SQUARE ROOT

[Saumyojit]

I found this on a page :
The square root algorithm is set up so that we take the square root of a number in the form of (X + R)^2. The square root of this number is obviously (X + R).

X represents the current approximation for the square root, and R represents the remainder of the number left over from the approximation.


suppose my no is 10.89 i can assume the square root will be (3. something)^2 can also be written in (X+R) form i.e (3+0.xxxx) . So here X means 3 and R means 0.something am i interpretting wrong?

Our approximation will always be the correct square root of the number truncated (not rounded) to the number of digits in our approximation
What do this line mean?
@Dr.Peterson

 

[Dr.Peterson]

There are several different ways to calculate the square root, so I can't be positive which algorithm is being discussed. It would be helpful if you showed the context. From the mention of the number of digits, it sounds like it's the digit-by-digit method that looks something like long division, rather than, say, the divide-and-average method.

The wording of what you quote could be better. But taking your example to start with, we want to find the square root of 10.89, and we start with the "guess" of 3: (3 + r)^2 = 10.89. This first approximation, 3, is correct up to the decimal point; that is, whereas the actual square root is 3.3, when you truncate that (that is, round down) to the nearest whole number, you get 3. "Truncate" means to chop off digits beyond a certain place, in this case the ".3", leaving the 3; it is not rounding to the nearest whole number (in either direction), but to the nearest whole number less than 3.3.

Taking a more typical example, if they were taking the square root of 10, the correct answer is 3.1622776601683793319988935444327..., so the estimates in turn would be 3, 3.1, 3.16, 3.162, and so on. That is, the process produces one digit at a time.

If I were writing the line you quoted, I would at least add a comma: "Our approximation will always be the correct square root of the number, truncated (not rounded) to the number of digits in our approximation"; it is the square root, that is truncated, not the given number as I first thought when I read it.

For some details on this method and a couple others, see this page I wrote. The method you appear to be learning is the third.

 

[Saumyojit]

PLease read carefully

1. R represents the remainder of the number left over from the approximation ---> this part means the fractional part of the square root right? WHY Number in this above line means the square root ? Thats because square root of a number is in the form of (X + R)^2 so the R part has to be in the square root so thats why number is the square root. Am i right?
2. What is this "left over from the approximation" line means? But my intution tells me that if my no is 10.81 and i approximate 3 in the first stage of thinking then the remainder R or the left over from the approximation will be 10.81- 3^2=1.81 but at the same time i am saying to myself that cannot be because R has to be a part of square root.MAYBE I am thinking of the normal remainder that we are generally left with when something is complelety divisble.

2.Why we are truncating 3 or why they are saying this line "Our approximation will always be the correct square root of the number truncated (not rounded) to the number of digits in our approximation ". I know that if something is truncated it is equal to only the non fractional part of the number i.e no of digits in our approximation .That means if my square root is 106.25 then digits of approximation is 3 i.e 106
Am i right??

Why they are saying it will be the correct square root of the number . if my number is 10.89 then if i truncate 3.3 to 3 then it becomes the square root of 9 not 10.89 then it is not correct


Why approximation means only the integer part only thats because if i know a no suppose 147 then it square root in my first approxmitation for surely will be 12 as i know square of 12 is close to 147 thats why right?

please calrify my doubts and wrong thinkings

 

[Saumyojit]

@Dr.Peterson What does this line mean "rounding to the nearest whole number (in either direction), but to the nearest whole number "
3.3 suppose taking the first case would be 3 or 3.6 would be 4
2nd case : 3.3 would be 3 or 3.6 would be 4
am i right? if i am right there is no difference?

 

[Dr.Peterson]

@Dr.Peterson What does this line mean "rounding to the nearest whole number (in either direction), but to the nearest whole number "
3.3 suppose taking the first case would be 3 or 3.6 would be 4
2nd case : 3.3 would be 3 or 3.6 would be 4
am i right? if i am right there is no difference?

