Puzzle of the Day [modified .2 ]
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Easy
0EXE (or) == (or) ENTER(or whatever is the calculators equivalent) et voila, 0 the correct answer If you want a serious answer then: 1) You can use the iteration: xn = (1/2)(x(n-1) + a/x(n-1)) where a is the number you're trying to find the root of, x0 should be as close to the root as possible. 2) You could probably use the Newton-Raphson method although this may involve powers (havn't checked, I just thought about it). This does work 3) You can use the Binomial Theorem 4) You can use "guessing", i.e. a trail and error method. 5) You could use logs (ok, this doesn't use a "basic" calculator).
The Welsh will always support two teams: The Welsh, and anyone playing England :)
Last modified: 23 June 2006 18:18:05 -- Added some more answers
Ed.Poore wrote:
If you want a serious answer
yes, assume you are in the examination hall and your standard calculator got broken with the power and root key and you have to find a root what will you do ? steps ? Rule # 17: Omit Needless Words - Strunk, William, Jr. ------------------------------------------------------------- Universal DBA | Ajax Rating
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Ed.Poore wrote:
If you want a serious answer
yes, assume you are in the examination hall and your standard calculator got broken with the power and root key and you have to find a root what will you do ? steps ? Rule # 17: Omit Needless Words - Strunk, William, Jr. ------------------------------------------------------------- Universal DBA | Ajax Rating
Suppose the easiest would be the Newton-Raphson, or the iterative one (essentially the same as the Newton-Raphson), since they both converge quickly thus requiring the fewest steps. In a real exam though: 1) I was allowed my Graphics calculator so there several thousand more ways than one to skin this particular cat. 2) I'd leave it as something like sqrt(2), of course cancelling it out by removing factors which are square numbers. That is the most exact way and since I did further it's also easier to work with if you're doing a long calculation.
The Welsh will always support two teams: The Welsh, and anyone playing England :)
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Ed.Poore wrote:
If you want a serious answer
yes, assume you are in the examination hall and your standard calculator got broken with the power and root key and you have to find a root what will you do ? steps ? Rule # 17: Omit Needless Words - Strunk, William, Jr. ------------------------------------------------------------- Universal DBA | Ajax Rating
Seeing as you want the exact steps, assuming that the calculator has only a broken power & root button (but is a scientific calculator (or at least can use a function like Ans)). Let r be the number whose root you want to find (e.g. 5) Let a be r/2 So enter in the calculator: 5 / 2 EXE (resulting in 2.5 being displayed) (0.5 * (Ans + (5 / Ans))) EXE (0.5 * (Ans + (5 / Ans))) EXE ... and so on until the accuracy required is obtained (should be < 5 iterations for ~6 places). If calculator does not support Ans button then jot it down or put it in memory to re-use. (Sorry can't remember how M+ M- etc work anymore since it's years since I used them).
The Welsh will always support two teams: The Welsh, and anyone playing England :)
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Seeing as you want the exact steps, assuming that the calculator has only a broken power & root button (but is a scientific calculator (or at least can use a function like Ans)). Let r be the number whose root you want to find (e.g. 5) Let a be r/2 So enter in the calculator: 5 / 2 EXE (resulting in 2.5 being displayed) (0.5 * (Ans + (5 / Ans))) EXE (0.5 * (Ans + (5 / Ans))) EXE ... and so on until the accuracy required is obtained (should be < 5 iterations for ~6 places). If calculator does not support Ans button then jot it down or put it in memory to re-use. (Sorry can't remember how M+ M- etc work anymore since it's years since I used them).
