Questions about this video on Taylor series

[PainterGuy]

Hi,

I was watching a video on the origin of Taylor Series shown at the bottom.

Question 1:
The following screenshot was taken at 2:06.

The following is said between 01:56 - 02:05:
Halley gives these two sets of equations for finding nth roots which we can generalize coming up with one irrational and one rational form.

Why are two forms, rational and irrational, needed? Could you please help me with it?Question 2:
The following screenshot was taken at 2:25.

The following is said between 01:56 - 02:24.
Halley gives these two sets of equations for finding nth roots which we can generalize coming up with one irrational and one rational form. Let's look at the irrational. Let's use our earlier example of 1337 and we'll still use the values of eleven and six for a and b. Plug everything in. We get this number. Now we can repeat the calculation using that number for a and whatever would be the remainder we call b. Plug everything in. We get this number which is accurate to fifteen digits.

I don't see how the author comes up with b=0.00000449553611. Also, 1337 ≠ {11.0165041631289^3 + 0.00000449553611}. Could you please help me with it? Thank you!

 

[jedishrfu]

11.0165041631289^3 + 0.00000449553611 computes to 1337.00000899 which is pretty darn close.

so a = 11.0165041631289

so b = 11.0165041631289^3 - 1337 = 0.00000449553

using google calculator.

However, the video narrator's calculator must compute to a higher digital precision.

 

[PainterGuy]

Thank you.

so b = 11.0165041631289^3 - 1337 = 0.00000449553

b = 11.0165041631289^3 - 1337 = 0.00000449553
⇒ 11.0165041631289^3 - 1337 - 0.00000449553 = 0
⇒ 1337 = 11.0165041631289^3 - 0.00000449553
⇒ (1337)^1/3 = {11.0165041631289^3 - 0.00000449553}^1/3

In the video it's:
(1337)^1/3 = {11.0165041631289^3 - 0.00000449553}^1/3

Where am I going wrong?

 

[Mark44]

Thank you.
b = 11.0165041631289^3 - 1337 = 0.00000449553
⇒ 11.0165041631289^3 - 1337 - 0.00000449553 = 0
⇒ 1337 = 11.0165041631289^3 - 0.00000449553
⇒ (1337)^1/3 = {11.0165041631289^3 - 0.00000449553}^1/3

In the video it's:
(1337)^1/3 = {11.0165041631289^3 - 0.00000449553}^1/3

Where am I going wrong?

Are you claiming that the last two lines are different? They look exactly the same to me.

 

[berkeman]

They look exactly the same to me.

Well, the minus sign is red in one of them, after all...

 

[Mark44]

They look exactly the same to me.

Well, the minus sign is red in one of them, after all...

Modulo the color of the minus sign, of course...

 

[PainterGuy]

Thank you! :)

It should have been a "+" sign instead.

In the video it's:
(1337)^1/3 = {11.0165041631289^3 + 0.00000449553}^1/3

Where am I going wrong?

 

[pbuk]

Thank you! :)

It should have been a "+" sign instead.

In the video it's:
(1337)^1/3 = {11.0165041631289^3 + 0.00000449553}^1/3

Where am I going wrong?

The video is wrong, $11.0165041631289^3 > 1337$ so in the next iteration you clearly need to subtract the difference. The video is also inconsistent in the use of the equals sign when it should be $\approx$.

Use a proper book, not carelessly prepared and inaccurate videos.

 

[PainterGuy]

Thank you!

Could you please also comment on Question #1 from my first post?

 

[pbuk]

Question 1:
The following screenshot was taken at 2:06.

View attachment 265456

The following is said between 01:56 - 02:05:
Halley gives these two sets of equations for finding nth roots which we can generalize coming up with one irrational and one rational form.

Why are two forms, rational and irrational, needed? Could you please help me with it?

It's not that two forms are needed, the point is that Halley provides equations for the upper and lower bounds one of which they (both the video and Halley) refer to as a 'rational form' and the other an 'irrational form'.

Bear in mind

1. Halley wrote in Latin
2. I have no idea how good the translation is that the video is quoting from
3. I do have an idea how good the video is, and as mentioned above I don't think it is very good
4. Mathematics has changed a lot in 300 years

What is your motivation for analysing this in depth? This particular point is of historical interest only, and if you want to learn History of Maths don't try to do it from a video produced by a computer science postdoc. If you just want to lean computational root finding methods, move on.

 

[PainterGuy]

Use a proper book, not carelessly prepared and inaccurate videos.

Actually I did let the presenter know about the mistake and I thought it'd be a good idea to quote his reply for the sake of completeness.

That b value is likely negative (I'll go through this again just to be sure). This would not be the first time I have made a mistake in a video. I actually have made a video about mistakes in videos and I will likely make another mistakes video soon. There are actually two other mistakes in that Taylor Series video that nobody has commented on but I noticed after making it. The fact that James Gregory was 36 (not 37) when he died and an incorrect notation of the numerator when formalizing the series.

...

I took a closer look and indeed that b value should be negative, however that is not where the story ends. If you used the positive b value then the resulting approximation would be 11.0165041754762 and when cubed giving 1337.0000089910657 off by 6 decimal places. But if you used the correct negative value for b, it results in the value given in the video accurate to 15 places. In short, I used the negative value in my calculation but wrote it as positive on the slide. I should mention that Halley also makes a numerical error in his paper, which the translation points out, and Gregory also made some mistakes in his letter to Collins. As the saying goes, even monkeys fall from trees.

What is your motivation for analysing this in depth?

Just out of interest.I have one last question. Would really appreciate if you could help me with it.

The square root of "2" is an irrational number. Why is the form in yellow called irrational? Is the irrational form in yellow always going to result into an irrational number? Would there be any non-zero values of "a" or "b" which would make the so-called irrational form a rational one, or which would make the so-called rational form an irrational one?

 

[pbuk]

The square root of "2" is an irrational number. Why is the form in yellow called irrational?

Because it has a term under a square root. Incidentally Halley's original paper (in Latin) is https://www.jstor.org/stable/102449, this is on page 141.
And while I'm giving links, here are more authoratitive links describing these methods:

• https://mathworld.wolfram.com/HalleysMethod.html
• https://mathworld.wolfram.com/HalleysIrrationalFormula.html

Is the irrational form in yellow always going to result into an irrational number? Would there be any non-zero values of "a" or "b" which would make the so-called irrational form a rational one, or which would make the so-called rational form an irrational one?

Remember we are using these formulae to approximate a root, and Halley was doing these computations by hand, so we will always be working with rational approximations: $11.0165041631289 = \frac{110165041631289}{10^{13}}$.

It would be useful for you to work out the answers to your questions yourself, I'd start with the rational form:

Can we choose $a, b \in \mathbb R, n \in \mathbb N$ to make Halley's rational formula yield an irrational number? What about $a, b \in \mathbb Q$ (noting the fact that our approximate terms are always rational)?