# Nearly half a million



## Kefirka

There is a test on the internet. 

The text: 
_"... If you added all the technical, scientific and navigation terms to the number of words in the English dictionary it would practically double the total to make nearly a million."
_
One of the questions of the test:
_"Not including technical, navigational and scientific words, approximately how many English words are there?"
_
_1.Nearly half a million
2.About half a million
3.700,000,000
_
I know the right answer: _"About half a million"_. But I don't understand the answer. "About half a million" could be "half a million and 100", but double of "half a million and 100" is "a million and 200". As I know, "nearly a million" means "less than a million". How could "a million and 200" be equal to "less then a million" ?

Why is "_Nearly half a million_" an incorrect answer ?


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## Kryptonite1303

I too thought (1) was the answer and I agree with your logic.


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## PaulQ

The example you give falls into the category of "Stupid Questions and Answers set by People who did not Think." 

_1.Nearly half a million = a little less than 500,000
2.About half a million = __a little more or less than 500,000,_ _or even 500,000 itself._
_ 3.700,000,000 = one more than __699,999,999_


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## Kefirka

PaulQ, could you explain why "_a little less than 500,000_" is wrong ?


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## Keith Bradford

Why?  It's entirely logical.

The full total is _nearly a million_ (say, 950,000).
This is _practically double (i.e. almost double) _the non-technical total (say, 1.9 times).
Therefore the non-technical total is approx. 950,000/1.9 = 500,000.
Therefore answer 2 is correct.


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## Kefirka

Keith Bradford, thank you very much ! I understand now. I didn't know that "_practically double = "almost double_" (in fact, I didn't think about that)


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## JulianStuart

Keith Bradford said:


> Why?  It's entirely logical.
> 
> The full total is _nearly a million_ (say, 950,000).
> This is _practically double (i.e. almost double) _the non-technical total (say, 1.9 times).
> Therefore the non-technical total is approx. 950,000/1.9 = 500,000.
> Therefore answer 2 is correct.


Nearly half a million could represent, say, 480,000.
Practically double this might be 950,000.
950,000 is nearly a million.
The question was poorly thought out.


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## Keith Bradford

Yes, but you're arguing backwards, from the unknown to the known. There is only one answer which must be right (no.2). 

Only in specific circumstances might no.1 be right; in other circumstances the arithmetic will give answers greater than half a million. (E.g. 930,000/1.8 = 516,00 which makes answer no.1 quite wrong.)


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## JulianStuart

For 2, about half a million _could _be somewhat more than half a million. Let's say 520,000.  Then if you almost double it (x by 1.96), you could get to 1,020,000 which is clearly more  than a million and definitely more than "nearly a million" - the latter  always meaning less than (but not by much) a million. Therefore it is 2  that could exceed the million.  Similarly. for 1, if we say that "nearly half a million" always means less than 1/2 million so if you actually double it, you will not exceed 1 million, ever.  So 2 can sometimes be wrong, but 1 never.

_"... If you added all the technical, scientific and navigation terms (Y)  to the number of words in the English dictionary (X) it would practically  double the total to make nearly a million."_
Therefore X is practically equal to Y.  Therefore X+Y is nearly (i.e. a bit less than) 1 million.  However, X+Y is  <2X  - I take practically also to mean almost, but still less than, 2X.   Therefore,  X must be < 1/2 million.  What are you seeing that's  different from this algebra?

It may hinge on the precision of the mathematics represented by the vague word "practically"*   In any case, a poorly designed test.

*If X is 502,000 and Y is 497,000 technically we meet the mathematical criteria of "almost doubling" and "almost a million" and 2) _could_ be right.  However, if X is always <500,000 (i.e nearly half a million, as in 1) we will never contravene the algebra (we cannot exceed 1 million unless we _more than double_ X)  so 1 is never wrong.


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## Keith Bradford

I'll say it again: you're arguing backwards, from the unknown to the known.

The only known data are:

_"... If you added all the technical, scientific and navigation terms to the number of words in the English dictionary it would practically double the total to make nearly a million."

_Total = (let's say) between 900,000 and 1 million
Tech = (let's say) between 40% and 50% of that. *
Therefore Total - Tech = between 450,000 and 600,000. = answer no.2.

