## Roots

Discussions concerned with knowledge of measurement, properties, and relations quantities, theoretical or applied.

### Roots

Why the hell is there TWO ways of writing roots? What is the point?

What is the reason for two different forms of notion?

I have never really thought about this before, but assume there is some practical and/or historical reason for this?

Does anyone know the reason. Practically it seems obvious that is easier to write one more than the other clearly(writing to the power can geta little cluttered sometimes), other than that I don't see the point.

Resident Member

Posts: 5570
Joined: 14 Mar 2012

### Re: Roots

Can you provide an example? Are you talking about writing it as a fraction?

I think the fraction form might be useful to move something to the other side of a division sign/line by adding a negative.

And the notation form where you have the check mark looking thing can be useful for bringing all like roots under the same umbrella. I suppose you could combine them in fraction form as well though.

That's my thoughts after not having done much if any math of that type for a couple years now.

zetreque
Forum Moderator

Posts: 3723
Joined: 30 Dec 2007
Location: Paradise being lost to humanity
Blog: View Blog (3)

### Re: Roots

It might be easier to visualize.

Sqrt(2*2*2*2) instead of 2^1/2 * 2^1/2 * 2^1/2 * 2^1/2

easier to write too.

zetreque
Forum Moderator

Posts: 3723
Joined: 30 Dec 2007
Location: Paradise being lost to humanity
Blog: View Blog (3)

### Re: Roots

∛2² = 2^(2/3)

Why use one over the other? It seems to me that ∛4 and 3√4 are easy to confuse at a glance. I am question the use of the √ sign.

Your example above can be written 4x^(1/2), in your case where x=2 right?

Does the use of √ exist simply because it is quicker to write a large "tick" rather than bracketing and then having to write fractions? If so then it's plain crazy! haha!

I am serious about this? IS there a mathematical reason? Or is this just a matter of writing convenience? I cannot see a reason for using two different ways of writing the same thing.

At least with 1/x and x^-1 I understand it makes equations easier to juggle around rather than having to write out things like 1/2/3 and instead write 2/3^-1 then understand that it means 3/2.

Honestly using √ ends up distracting me quite a lot. I find myself stopping in my tracks and losing my stream of thought. It would help a great deal if I understand its use in order to assimilate it into my mathematical lexicon without having to wrestle with the bloody thing all the time (I am a stubborn bastard, I need to know the reason if there is one!)

Resident Member

Posts: 5570
Joined: 14 Mar 2012

### Re: Roots

may you possibly accept, Badger, that the idea of the square root was already known to Pythagoras, and that the whole idea of irrational numbers was got at (by him) through that concept?

so, the square root symbol is "mathematical" because it was introduced, like the times (x) or divide (: or fraction line) as the symbol for a primary mathematical operation.

You yourself say:
BadgerJelly » August 24th, 2017, 8:04 am wrote:∛2² = 2^(2/3)
At least with 1/x and x^-1 I understand it makes equations easier to juggle around rather than having to write out things like 1/2/3 and instead write 2/3^-1 then understand that it means 3/2.

Apart from the fact that you should use parentheses [because actually 1/2/3 = 1/6 because it is computed 1/2 and then /3: the thing you refer to is written 1/(2/3); similarly, 2/3^-1 = 6 because powers have precedence on multiply and divide, and it is computed 2/(3^-1): the thing you refer to is written (2/3)^-1], you are yourself giving the reason why we use the root symbol: it is the same reason why we use the fraction sign. It is not needed at all; the multiplication and powers suffice. Still, when you just wish to indicate a/b, it is simpler to write it this way.

neuro
Forum Moderator

Posts: 2621
Joined: 25 Jun 2010
Location: italy

### Re: Roots

Neuro -

So it is essentially a tradition? I guess if that is the case I'll just have to learn to be more accustomed to it like the ampersand and and. Actually, that comparison has just made the whole thing a little more easier to swallow :)

Sorrt about the mistakes. I am used to writing by hand with math symbols. I guess I'll just have to learn to be much more diligent with my equations when writing on a computer.