You have seriously misquoted me, omitting the essential words before and after the underlined part you quote. I said, "Truncate" means to chop off digits beyond a certain place, in this case the ".3", leaving the 3; it is not rounding to the nearest whole number (in either direction), but to the nearest whole number less than 3.3." That is, truncation is NOT rounding to nearest, BUT rounding to the nearest number LESS than the given number. Obviously "rounding to the nearest whole number" and "rounding to the nearest whole number" are the same thing! But "rounding to the nearest whole number" is very different from "rounding to the nearest whole number less ..."

Rounding 3.3 to the nearest whole number --> 3​

Rounding 3.3 down to the nearest whole number = truncating --> 3​

Rounding 3.16 to the nearest tenth --> 3.2​

Rounding 3.16 down to the nearest tenth = truncating --> 3.1​

 

[Saumyojit]

rounding down means truncating
rounding off means we have to see the next digit whether it is greater than equal to 5 or not. if yes then round up eg : 3.16 nearest tenth will give 3.2 Now if no 3.06 rounding down to nearest tenth will give 3.0 ?
if no is 3.96 rounding up to nearest tenth will give me 3.9?
@dr peterson please give reply also to the above doubts that i have

 

[Dr.Peterson]

PLease read carefully

1. R represents the remainder of the number left over from the approximation ---> this part means the fractional part of the square root right? WHY Number in this above line means the square root ? Thats because square root of a number is in the form of (X + R)^2 so the R part has to be in the square root so thats why number is the square root. Am i right?
2. What is this "left over from the approximation" line means? But my intution tells me that if my no is 10.81 and i approximate 3 in the first stage of thinking then the remainder R or the left over from the approximation will be 10.81- 3^2=1.81 but at the same time i am saying to myself that cannot be because R has to be a part of square root.MAYBE I am thinking of the normal remainder that we are generally left with when something is complelety divisble.

They are saying that X+R is the actual square root of some given number N; X is the approximation, and R is the difference between that and the actual square root, which they call a remainder. We could say that [MATH]R = \sqrt{N} - X[/MATH].

Their English is horribly written; you're right that the only way to figure out what they mean is to pick the interpretation that makes sense. So, yes, "number" in that line refers to the square root, not to the given number I called N.

And, no, R is not 10.81- 3^2=1.81, but the difference between the exact square root (which you don't know) and your approximation. In your example, the exact square root is 3.2878564445547193020469184596699..., so R is the difference between that and 3, namely 0.28785644455471930204691845966992... . Part of the point of any explanation of the method is that you are always working toward a number that you will never be able to write out exactly, so they've given that a name, X+R.

2.Why we are truncating 3 or why they are saying this line "Our approximation will always be the correct square root of the number truncated (not rounded) to the number of digits in our approximation ". I know that if something is truncated it is equal to only the non fractional part of the number i.e no of digits in our approximation .That means if my square root is 106.25 then digits of approximation is 3 i.e 106
Am i right??

Why they are saying it will be the correct square root of the number . if my number is 10.89 then if i truncate 3.3 to 3 then it becomes the square root of 9 not 10.89 then it is not correct

Why approximation means only the integer part only thats because if i know a no suppose 147 then it square root in my first approxmitation for surely will be 12 as i know square of 12 is close to 147 thats why right?

Truncation, as I demonstrated previously, is not always truncation to a whole number, taking the "non-fractional" part. You can truncate 3.16 to the tenths place and get 3.1; in that case, "the number of digits in our approximation" is 2 (3, 1). I would have said, "to the current place", in this case the tenths.

The "correct square root of the number" is the square root that you don't know yet. And what they are saying here is not that the approximation IS the correct square root, but that it is the RESULT of truncating the correct square root. I already said that I would have added a comma, at the least, to make it easier to tell what they mean! The approximation 3 is equal to "the exact root, 3.3, truncated to a whole number".