(*I think it's at this point that your maths falls down. If you "practically double" (which in normal English means "nearly double") the normal vocabulary, you multiply it by something between, say 1.8 and 2. So when you reverse the calculation as you are trying to, you must divide by less than 2. This will only give the result "_1. Nearly half a million" _in half the circumstances. An answer that is correct in only half the circumstances is a wrong answer, especially when there is an answer that is always correct: "_2. About half a million"._


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## JulianStuart

It still hinges on the meaning of practically.  I think "would practically double"  means "would almost double" so that the tech number can't be _bigger_ than the "normal" number (or Y < X).  Otherwise the statement would be "If you added the tech words it would more than double the total - that would state that Y > X)"  wouldn't it?  That's the logic I had in mind.  

This is in direct conflict with your interpretation of "Tech = (let's say) between 50% and 60% of that. "


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## PaulQ

_If you added all the technical, scientific and navigation terms to the number of words in the English dictionary it would practically double the total to make nearly a million."_

_ (wt+ws+wn)+W _*<~*_1,000,000_ (Vbulletin will not allow the "less than but approximately equal to" notation)

_(wt+ws+wn)_*<~*_W_

_(wt+ws+wn) is nearly half a million _
_W is nearly half a million_

_nearly half a million + nearly half a million = nearly one million_

_Any number that is nearly one million is also describable as “about a million.”_


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## wandle

The distinction between dictionary words and technical words is not valid, but if we accept it for the purpose of the argument, then let us say _*x *_is the number of dictionary words and *y* is the number of technical words.

We are told that *x + y* is nearly a million.
We are also told that nearly *2x* is nearly a million.
It follows that that *x + x* is a million.
Thus *x* is half a million and *y* is nearly half a million.


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## JulianStuart

Yes, but I'm sure I read, on the internet somewhere, that "practically double" means at least 1.92x and that "nearly X" means at least 0.95X so therefore Y must be smaller than X if it didn't get to 2x when Y was added to it 

All this just goes to show how poor a test it was!


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## pwmeek

I think the tricky part of the question is "practically double". To me, this means "almost, but not quite double". This means that the s,t,n terms are fewer than the rest of the words in the English languagre.

Now we have to find a number that when doubled (and then be reduced by a bit) is "nearly a million". 

(1) is already reduced a bit from the required number. Adding it to a slightly smaller number is getting a fair way below the mark

(2) is about half of the full million. Adding the lesser number of s,t,n terms adds up to "nearly a million"

(3) is laughable

Only (2) is correct. You might argue for (1), but the whole point of multiple-choice tests is to pick the answer that the test-writers were looking for.  Clearly, this is (2).

Nit-picking about how some less-suitable answer could be stretched to be correct will *always* get you marked wrong.

EDIT: When answering multiple-choice questions, the first thing you must ask yourself is not: "What is the answer?" but: "Why did they ask this question?" When analyzed, this is really the answer to what the original poster is asking: "Why is (1) wrong?"


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## JulianStuart

Multiple choice often results in multiple answers - especially ones we see on this forum.  Usually, they are to do with English.  This one, not so much


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## Lyndon

JulianStuart said:


> Multiple choice often results in multiple answers - especially ones we see on this forum. Usually, they are to do with English. This one, not so much



As I see it, the question *is* to do with English. It seems to test the student's ability to distinguish between 'nearly' (a little less than) and 'about' (a little more than or less than).

Effectively, the question is "What is half of _nearly a million_ ?" and the only possible answer is "_nearly half a million_".

_"About half a million"_ cannot be the correct answer because it admits the possibility that half of "_nearly a million_" could be greater than "_half a million_", which is clearly wrong.


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## Lyndon

You're right, it's a typing error. I meant to say "_nearly half a million_". Thank you.


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## Beryl from Northallerton

wandle said:


> We are also told that nearly *2x* is nearly a million.
> It follows that that *x + x* is a million.



Did you mean to write that?


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## Kefirka

*Lyndon*, there is no "What is half of _nearly a million ?_". There is "What is _practically_ (nearly) half of _nearly a million _?".


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## wandle

Beryl from Northallerton said:


> Did you mean to write that?


Since 'nearly double the total of dictionary words' (nearly *2x*) is equal to 'nearly a million', it follows that double the total of dictionary words (*2x*) is equal to a million.


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## Beryl from Northallerton

Kefirka said:


> *Lyndon*, there is no "What is half of _nearly a million ?_". There is "What is _practically_ (nearly) half of _nearly a million _?".