Resident Member

Posts: 5570
Joined: 14 Mar 2012

### Re: Roots

Thank you for that explanation of why there are two ways of writing roots. For the life of me, I could not figure out that question because I was thinking about the thread for universal languages and couldn't think of a second way to spell "roots".

Math excites me but I don't understand it. However, 1/2/3 = 1/6 is awesome. I think may be math is being taught better today than in the past, because I think my great grandson has a better understanding of math as an abstract than the grade school math I remember. My understanding of math is very concrete like during the time of history when there was no zero because a number represented what exists, not what does not exist. When we move from the concrete to the abstract it starts looking magic to me and that is delightful. I would wish I could understand this magic.

So it is essentially a tradition?

1/2/3 = 1/6 is seeing the same thing in different ways and while I don't understand math, it seems to me the different ways of writing roots, is also different ways of seeing the same thing.
Athena
Banned User

Posts: 1865
Joined: 22 Apr 2012
Location: Eugene, Oregon

### Re: Roots

Athena, just for your excitement, I may point your attention to the fact that the quite trivial number -1 can be written by making use of exponentials and imaginary numbers (and pi, by the way), as:

exp(π·í) = exp(-π·í) = i^2 = -1

and, as a consequence, pi can be computed as

π = 4·arctan(1) = -i·log(-1)

This may now lead Badger to ask why the hell do we need pi if it can be indicated as the logarithm of -1 times -i, but possibly this is the nice thing of maths: in the end, it is quite redundant!

neuro
Forum Moderator

Posts: 2621
Joined: 25 Jun 2010
Location: italy

### Re: Roots

Athena » August 24th, 2017, 10:08 am wrote: For the life of me, I could not figure out that question

Possibly because it is trite and pointless and the poser couldn't either.

Thank you for that explanation of why there are two ways of writing roots.

What explanation? No matter. The answer is irrelevant.
rajnz00
Member

Posts: 270
Joined: 28 Dec 2016

### Re: Roots

neuro » August 25th, 2017, 8:07 am wrote:Athena, just for your excitement, I may point your attention to the fact that the quite trivial number -1 can be written by making use of exponentials and imaginary numbers (and pi, by the way), as:

exp(π·í) = exp(-π·í) = i^2 = -1

and, as a consequence, pi can be computed as

π = 4·arctan(1) = -i·log(-1)

This may now lead Badger to ask why the hell do we need pi if it can be indicated as the logarithm of -1 times -i, but possibly this is the nice thing of maths: in the end, it is quite redundant!

Thank you, that was delightful to me. And unfortunately I have no idea what n = 4.arctan(1) = i.log(-) means. Perhaps this is like a child being delighted by the process of knitting and wanting to know how to do it, but finding it quite impossible to figure out how to do it, even when it is demonstrated several times. Some things are so simple once they are learned, but the challenge of learning can seem impossible. But I do get, you are talking about the relationship between the number 1 and Pi.

Now if the discussion between those of us who do not know and those of us who do know continues and it is okay to be as a child full of wonder, perhaps those of who don't know will transition into being among those who do know.
Athena
Banned User

Posts: 1865
Joined: 22 Apr 2012
Location: Eugene, Oregon

### Re: Roots

rajnz00 » August 25th, 2017, 11:43 am wrote:
Athena » August 24th, 2017, 10:08 am wrote: For the life of me, I could not figure out that question

Possibly because it is trite and pointless and the poser couldn't either.

Thank you for that explanation of why there are two ways of writing roots.

What explanation? No matter. The answer is irrelevant.

There is an answer for me. I was looking for different letters used to spell the word "roots" and I would never find those letters because the OP is not about a written word but is about math. See for me, the mystery of what was being talked about has been resolved.

