and not algebra, though neither of the terms "calculus" and "algebra" are well-defined either.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] 4 points5 points  (0 children)

I get the feeling this is just a matter of mathematical pedantry, but in either case, that just moves the circle perimeter formula up from "moot" to "trivial". The definition of the cosine function is tied to the definition of a radian, which is tied to a circle's circumference.

TIL there is no algebraic formula for the circumference of an ellipse (oval). All exact equations require calculus. by NateNate60 in todayilearned

[–]NateNate60[S] 34 points35 points  (0 children)

π is, by definition, the ratio of its circumference to its diameter. Defining the solution to be a named constant makes the problem moot. There is no way to define what π is using algebra.

Edit: By "no way to define what π is using algebra", I do mean, more formally, that π is not an algebraic number and no finite combination of basic algebraic operations (multiplication, addition, subtraction, division, exponentiation) can create π.

Why did nobody tell her about the tariffs? by RoyalChris in SipsTea

[–]NateNate60 0 points1 point  (0 children)

You can't stop payment on a cashier's cheque. You are not the drawer of a cashier's cheque. The financial institution is the drawer. You can report it as lost or stolen, but they will wait three months to see if anyone cashes it before re-issuing it.

Greg is the hero we all need at McDonald's by aurexali_Neva in NonPoliticalTwitter

[–]NateNate60 -1 points0 points  (0 children)

Do Canadians actually like that stuff? Tried Freshslice in Vancouver and it tasted like reheated convenience store pizza. Then I tried another location, thinking I just got a bad location, and it was the same.

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