Yes, well spotted. There is also:_ 'approximately'... _"What is _practically_ (nearly) half of _nearly a million _?"


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## Beryl from Northallerton

wandle said:


> Since 'nearly double the total of dictionary words' (nearly *2x*) is equal to 'nearly a million', it follows that double the total of dictionary words (*2x*) is equal to a million.



Thanks for expanding your algebra. I can now understand that you really did 'mean to write that'. Does this mean then that none of the (multiple choice) answers were right?


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## Lyndon

Kefirka said:


> *Lyndon*, there is no "What is half of _nearly a million ?_". There is "What is _practically_ (nearly) half of _nearly a million _?".




Are you trying to split a hair between my 'effectively' and your 'practically' ?


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## wandle

Beryl from Northallerton said:


> Thanks for expanding your algebra. I can now understand that you really did 'mean to write that'. Does this mean then that none of the (multiple choice) answers were right?


No, it shows that (2) is correct.
The question "Not including technical, navigational and scientific words, approximately how many English words are there?" means (in the terms which it assumes) 'Approximately how many dictionary words are there?'
Answer: About half a million.
(The question asks specifically for an approximation.)


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## Bender_Bending_Rodriguez

"About.." is the correct answer. The number of dictionary words can be over 500,000 and still be practically doubled to be nearly a million. That's all you need to know. As long as it can be over or under, "around" is the best choice.


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## Beryl from Northallerton

wandle said:


> (The question asks specifically for an approximation.)



Yes, I'd noticed. But surely 'nearly a half a million' is also an approximation. Why do you favour the one approximation over the other?


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## Loob

Erm....

There are at least two internet versions of this test - here and here - which list the three options as (my red highlighting):
a) nearly *a million*
b) about half a million
c) 700,000.

So maybe "nearly half a million" in Kefirka's source is a misprint?


_(Runs away v fast and hides under table)_


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## wandle

Beryl from Northallerton said:


> Yes, I'd noticed. But surely 'nearly a half a million' is also an approximation. Why do you favour the one approximation over the other?


If no approximation were asked for in the question, or given in the answer, the result, as shown algebraically, would be half a million. However, it is hardly realistic to suggest that the number of words in the dictionary would be exactly 500,000. Therefore the question asks for an approximate figure. There is nothing in that to favour either a higher or a lower amount. Therefore: 'about'.


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## Valvs

Kefirka said:


> *Lyndon*, there is no "What is half of _nearly a million ?_". There is "What is _practically_ (nearly) half of _nearly a million _?".


Well, actually, the real question is "what is *slightly more *than a half of nearly a million".
See, if _x_ *nearly doubled *equals _y_, then _x_ is not nearly half of _y_, it is always *slightly greater *than half of _y_.
And I think it naturally follows that the best answer is "about."

Edit: Oh, and if Loob is correct about this being a misprint, than the entire discussion is pointless.


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## Beryl from Northallerton

Loob said:


> Erm...._(Runs away v fast and hides under table)_


 Outrageous!


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## Bender_Bending_Rodriguez

> Since 'nearly double the total of dictionary words' (nearly *2x*) is equal to 'nearly a million', it follows that double the total of dictionary words (*2x*) is equal to a million.


That's not true. You're assuming "practically" divided by "nearly" = 1.


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## lixiaohejssz

I agree with Keith Bradford and any other proponent.


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## wandle

Bender_Bending_Rodriguez said:


> You're assuming "practically" divided by "nearly" = 1.


'Practically' and 'nearly' in the present context are words with the same meaning; at least, there is (as a matter of English) no discernible difference.


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## Beryl from Northallerton

Bender_Bending_Rodriguez said:


> You're assuming "practically" divided by "nearly" = 1.



And you're saying it doesn't?


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## Bender_Bending_Rodriguez

They could mean almost anything. What if "practically" is 0.95 and "Nearly" is 0.99? What you said only works if they're the exact same number, but they almost definitely wouldn't be.


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## wandle

Bender_Bending_Rodriguez said:


> What you said only works if they're the exact same number, but they almost definitely wouldn't be.


They are words, not numbers.


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## Bender_Bending_Rodriguez

wandle said:


> They are words, not numbers.


Exactly. So you can't do math with them.