Laugh, I am accused of being close minded, but the opposite is true. I have the open mind of a child, not the closed mind of an adult. I do not know and must rely on creative thinking in my effort to know, something like Einstein's imagination experiments. There is nothing in my head to know the meaning of exp(π·í) = exp(-π·í) = i^2 = -1 . Only the last part i^2 = -1 comes close to something I understand. I understand math as a language I do not know, and I can not read or write, any more than I read Egyptian hieroglyphs, but the concept of "language" might be a bridge from what I know to what I do not know.

If I were to teach math I would teach it as a language. What does ^ mean? Give me the word and may be I will understand. If we think translating Egyptian hieroglyphs and translation the meaning of exp(π·í) then we might have a way of communicating that equals learning that could be easier for females who are verbal but tend to struggle with math. I have struggled with this for years now, but without the help of an understanding teacher. I have books and DVD's of college lectures explaining math and I know Pi is really awesome and like magic, those who understand the power of Pi can understand many, many things, but still this is magic to me, not something I understand. Like a child who stares at the knitting growing as it moves from one needle to the other, the doing is still a mystery to me, and am comforted knowing not everyone finds it easy to learn how to knit but me it is almost as natural as breathing.
Athena
Banned User

Posts: 1865
Joined: 22 Apr 2012
Location: Eugene, Oregon

### Re: Roots

What does ^ mean?

It means "to the power of" or the opposite of root. Mathematical shorthand, without which it would become too cumbersome.
rajnz00
Member

Posts: 270
Joined: 28 Dec 2016

### Re: Roots

x^2 = x2

zetreque
Forum Moderator

Posts: 3723
Joined: 30 Dec 2007
Location: Paradise being lost to humanity
Blog: View Blog (3)

### Re: Roots

rajnz00 » August 25th, 2017, 6:38 pm wrote:
What does ^ mean?

It means "to the power of" or the opposite of root. Mathematical shorthand, without which it would become too cumbersome.

Okay, and what does "to the power of" mean?

I found this explanation

https://www.mathsisfun.com/exponent.html
The exponent of a number says how many times to use the number in a multiplication.

In 82 the "2" says to use 8 twice in a multiplication,
so 82 = 8 × 8 = 64
In words: 82 could be called "8 to the power 2" or "8 to the second power", or simply "8 squared"

But now the word "exponent" is a foreign word and the explanation would be incomprehensible without demonstrating the meaning. It is like looking up the definition of a word in the dictionary and having to look up the meaning the of words used to define the first word.

And what is the power of 2? I think I know what is meant, but I find the concept of numbers having power interesting. What gives any number power? I think only a man would come up with the notion that numbers have power and how far is this from thinking words and magic spells have power? However, Pi surely seems to have the power of a magic spell.

And if two 8's equal 64, what does that have to do with the number 82? 82 = 8x8=64? Huh?

No, I am not smoking pot or using LSD. But seriously
In 82 the "2" says to use 8 twice in a multiplication,
so 82 = 8 × 8 = 64
does not make sense to me. 8x8=64 makes sense, but why is the number 82 thrown in there?
Athena
Banned User

Posts: 1865
Joined: 22 Apr 2012
Location: Eugene, Oregon

### Re: Roots

Athena, 82 does not mean 8x8. It just means 82
82 where 2 is in superscript is quite different and does mean 8x8
And if two 8's equal 64, what does that have to do with the number 82? 82 = 8x8=64? Huh?

Two 8's does not equal 64, it equals 16 and it has nothing to do with 82, which does not = 64
No, I am not smoking pot or using LSD.

Maybe you should :)
rajnz00
Member

Posts: 270
Joined: 28 Dec 2016

### Re: Roots

Obviously they missed out the ^ between 8 and 2. This kind of thing happens a lot when people who don't understand the notation try to replicate next (I remember in A-level math and physics books a few answers were wrong because it was clear the printing company had no idea what they were printing. Especially with units of measure.)