Each approximation in the algorithm has one more digit than the previous approximation. The first is just the integer part of the root; the second is everything through the tenths place (which for this example is all there is); and so on.


Now, I just did a search to locate the original page you are referring to; it would be really helpful if people told us their sources so we could see the context! It's from a site with pages written by students, which are generally not bad but are not by really experienced teachers. Here is the paragraph you are asking about, quoted completely:

The square root of a number, N, is the number, M, so that M² = N. The square root algorithm is set up so that we take the square root of a number in the form of (X + R)². The square root of this number is obviously (X + R). X represents the current approximation for the square root, and R represents the remainder of the number left over from the approximation. Our approximation will always be the correct square root of the number truncated (not rounded) to the number of digits in our approximation. If we expand our number (X + R)² it will equal X² + 2RX + R². This gives us the basis for our derivation of the square root algorithm.​

Note, first, that you omitted the first sentence, which tells us that the author did in fact name the number N. It would have been better if he had mentioned both M (the correct/exact root) and N (the given number) more throughout this for clarity, which would have taken care of some of your confusion.

Anyway, don't worry too much if you had trouble following this; it's definitely not the best example of English to learn from, though it can be understood with effort. (I've probably written something occasionally that was hard to follow, too.)

 

[Dr.Peterson]

rounding down means truncating
rounding off means we have to see the next digit whether it is greater than equal to 5 or not. if yes then round up eg : 3.16 nearest tenth will give 3.2 Now if no 3.06 rounding down to nearest tenth will give 3.0 ?
if no is 3.96 rounding up to nearest tenth will give me 3.9?
@dr peterson please give reply also to the above doubts that i have

Yes, 3.16 rounds to 3.2, and 3.06 rounds down to 3.0.
No, 3.96 rounded up would be 4.0; 3.9 is less than 3.96.

 

[Saumyojit]

if i round down 7 then my no would be 6 or 7 itself?
@Dr.Peterson

 

[Saumyojit]

@Subhotosh Khan
please reply to my above doubt

 

[Deleted member 4993]

@Subhotosh Khan
please reply to my above doubt

You have a habit of asking for help from particular helpers - excluding others. So I did not follow your post. I think Dr. P has has given an excellent explanation to the solution of your problem. All you have to do now is to sit down with pencil & paper and follow his instructions.

 

[Saumyojit]

You have a habit of asking for help from particular helpers - excluding others. So I did not follow your post. I think Dr. P has has given an excellent explanation to the solution of your problem. All you have to do now is to sit down with pencil & paper and follow his instructions.

i am sorry u think that way . I am open to everyone's opinion it is just that dr P explanation suits me the most somehow .
I think if round down 7 the answer will be 6 am i right?
AS doc says "truncation is NOT rounding to nearest, BUT rounding to the nearest number LESS than the given number "

the nearest whole no less than 7 is 6 .

 

[Dr.Peterson]

if i round down 7 then my no would be 6 or 7 itself?
@Dr.Peterson

I think if round down 7 the answer will be 6 am i right?
AS doc says "truncation is NOT rounding to nearest, BUT rounding to the nearest number LESS than the given number "

the nearest whole no less than 7 is 6 .

If you round 7 "down" to the nearest integer, the result would be 7 itself.

That is a slight peculiarity of the language, but we all know what it means, don't we? Rounding to an integer means finding the nearest integer, and if the number is already an integer, we don't have to do anything.

Adding the word "down" doesn't change this; it just means we only look in one direction. Technically, we should say "round to the nearest integer less than or equal to the given number", but we shorten that to "round down". I apologize for saying only "less than", but "or equal" wasn't needed in the case under discussion, and I expect you to have learned about rounding.