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## Kefirka

*Loob*, how very strange !
I got it there: http://quiz.kaplaninternational.com/quiz/English%20Level%20Test
Perhaps the test was too complicated, so someone decided to change it ?

*Lyndon*, now I'm trying to find the difference between  'effectively' and  'practically'


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## Beryl from Northallerton

lixiaohejssz said:


> I agree with Keith Bradford and any other proponent.



Are you hedging?


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## Beryl from Northallerton

Kefirka said:


> *Lyndon*, now I'm trying to find the difference between  'effectively' and  'practically'



You'll search in vain I fear.


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## Lyndon

Kefirka said:


> *Lyndon*, now I'm trying to find the difference between 'effectively' and 'practically'




Then, I suggest you type "hairsplitting" or "split hairs" into the Dictionary search box at the top of the page, and ...


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## wandle

Bender_Bending_Rodriguez said:


> Exactly. So you can't do math with them.


If so, the question would be insoluble:  in fact, nonsensical.
However, algebra allows us to calculate without specifying quantities. See posts 13 and 23. 
In both places, 'nearly' is substituted for 'practically', because the meaning is equivalent.
The resulting equation is:
Nearly double the total of dictionary words     is equal to     nearly a million.
Since 'nearly' is the same on both sides, we can remove it from the equation. Result:
Double the total of dictionary words     is equal to     a million.
Divide both sides by two:
The total of dictionary words     is equal to     half a million.
Express this as an approximate figure:
Answer: 'about half a million'.


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## Loob

Kefirka said:


> *Loob*, how very strange !
> I got it there: http://quiz.kaplaninternational.com/quiz/English Level Test
> Perhaps the test was too complicated, so someone decided to change it ?


Your link takes me to a page asking me to fill in my email address, Kefirka, so I can't see what wording you're looking at.  Are you saying the wording has been changed since you first looked at it?

If so, that wouldn't surprise me, as the first link I gave was to a Kaplan International version.  If it has been changed, it'll be for a reason.  
_My money's still on "nearly half a million" being a misprint - but I'll say that quietly, because I don't want to have to go back under the table. _


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## Kefirka

*Lyndon*, yes, I did that 

*Loob*, it's easy, you can put a fake email address and a random name there  But the answers did not change. 
I meant that possibly someone moved the test to their site and changed the answers...


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## Beryl from Northallerton

wandle said:


> Since 'nearly' is the same on both sides, we can remove it from the equation.



(Okay, so I'm having to write this down on paper now.) At this point are you dividing through by 'nearly', or are you subtracting it from both sides?


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## Beryl from Northallerton

Loob said:


> _My money's still on "nearly half a million" being a misprint - but I'll say that quietly, because I don't want to have to go back under the table. _


To have spilt the beans once, Miss Loob, may have been regarded as a misfortune....


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## wandle

Beryl from Northallerton said:


> (Okay, so I'm having to write this down on paper now.) At this point are you dividing through by 'nearly', or are you subtracting it from both sides?


I see it as:  nearly (double the total of dictionary words) is equal to nearly (a million).
For 'double the total of dictionary words' we can put *2x*.
'Nearly *2x*' means 'a figure which is close to but less than *2x*'.
'Nearly a million' means  'a figure which is close to but less than a million'.
Since the operator 'nearly' is the same on both sides, we can remove it.
It does not need to be calculated.


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## Valvs

wandle said:


> Since the operator 'nearly' is the same on both sides, we can remove it.


Only the *word *nearly is the same on both sides.

In an algebraic equation, we can only eliminate variables that we know represent *identical *numbers.
The expression _2xa=xb_ resolves into _2a=b_ because _x_ within the same equation or system of equations is presumed to always represent the same (albeit unknown) number. Identical unknowns are not the same as approximations. 
_"xa _divided by _xb" _is always equal to "_a _divided by _b_." But _approximately 10_ divided by _approximately 5_ does NOT always equal to 2: the only thing we can safely say is that _*approximately *10 _divided by_ *approximately *5 _equals to _*approximately *2_. In reducing an equation or formula, you can eliminate variables or precise numbers, but you cannot eliminate such symbols as "approximately equals", "greater than", or "lesser than". "Nearly a million" divided by "nearly two" is equal to "approximately half a million".


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## wandle

In the expression 
nearly 2x equals nearly a million
I see no reason to attach a different value to 'nearly' in the two cases.
However, if we do, the problem becomes insoluble.


