Were you taught square root extraction at school?

[epenguin]

And what are the methods?

This was stimulated by

Homework Statement

Determine a positive rational number whose square differs from 7 by less than 0.000001 (10^(-6))

This question could be rephrased as "use the method for finding square roots you were taught in school to find √7 to 6 decimal places".

If you were taught at school. If you were not taught at school you were luckier than me, I was. It has had a negative energy for me for a long time since. I forgot by the next school year and it became something I couldn't touch. But square roots, inevitable in quadratic equations, trig, calculus, SHM and oscillations were all OK because by that time I had a slide rule, logarithms, tables (long before pocket calculators). But the arithmetical method was like a sinister room with something lurking that I've always hurried past, swept under carpet. Sure I'm not the only one? Actually the slide rule, logarithms, tables would not have been up to solving the above problem (but I would have smugly said never needed practically).

The trouble was that although the method clearly worked it was never explained why. I mention all this because I am curious to know how many people were taught the same way? And how many have been taught nothing at all about it at school?

In much later years I thought about it and I thought oh well you can just use for e.g. √7: if x is an approximate root let (x + a) be the exact root. Then (x + a)² = 7 .
Now (x + a)² = x² + 2ax + x²
Approximate this as x² + 2ax
Then the difference between x² and 7 is approximately 2ax and you can get a (involves division )
you just continue that getting closer and closer.
Actually doing it, maybe for the first time and starting with x = 2 - very crude, I would better have started with 3 - I got by third step to 2.64575 which squared is 6.99999. Sounds a bit too good actually maybe there is some fluke.

It can't be the school method quite as there can be subtractIons which I don't remember. I think in any case it is not efficient and a hard slog to get to the precision required in Curieuse's question.
You could program a calculator to do that in a wink and then inefficiency would not matter.
- except I believe it does sometimes and there are applications like statistics and stock flow control and for all I know protein or other dynamics where they have to be calculating zillions of square roots per second.

So what are the methods?

I am also curious about this: I looked into calculating π recently and the most obvious method, same as Archimedes I think, involved calculating a lot of square roots. Quite time consuming. Did he do that? It also seems easier with decimals than rational fractions, or maybe it is just more natural for us.

 

[Mark44]

I learned how to calculate square roots in school along about 8th grade. This was well before the advent of pocket calculators and personal computers and such. The method I learned is shown here -- https://en.wikipedia.org/wiki/Methods_of_computing_square_roots -- in the section titled "Examples."

 

[micromass]

I never learned to calculate square roots. I think it is useless to do it in a young age. We have calculators right now, so being able to do it efficiently is clearly not very important. The only fun thing is to understand why the method works, which would not be suitable to teach young children. I think once you're in a calculus class, they should teach you different ways of calculating roots, and sines, and exponentials, etc.

 

[Mark44]

I never learned to calculate square roots.
I think it is useless to do it in a young age.

It wasn't useless when I learned it since the only calculating devices that were readily available were slide rules.

We have calculators right now, so being able to do it efficiently is clearly not very important. The only fun thing is to understand why the method works, which would not be suitable to teach young children. I think once you're in a calculus class, they should teach you different ways of calculating roots, and sines, and exponentials, etc.

 

[micromass]

It wasn't useless when I learned it since the only calculating devices that were readily available were slide rules.

Sure. Actually, working with a slide rule is fun. I think that should still be taught to give more intuition for the logarithm.

 

[Mark44]

Sure. Actually, working with a slide rule is fun. I think that should still be taught to give more intuition for the logarithm.

I agree. It also gives people a better sense of orders of magnitude for their answers, actually needing to figure out where the decimal point goes.

 

[epenguin]

Sure. Actually, working with a slide rule is fun. I think that should still be taught to give more intuition for the logarithm.

There might be an idea here for some teacher - my Dad told me his teacher had a half-wall-sized slide rule fixed up in the classroom and used that for teaching. This would have been around 1920 ±. I guess individual ones not so easy to find for a class nowadays.

 

[Mark44]

There might be an idea here for some teacher - my Dad told me his teacher had a half-wall-sized slide rule fixed up in the classroom and used that for teaching. This would have been around 1920 ±. I guess individual ones not so easy to find for a class nowadays.

I think the big ones are still around. One of the classrooms I used in the 80's and 90's had one that was about 5 feet long.