Resident Member

Posts: 5570
Joined: 14 Mar 2012

### Re: Roots

Athena » August 26th, 2017, 3:28 am wrote:
Okay, and what does "to the power of" mean?

Obviously it is mathematical slang.
But it has much more sense than you realize at first sight.

If I said "8 is 8" and "8 squared is 64", then this may make some sense: if you have a room of 8x8 feet you have 64 square feet (8x8, 8², 8^2)

And if the room is also 8 feet high, then you have 8x8x8=512 cube feet, so I could say 8 cubed is 512. And I short-hand write it as 8^3, 8³.

Then you may have noted that mathematicians do not stop in front of anything, so they say if it makes sense to talk about 8 squared = 8x8 = 64 = 8² = 8^2, and 8 cubed = 8x8x8 = 512 = 8³ = 8^3, why shouldn't we be able to write 8x8x8x8x8x8 = 8^6, and why shouldn't we be permitted to write 8^0.2, or 8^(-1/2) or 8^pi?

So they did. Computing 8^pi actually is not so trivial, but there is a way (or actually a number of ways), such as computing the exp(pi·log(8)), or, if you are content with an approximation, taking a mathematical ruler (did you ever see one? they are magic!), measure the distance between 1 and 8 on the, multiply it times 3.14, and look for the number on the ruler that is located at such distance from 1.

neuro
Forum Moderator

Posts: 2621
Joined: 25 Jun 2010
Location: italy

### Re: Roots

rajnz00 » August 25th, 2017, 8:51 pm wrote:Athena, 82 does not mean 8x8. It just means 82
82 where 2 is in superscript is quite different and does mean 8x8
And if two 8's equal 64, what does that have to do with the number 82? 82 = 8x8=64? Huh?

Two 8's does not equal 64, it equals 16 and it has nothing to do with 82, which does not = 64
No, I am not smoking pot or using LSD.

Maybe you should :)

Oh banana's. How do you get that little 2, so that the 2 means X the 8 instead of two ones?
Athena
Banned User

Posts: 1865
Joined: 22 Apr 2012
Location: Eugene, Oregon

### Re: Roots

neuro » August 27th, 2017, 7:12 am wrote:
Athena » August 26th, 2017, 3:28 am wrote:
Okay, and what does "to the power of" mean?

Obviously it is mathematical slang.
But it has much more sense than you realize at first sight.

If I said "8 is 8" and "8 squared is 64", then this may make some sense: if you have a room of 8x8 feet you have 64 square feet (8x8, 8², 8^2)

And if the room is also 8 feet high, then you have 8x8x8=512 cube feet, so I could say 8 cubed is 512. And I short-hand write it as 8^3, 8³.

Then you may have noted that mathematicians do not stop in front of anything, so they say if it makes sense to talk about 8 squared = 8x8 = 64 = 8² = 8^2, and 8 cubed = 8x8x8 = 512 = 8³ = 8^3, why shouldn't we be able to write 8x8x8x8x8x8 = 8^6, and why shouldn't we be permitted to write 8^0.2, or 8^(-1/2) or 8^pi?

So they did. Computing 8^pi actually is not so trivial, but there is a way (or actually a number of ways), such as computing the exp(pi·log(8)), or, if you are content with an approximation, taking a mathematical ruler (did you ever see one? they are magic!), measure the distance between 1 and 8 on the, multiply it times 3.14, and look for the number on the ruler that is located at such distance from 1.

Now talking about the power of a number is more fun when we are speaking of square and cube or of monad, dyad, triad, etc.. Our language seems inadequate to Greek because it does not hold the richness of meaning that the Greek symbols have, neither in symbols for numbers nor for the letters of the alphabet (alpha, beta). I wonder if we can achieve the same degree of genius with an inferior understanding of numbers and letters?

You totally lost me on the explanation of 8^pi but I am turned on (excited). Pi is a total turn on and just out of my comprehension, like a deer hiding in the forest. I know it is there but can not see it. It would be great if we could sit in the same room and we could work on my understanding of it. Hum, maybe I should go to Khan math and see if there are math problems, that can help me comprehend pi?