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## Beryl from Northallerton

wandle said:


> I see it as:  nearly (double the total of dictionary words) is equal to nearly (a million).
> For 'double the total of dictionary words' we can put *2x*.
> 'Nearly *2x*' means 'a figure which is close to but less than *2x*'.
> 'Nearly a million' means  'a figure which is close to but less than a million'.
> Since the operator 'nearly' is the same on both sides, we can remove it.
> It does not need to be calculated.



No, but hang on a minute. Doesn't that mean that if a child is nearly a teenager, and a teenager is nearly an adult, but a teenager equals a teenager, then we can remove 'nearly a teenager' leaving 'a child is an adult'. Surely that can't work?


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## Valvs

wandle said:


> I see no reason to attach a different value to 'nearly' in the two cases.



"Nearly a million" is an APPROXIMATE number by definition. The word "nearly", in and by itself, simply cannot have any specific, precise value attached to it. It is not the same as an "x" in an algebraic expression. It is not a variable. It is more like the sign "<" (less then).
And the expression "<5/<5 = 1", generally speaking, is not true. 
Mathematical expression can include approximate values just fine, but you should know how to treat them. In most cases, the solutions to equations containing approximate values are also approximate. That does not mean that those equations are insoluble. "Approximately half a million" can be a valid answer even in math, depending on how the original problem is formulated.


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## Beryl from Northallerton

Valvs said:


> The expression _2xa=xb_ resolves into _2a=b_ because _x_  within the same equation or system of equations is *presumed* to always  represent the same (albeit unknown) number.


Who's to say that this presumption is warranted? If x is a variable, then why would we assume that it's constant?


Valvs said:


> ...the only thing we can safely say is that _*approximately *10 _divided by_ *approximately *5 _equals to _*approximately *2_.


But that's nearly exactly what *wandle* is precisely saying, only with different numbers and words - you've cleverly replaced his 'nearly' with your 'approximately'. In his view (unless I've misunderstood, which is very possible, I admit) he goes on to show that by removing the 'approximately' throughout the equation, we are left with '10 divided by 5 equals to 2' - which is true enough.


Valvs said:


> Only the *word *nearly is the same on both sides.





Valvs said:


> "Nearly a million" is an APPROXIMATE number by definition.


'Nearly a million', isn't a number at all though - it's an unspecified range, surely?


Valvs said:


> The word "nearly", in and by itself, simply cannot have any  specific, precise value attached to it. It is not the same as an "x" in  an algebraic expression


Unless you specify that x='nearly'


Valvs said:


> And the expression "<5/<5 = 1", generally speaking, is not true.


But that's nearly precisely what you said above, only with different words and numbers: 





> the only thing we can safely say is that _*approximately *10 _divided by_ *approximately *5 _equals to _*approximately *2_





> Mathematical expression can include approximate values just fine, but  you should know how to treat them. In most cases, the solutions to  equations containing approximate values are also approximate.


Well now you're talking my kind of language - this is exactly my beef with *wandle* who moves from an approximate solution to an exact solution then back to an approximate solution at the drop of what I consider to be a magician's hat.


> That does not mean that those equations are insoluble.


So what's the solution?


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## PaulQ

You have no idea how much I am enjoying this mathematical discussion. 

Let me help you all with some information:
http://oxforddictionaries.com/words/how-many-words-are-there-in-the-english-language


> How many words are there in the English language?
> 
> *There is no single sensible answer to this question.* (My emphasis) It's impossible to count the number of words in a language, because it's so hard to decide what actually counts as a word. Is dog one word, or two (a noun meaning 'a kind of animal', and a verb meaning 'to follow persistently')? ...
> 
> The Second Edition of the 20-volume  _Oxford English Dictionary_ contains full entries for 171,476 words in current use, and 47,156 obsolete words. To this may be added around 9,500 derivative words included as subentries.
> 
> This suggests that there are, at the very least, a quarter of a million distinct English words, ... If distinct senses were counted, the total would probably approach three quarters of a million.


I don't think that three quarters of a million is "*nearly*" half a million... 

Carry on...


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## Cagey

The thread is wandering a bit from the original language question. Though it doesn't seem possible to reach agreement on the answer, the Kefirka now has many helpful responses to make use of.   

Thank you all for your participation.  Thank you, Kefirka, for an interesting question. 

I am closing this thread. 
Cagey, moderator.


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