The Welsh will always support two teams: The Welsh, and anyone playing England :)
your solution is quite good , but it won't give a result instantly in a calculator, defies the purpose of the calculator Rule # 17: Omit Needless Words - Strunk, William, Jr. ------------------------------------------------------------- Universal DBA | Ajax Rating
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Seeing as you want the exact steps, assuming that the calculator has only a broken power & root button (but is a scientific calculator (or at least can use a function like Ans)). Let r be the number whose root you want to find (e.g. 5) Let a be r/2 So enter in the calculator: 5 / 2 EXE (resulting in 2.5 being displayed) (0.5 * (Ans + (5 / Ans))) EXE (0.5 * (Ans + (5 / Ans))) EXE ... and so on until the accuracy required is obtained (should be < 5 iterations for ~6 places). If calculator does not support Ans button then jot it down or put it in memory to re-use. (Sorry can't remember how M+ M- etc work anymore since it's years since I used them).
The Welsh will always support two teams: The Welsh, and anyone playing England :)
Ed.Poore wrote:
Sorry can't remember how M+ M-
Works just fine with the chessy Windows Calculator in Standard Mode. After each step, put the result in memory using the MS (Memory Save/Store) button. To get the result from the previous step, use the MR (Memory Recall) button.
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your solution is quite good , but it won't give a result instantly in a calculator, defies the purpose of the calculator Rule # 17: Omit Needless Words - Strunk, William, Jr. ------------------------------------------------------------- Universal DBA | Ajax Rating
Quartz... wrote:
but it won't give a result instantly in a calculator
Neither does the ^ or sqrt button, it performs the ln comparison to solve roots. It just happens much more quickly. Basically it all depends on how accurate you want to be.
The Welsh will always support two teams: The Welsh, and anyone playing England :)
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Ed.Poore wrote:
Sorry can't remember how M+ M-
Works just fine with the chessy Windows Calculator in Standard Mode. After each step, put the result in memory using the MS (Memory Save/Store) button. To get the result from the previous step, use the MR (Memory Recall) button.
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Can You Find ROOT of a number on a standard calculator WITHOUT touching the root sign or power sign a power sign also acts like a root sign and believe me you can Easy huh! try it ? tell me the EXACT steps * Easy looking problems are sometimes the most challenging ones Rule # 17: Omit Needless Words - Strunk, William, Jr. ------------------------------------------------------------- Universal DBA | Ajax Rating -- modified at 19:15 Friday 23rd June, 2006
Easy. Defining x = ea => a = ln(x) sqrt(x) = exp(a / 2) sqrt(x) = exp(ln(x) / 2) Similarly, you can go for other roots (n) by: nth_root(x) = exp(ln(x) / n)
Ryan
"Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late" John Nichol "Point Of Impact"
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Can You Find ROOT of a number on a standard calculator WITHOUT touching the root sign or power sign a power sign also acts like a root sign and believe me you can Easy huh! try it ? tell me the EXACT steps * Easy looking problems are sometimes the most challenging ones Rule # 17: Omit Needless Words - Strunk, William, Jr. ------------------------------------------------------------- Universal DBA | Ajax Rating -- modified at 19:15 Friday 23rd June, 2006
Use of Bakhshali Formula here[^] ofcourse as suggested by ed.poore and a few others we have to do a guess work , and yes this bakshali formula is actually a version of newtons method and as you will see its uses two iterations in a single step q = A ^ 2 + b = A * A + b example 41 = 6 ^ 2 + 5 = 6 * 6 + 5 Bakhshali Formula says sqrt(q) = sqrt(A^2 + b) = A + b/(2A) - (b/(2A))^2/[2(A + b/(2A))] which comes down to Sqrt(q) = sqrt(A^2 + b) = A + b/(2A) - (b/(2*A))*(b/(2*A)) / (2*(A + b/(2*A))) sqrt(41)[^] = 6 + 5/(2*6) - ((5/(2*6))*(5/(2*6)))/(2*(6 + 5/(2*6)))[^] Mathematical operations used = /, * , + , - have a nice weekend Rule # 17: Omit Needless Words - Strunk, William, Jr. ------------------------------------------------------------- Universal DBA | Ajax Rating
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Quartz... wrote:
but it won't give a result instantly in a calculator
Neither does the ^ or sqrt button, it performs the ln comparison to solve roots. It just happens much more quickly. Basically it all depends on how accurate you want to be.