 

[SteamKing]

I think the big ones are still around. One of the classrooms I used in the 80's and 90's had one that was about 5 feet long.

The physics class room at my HS had one of these giant slide rules above the chalk board. I never saw the instructor use it, 'cuz calculators had just become available at prices a student could afford.

As far as extracting square roots by hand, I learned this from my father, who was an engineer. There's also a manual method of extracting cube roots, but it is much more cumbersome, even with a calculator to help with the arithmetic.

Sure, everybody has a calculator to crunch numbers, but such devices do little to engender understanding the nitty gritty of arithmetic. This isn't "fun" (since when did everything have to be "fun" to be worthwhile?), bit it is necessary later as one takes more advanced math courses. I haven't seen any math geniuses claim they owe their success to the kind of calculator they used.

Things like long division, extracting roots by hand, etc., expose one to what an algorithm is and why such things are useful even for non-math related things. All of this is suspect, even down right discouraged in modern pedagogy, because one learns algorithms by rote, as if that's the worst thing in the world. After all, we learn our names and our addresses by rote (unless they're tattooed on the back of our necks), so it can't be that bad.

 

[Mark44]

Things like long division, extracting roots by hand, etc., expose one to what an algorithm is and why such things are useful even for non-math related things. All of this is suspect, even down right discouraged in modern pedagogy, because one learns algorithms by rote, as if that's the worst thing in the world. After all, we learn our names and our addresses by rote (unless they're tattooed on the back of our necks), so it can't be that bad.

One of the other instructors where I taught for 18 years had that bias against rote learning, which he called "vacuous drill," a pejorative if I've ever heard one. He used that phrase so often that I suggested he could save time by referring to it as VD.

 

[Curieuse]

This question could be rephrased as "use the method for finding square roots you were taught in school to find √7 to 6 decimal places".

Yes i should've done that really! But being blinded by all the algebra i failed to see it was the decimal which was to just be limited at some point! :P
I learned the method explained in this page
http://www.basic-mathematics.com/square-root-algorithm.html
I have no idea why it works, but I'll try to debunk it! Seems like a continuous fractions thing.. i don't know!

 

[Mark44]

Yes i should've done that really! But being blinded by all the algebra i failed to see it was the decimal which was to just be limited at some point! :P
I learned the method explained in this page
http://www.basic-mathematics.com/square-root-algorithm.html
I have no idea why it works, but I'll try to debunk it! Seems like a continuous fractions thing.. i don't know!

You're not going to have any success in trying to debunk this method, because it works. The technique is not based on continued fractions - my understanding is that it is a variant of Newton's method of finding the root of a quadratic polynomial.

 

[micromass]

Yes i should've done that really! But being blinded by all the algebra i failed to see it was the decimal which was to just be limited at some point! :P
I learned the method explained in this page
http://www.basic-mathematics.com/square-root-algorithm.html
I have no idea why it works, but I'll try to debunk it! Seems like a continuous fractions thing.. i don't know!

God, I absolutely hate explanations of algorithms like this without detailing why they work. It reminds me of elementary school where I had to learn division using an algorithm that I had no clue where it came from. I did not like that. (not that I would have understood the explanation behind the algorithm at that time).

 

[SteamKing]

Yes i should've done that really! But being blinded by all the algebra i failed to see it was the decimal which was to just be limited at some point! :P
I learned the method explained in this page
http://www.basic-mathematics.com/square-root-algorithm.html
I have no idea why it works, but I'll try to debunk it! Seems like a continuous fractions thing.. i don't know!

That is the method which is being discussed here.

Actually, epenguin gave a pretty good explanation in Post #1 of this thread. The hand method of extracting a square root is based on the expansion of the binomial (x+a)².

 

[SteamKing]

Now that people are curious about these methods and algorithms, let's pull back the curtain a little bit.

I came across these pages from the NIST website, no less, which explains a little of the algebra behind the manual extraction of square roots:

http://xlinux.nist.gov/dads/HTML/squareRoot.html

For the terminally curious, there is a companion page on the similar (but more complicated) method to extract cube roots manually:

http://xlinux.nist.gov/dads/HTML/cubeRoot.html

 

[Curieuse]

You're not going to have any success in trying to debunk this method, because it works.

Decode
My bad!