I bought a whole book explaining pi but it might as well be written in Russian because I can read the word but not grasp their meaning. Very frustrating!
Athena
Banned User

Posts: 1865
Joined: 22 Apr 2012
Location: Eugene, Oregon

### Re: Roots

Athena -

I have just started a course of Coursera site called "Mathematical Thinking". It may help you get a better grasp? (FREE course by the way. There are LOTS of courses on there that might tickle your fancy other than that.)

Resident Member

Posts: 5570
Joined: 14 Mar 2012

### Re: Roots

Athena,
I was not "explaining" 8^pi to you, I was just having fun in writing things that are correct but cannot be understood if do not know them in advance. :°)

However, you may look at this picture of a log ruler:
can you see the harmony in this picture?
the bottom scale moves by the same space for an equal step (0.1)
whereas the top scale move by the same space for an equal multiplication (e.g. doubling: 1, 2, 4, 8, 16).

Now try this:
measure the distance between 1 and 3 on the top scale.
Now move to the right 1, 2, 3, 4 times this same distance: you will fall (on the top scale) onto 3, 9, 27, 81, which are 3^1, 3^2, 3^3, 3^4.
Now assume that the distance you measured, D, is 5 cm.
Multiply this by pi, or just by 3.14 (you get some value X=D·3.14, in the hypothesis of D=5cm, X=15.7 cm)
Move from 1 to the right by the value X you just computed (on the top scale).
You will get pretty close to the value 31.5 which actually equals 3^pi.
See the magics?

neuro
Forum Moderator

Posts: 2621
Joined: 25 Jun 2010
Location: italy

### Re: Roots

Athena » August 26th, 2017, 3:28 am wrote:
Okay, and what does "to the power of" mean?

In plain English it means "multiplied by itself"

Thus if you have Xn, then in English it is saying "X is multiplied by itself n times"

Thus if X = 2 and n = 3
The mathematical expression is 23
Or in English it is saying "two is multiplied by itself three times", which works out to 8
2x2x2 = 8
How do you get that little 2....?

That "little 2" is known as a superscript. It is got by writing 2 on the top right hand corner of the number, instead of alongside it. You get it by writing it with a pencil on a piece of paper. If you are asking how do you get it on your computer, you will have to ask someone else. Hopefully you are not one of those who has forgotten how to write with your hand while not having quite learnt the computer.
rajnz00
Member

Posts: 270
Joined: 28 Dec 2016

### Re: Roots

I was meaning how is it done with the computer. We might be able to talk math better with internet computers designed for that purpose.

I understood "to the power of" means "multiplied by itself", but wondered why this is called a power? Now if we are speaking of square and cube, and also of the mathematical archetypes of nature, the word "power" makes more sense, but thinking like that is taboo. Laugh, never mind me. I have strange notions and more imagination than knowledge and something really awful happens when I try to understood math. Like I can get Sumerians counted by 12's instead of 10's, but my brain doesn't grasp how that is done. It sort of understands, but not really.

But back to the subject, I am clear that a good knowledge of math means being able to see things in many different ways and this is compliant with having more than one expression for roots. New math seems to say everything can be done in many different ways. It is seeing 6 can be divided by 1, 2,3, and 6 and more. It is knowing we have two even numbers on our right hand and three even numbers on our left hand on a deeper level than just understanding the words to express this. To really know math is to be intimate with it and I am not there yet, but I am okay with more than way to express roots.

Neuro, I know nothing about the log ruler!

Your log ruler is hard for me to see so I looked for another and found this link https://chilimath.com/lessons/advanced- ... thm-rules/ . Now I look at the link and my brain starts screaming like it were being tortured! It screams "oh my god what are doing to me? Stop it, stop it!"