 

[Saumyojit]

@Dr.Peterson in wikipeadia it is showing that "rounding down means taking the floor value of something . Truncation of positive real numbers can be done using the floor function" . y (floored value)is the largest integer that does not exceed x (original value)


their defination of truncating is : round towards zero (or truncate, or round away from infinity): y is the integer that is closest to x such that it is between 0 and x (included); i.e. y is the integer part of x, without its fraction digits.

u told that : You can truncate 3.16 to the tenths place and get 3.1 and here they are saying y is the integer part of x, without its fraction digits.That means they are truncating to the whole number place?

what is the meaning of x included? i want a example where x is being included. the one u showed above 7 is truncated to 7 itself that means including x

SO for postive nos suppose floor of 3.4 to the nearest ones place is 3 and trunc of 3.4 to the nearest ones place is also 3
but if the no is -3.4 floor of -3.4 to the nearest ones place will be -4 and trunc of -3.4 will be -3 because-3 is the closest whole no between x(-3.4) and 0

And if we round down 3.2 to nearest tenths place then the no remains same .
and if we floor 3.2 to nearest tenth place then also 3.2

if we round down that means take the floor of -3.2 to nearest tenths place then the no remains the same
if we truncate -3.2 to nearest tenths place then also no remains same?

 

[Saumyojit]

@Dr.Peterson tommorw morning i will chat with u . Now it is 1.13 am in india . tommorw at 09:00 am i will be online again. u please stay online . I know in usa it will be that time around 11 pm ...

 

[Dr.Peterson]

@Dr.Peterson in wikipeadia it is showing that "rounding down means taking the floor value of something . Truncation of positive real numbers can be done using the floor function" . y (floored value)is the largest integer that does not exceed x (original value)


their defination of truncating is : round towards zero (or truncate, or round away from infinity): y is the integer that is closest to x such that it is between 0 and x (included); i.e. y is the integer part of x, without its fraction digits.

u told that : You can truncate 3.16 to the tenths place and get 3.1 and here they are saying y is the integer part of x, without its fraction digits.That means they are truncating to the whole number place?

what is the meaning of x included? i want a example where x is being included. the one u showed above 7 is truncated to 7 itself that means including x

SO for postive nos suppose floor of 3.4 to the nearest ones place is 3 and trunc of 3.4 to the nearest ones place is also 3
but if the no is -3.4 floor of -3.4 to the nearest ones place will be -4 and trunc of -3.4 will be -3 because-3 is the closest whole no between x(-3.4) and 0

And if we round down 3.2 to nearest tenths place then the no remains same .
and if we floor 3.2 to nearest tenth place then also 3.2

if we round down that means take the floor of -3.2 to nearest tenths place then the no remains the same
if we truncate -3.2 to nearest tenths place then also no remains same?

Terminology often varies according to context, so we have to use our minds in reading. Interpret what you read in a way that makes sense, rather than looking up one definition and blindly applying it.

The definition you quote is for truncation, period -- no qualifiers. It is talking only about rounding to an integer. The article you are struggling to understand talks about truncation "to a specified number of digits"; I've already told you I would say it differently (to a specified place), but it is not impossible to understand what he means, is it? This is a broader (but not uncommon) use of the word, just as "rounding" can refer by default to rounding to a whole number, but also to any place you name. The floor function refers only to rounding down to an integer; truncation can be applied to places other than the ones place. We never talk about "flooring to the tenths place".

The phrase "between 0 and x inclusive" (the Wikipedia page you found seems to be using a different term than I am used to) means "between 0 and x, or equal to x; that is, in the interval (0, x], i.e. [MATH]0<n\le x[/MATH]. So if x itself is an integer, then truncation yields x itself. That's true of any rounding method.

@Dr.Peterson tommorw morning i will chat with u . Now it is 1.13 am in india . tommorw at 09:00 am i will be online again. u please stay online . I know in usa it will be that time around 11 pm ...

I don't do scheduled chats. And 11:00 is close to the time I get off the computer. I will be on when I choose to be on.