 

[epenguin]

I never learned to calculate square roots. I think it is useless to do it in a young age. We have calculators right now, so being able to do it efficiently is clearly not very important. The only fun thing is to understand why the method works, which would not be suitable to teach young children. I think once you're in a calculus class, they should teach you different ways of calculating roots, and sines, and exponentials, etc.

√

The age at which I learnt, or really didn't, square roots was about 11. (Not a bad teacher on the whole Mr. Ross.)

I think it's wrong, since you are going to use square roots a lot, to leave it a dark mystery. It can be done without calculus or such advanced things quite easily. Just use a picture at first. I'll take an easier one, √10 .

Here the area of the big square is 10, made up of a square of the next lowest perfect square, 9, and the bordering strip of unknown width x₁, but with total area 1.

However we can see that a reasonable approximation to the border area is the rectangular strips without the top right corner. Total area of these rectangles is obviously 6 × x₁. And as we require this border area to be 1, then x₁ = 1/6 ≈ 0.166667 (or other decimal approximation). So the length of the side of the large square is 3 1/6 = 19/6 ≈ 3.1666667. That squared is 10.02778, already not bad.

Make more squares: The outside one is the one we created before, side 3.1666667 and area 10.02778 and the inside one is one with area nearer to 10 we try to create by lopping off that border of area 0.0277. Which we can't do exactly but again the area of the narrow rectangles without the conrner square x₂ ×2 × 3.1666667 is an approximation to it - to 0.0277. So we can get x₂ .I got 0.004386, Subtracting that from the side (19/6) of the large square gave me 3.16228. and the square of that is 10.00002 - pretty good!

The method is essentially the same as those given by Curieuse and others, at least the idea is, you can have slightly different recipes. But who ever remembers the recipe? whereas I think if this had been illustrated to me with the pictures in school I would not have forgotten it and always been capable of reconstructing it.

But it gets better IMHO. At school I did these calculations and ever since I used for square roots decimal approximations of so many figures. I might not have done anything else if I had not heard some of Wildbergers lectures, who talks up rational numbers. And I seem to see the square root calculations are just as easy to do, if not more so, with rational rather than decimal approximations. (There is a nice cancellation on the way). Thus we've seen the second approximation to √10 is 19/6. The third approximation is
(2 × 19² - 1)/(12 × 19).
Or (19 - 1/(2×19))/6
I like it!*
OK that's just an approximation to √10 but it's exactly the approximation, not approximately! Which brought out something till now subliminal, that these decimal square roots had always been lacking in aesthetic and repulsive messy engineering numbers, OK all your measuring instruments have them, but these square root decimal numbers do contrast in aesthetic with the mathematical and physical theories that give rise to them and maybe I begin to see what Wildberger is on about.* I guess if I work this out generally it's going to be the continued fractions that were already mentioned?

 

[pbuk]

God, I absolutely hate explanations of algorithms like this without detailing why they work. It reminds me of elementary school where I had to learn division using an algorithm that I had no clue where it came from. I did not like that. (not that I would have understood the explanation behind the algorithm at that time).

I would generally agree with you, but for the particular instance of computation of a square root by Newton's method I disagree. Why? Because if we have an approximation $a$ to $\sqrt x$ then we know that a better approximation lies in the interval $[a, \frac xa]$ and so we can choose the midpoint of this interval $a' = \frac{\frac xa - a}{2}$ as our next approximation, and we find that this converges satisfyingly quickly.

This can be done with no knowledge of calculus, limits or even expansion of polynomials; I learned it aged 11 and it (and similar diversions) kept me interested in maths until more interesting and powerful things came along.

 

[SammyS]

...

In much later years I thought about it and I thought oh well you can just use for e.g. √7: if x is an approximate root let (x + a) be the exact root. Then (x + a)² = 7 .
Now (x + a)² = x² + 2ax + x²
Approximate this as x² + 2ax
Then the difference between x² and 7 is approximately 2ax and you can get a (involves division )
you just continue that getting closer and closer.
Actually doing it, maybe for the first time and starting with x = 2 - very crude, I would better have started with 3 - I got by third step to 2.64575 which squared is 6.99999. Sounds a bit too good actually maybe there is some fluke.

epenguin, Thanks for starting this thread.

It's not a fluke that your method converges fairly rapidly. As it turns out, the sequence of approximations you obtain with your method are exactly the same as results form Newton's method.