It (my brain) has zero understanding of "logarithm" so from there everything else is frighteningly incomprehensible. We are presenting math to children all wrong. It should not be "you have to learn this or you will be punished with shame and bad grades, and tortured with homework". It needs to be child's play, to prevent the brain from screaming like my brain screams when I try to understand math. Once the screaming starts, a child is not going to learn anything. Math teachers need to be seductive and enter through a child's sense of play.
Athena
Banned User

Posts: 1865
Joined: 22 Apr 2012
Location: Eugene, Oregon

### Re: Roots

Athena » August 30th, 2017, 11:42 am wrote:I understood "to the power of" means "multiplied by itself"

If you did, nowhere did you display that knowledge in that lengthy search of yours. Instead you managed to wind up with 2 8's = 82 =64.
You wanted to know what it meant in English. I have told you and you are unappreciative. Dont confuse yourself. Brevity is the soul of wit, as the great bard said.
I have strange notions and more imagination than knowledge and something really awful happens when I try to understood math.

That is apparent. Perhaps you should not try to. As you know mathematics is made of 50 percent formulas, 50 percent proofs, and 50 percent imagination, 100 % imagination just doesn't add up. And as for your fascination with pie, as you know the area of a circle is pi R square, but pies are not square they are round.
rajnz00
Member

Posts: 270
Joined: 28 Dec 2016

### Re: Roots

Athena:
in some computers and operative systems you hit number 2 while holding down the "AltGr" key and the small 2 up there appears.
Otherwise, you open word and look for the character menu, choose "superscript", and the following letters will appear smaller and displaced upward.
On this forum you can obtain it by hitting the "sup" button and writing the "2" between the [ sup]and[ /sup] tags that magically appear in your writing box: here --> 2

neuro
Forum Moderator

Posts: 2621
Joined: 25 Jun 2010
Location: italy

### Re: Roots

THANKS neuro 2

Resident Member

Posts: 5570
Joined: 14 Mar 2012

### Re: Roots

theoretically, the site supports TEX which allows you to print any kind of equations.
Recently the thing didn't seem to work properly, but it appears to have been fixed.

You would make a tag [ tex] .... [ /tex] and put the code in between.
The language is reasonably straightforward:
[ tex]\sum_{n=1}^\infty \frac{1}{n+1}[ /tex] yields
$\sum_{n=1}^\infty \frac{1}{n+1}$
Google "TEX HELP" to have an idea of what you can do.

Quite a number of effects can be directly obtained in the post editor by simply using BBCode: see http://www.sciencechatforum.com/faq.php?mode=bbcode.

neuro
Forum Moderator

Posts: 2621
Joined: 25 Jun 2010
Location: italy

### Re: Roots

neuro » August 31st, 2017, 3:25 am wrote:Athena:
in some computers and operative systems you hit number 2 while holding down the "AltGr" key and the small 2 up there appears.
Otherwise, you open word and look for the character menu, choose "superscript", and the following letters will appear smaller and displaced upward.
On this forum you can obtain it by hitting the "sup" button and writing the "2" between the [ sup]and[ /sup] tags that magically appear in your writing box: here --> 2

I have to try this.. 82 I got it! How wonderful is that?! Thank you

8 (obviously I didn't get my 2 when using the alt key. I don't see an AltGr key. )

I am not seeing the imagined computer in my head clearly, but in my imagination, I see a computer screen with math symbols and it functions like spell check. It has a voice that says encouraging things such as "try this" and shows you how the functions work. It is more like an art program and triggers a sense of creativity counteracting the intense fear and flight response some of us have to math. I am so close to enjoying math, but it is still like facing a frightening dragon instead of being in a land of flowers and wonder. Hey, that could make a good children's story that could inspire children to get past that dragon and into the wonderful world of math.

Neuro, thanks for that bbcode page. I bookmarked the page and hopefully will remember that when I want to try something new.
Athena
Banned User

Posts: 1865
Joined: 22 Apr 2012
Location: Eugene, Oregon