 

[Saumyojit]

@Dr.Peterson
@Subhotosh Khan

u said that "truncation can be applied to places other than the ones place " why so? if my no is 32.2 if i truncate 32.2 to ones places the no will be 32 right? Then i am able to apply truncation to ones place or u are telling from the context that if my digit is only one that is in the ones place trunc(7)=7 there is nothing to truncate beyond ones place

ANother case : if truncate 3.2 to nearest tenth it will result in 3.2 only as there is nothing to truncate beyond 2 (tenths place)

but u said i can apply to other places so if i truncate 32 to the tens place that means my no will be 3 .Beyond tens any digit will be chopped off.right?

so when truncating if i want something new value after truncation i need to have a digit beyond the place i am truncating?

Floor means any value that will be given inside it it will return the absolute value of the no. "THere will no such thing as flooring to tenths or hundreths or tens or hundreds" . floor (32)=32 floor(32.2)=32 floor(-32.2)=-33 (rounds towards negative infinity)
floor(-32)=-32
am i right?

 

[Dr.Peterson]

@Dr.Peterson
@Subhotosh Khan

u said that "truncation can be applied to places other than the ones place " why so? if my no is 32.2 if i truncate 32.2 to ones places the no will be 32 right? Then i am able to apply truncation to ones place or u are telling from the context that if my digit is only one that is in the ones place trunc(7)=7 there is nothing to truncate beyond ones place

ANother case : if truncate 3.2 to nearest tenth it will result in 3.2 only as there is nothing to truncate beyond 2 (tenths place)

but u said i can apply to other places so if i truncate 32 to the tens place that means my no will be 3 .Beyond tens any digit will be chopped off.right?

so when truncating if i want something new value after truncation i need to have a digit beyond the place i am truncating?

Floor means any value that will be given inside it it will return the absolute value of the no. "THere will no such thing as flooring to tenths or hundreths or tens or hundreds" . floor (32)=32 floor(32.2)=32 floor(-32.2)=-33 (rounds towards negative infinity)
floor(-32)=-32
am i right?

The page you initially asked about is using the word "truncate" not as the Wikipedia article you found does (for truncation to an integer), but for truncation to a specific place. I gave you examples of this: if we truncate 3.16 to the tenths place, we simply remove all digits to the right of the 1, leaving 3.1. If there are no digits to the right of the specified place, then the number is left unchanged. That is what truncation means in your source.

If we truncate 32 to the tens place, the result is 30. Here we can't literally "remove" digits, but change them to zero. Surely you know that is how any kind of rounding works.

The floor function has nothing to do with the absolute value. It yields the greatest integer less than or equal to the given number. Your examples there are correct.

 

[JeffM]

The text that you quoted very clearly says "truncated (not rounded)". So concentrate your attention on "truncation." In a text about manual derivation of square roots, you will be dealing with non-negative numbers, no? If you look up truncation on wikipedia, it gives you a formula for truncation of x to n decimal places if x is non-negative.

[MATH]\dfrac{\lfloor 10^nx\rfloor}{10^n}.[/MATH]
There is a reason why we use mathematical notation to discuss mathematics.

They also give a different formula for when x is negative, which is irrelevant to your manual derivation of square roots.

 

[Saumyojit]

The page you initially asked about is using the word "truncate" not as the Wikipedia article you found does (for truncation to an integer), but for truncation to a specific place. I gave you examples of this: if we truncate 3.16 to the tenths place, we simply remove all digits to the right of the 1, leaving 3.1. If there are no digits to the right of the specified place, then the number is left unchanged. That is what truncation means in your source.

If we truncate 32 to the tens place, the result is 30. Here we can't literally "remove" digits, but change them to zero. Surely you know that is how any kind of rounding works.

The floor function has nothing to do with the absolute value. It yields the greatest integer less than or equal to the given number. Your examples there are correct.

if in the case of 3.16 if we truncate to tenths place we can chop of the digits beyond tenths place but why in the case of 32 we can't literally "remove" digits, but change them to zero.Why not chopping off