The Welsh will always support two teams: The Welsh, and anyone playing England :)
i have added the solution its quite similar to yours here[^] have a good weekend Rule # 17: Omit Needless Words - Strunk, William, Jr. ------------------------------------------------------------- Universal DBA | Ajax Rating
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Easy. Defining x = ea => a = ln(x) sqrt(x) = exp(a / 2) sqrt(x) = exp(ln(x) / 2) Similarly, you can go for other roots (n) by: nth_root(x) = exp(ln(x) / n)
Ryan
"Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late" John Nichol "Point Of Impact"
Hey ryan Thats a great solution exp is actually the power button, i added later that without using the root ket or power key i have added a solution here[^] have a good weekend raj Rule # 17: Omit Needless Words - Strunk, William, Jr. ------------------------------------------------------------- Universal DBA | Ajax Rating
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Can You Find ROOT of a number on a standard calculator WITHOUT touching the root sign or power sign a power sign also acts like a root sign and believe me you can Easy huh! try it ? tell me the EXACT steps * Easy looking problems are sometimes the most challenging ones Rule # 17: Omit Needless Words - Strunk, William, Jr. ------------------------------------------------------------- Universal DBA | Ajax Rating -- modified at 19:15 Friday 23rd June, 2006
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Use of Bakhshali Formula here[^] ofcourse as suggested by ed.poore and a few others we have to do a guess work , and yes this bakshali formula is actually a version of newtons method and as you will see its uses two iterations in a single step q = A ^ 2 + b = A * A + b example 41 = 6 ^ 2 + 5 = 6 * 6 + 5 Bakhshali Formula says sqrt(q) = sqrt(A^2 + b) = A + b/(2A) - (b/(2A))^2/[2(A + b/(2A))] which comes down to Sqrt(q) = sqrt(A^2 + b) = A + b/(2A) - (b/(2*A))*(b/(2*A)) / (2*(A + b/(2*A))) sqrt(41)[^] = 6 + 5/(2*6) - ((5/(2*6))*(5/(2*6)))/(2*(6 + 5/(2*6)))[^] Mathematical operations used = /, * , + , - have a nice weekend Rule # 17: Omit Needless Words - Strunk, William, Jr. ------------------------------------------------------------- Universal DBA | Ajax Rating
sqrt x = log x/log 2 This I will do if only the sqrt key was not working. If the log key wasn't working either, I would have flunked! Abhishek
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sqrt x = log x/log 2 This I will do if only the sqrt key was not working. If the log key wasn't working either, I would have flunked! Abhishek
AbhishekBK wrote:
sqrt x = log x/log 2
Errr... no it's not. sqrt(9) = 3 log(9) / log(2) = 3.1699250014423123629074778878956
Ryan
"Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late" John Nichol "Point Of Impact"
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AbhishekBK wrote:
sqrt x = log x/log 2
Errr... no it's not. sqrt(9) = 3 log(9) / log(2) = 3.1699250014423123629074778878956
Ryan
"Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late" John Nichol "Point Of Impact"
er........... looks like I would have flunked anyway! :laugh: Abhishek
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er........... looks like I would have flunked anyway! :laugh: Abhishek
AbhishekBK wrote:
looks like I would have flunked anyway!
Hehe :)
Ryan
"Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late" John Nichol "Point Of Impact"
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Quartz... wrote:
ROOT of a number on a standard calculator WITHOUT touching the root sign or power sign a power sign also acts like a root sign
4/2 (4 divided by 2) 2 squared is 4 the square root of 4 is 2. what did i win? :laugh: ...yeah, i heard you.
david s_ wrote:
what did i win?
Square root of -2 Rule # 17: Omit Needless Words - Strunk, William, Jr. ------------------------------------------------------------- Universal DBA | Ajax Rating