Methods of computing square roots and Odd abundant number: Difference between pages

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There are a numbers of ways to calculate [[square root]]s of [[number]]s, and even more ways to [[Estimation|estimate]] them.
An '''odd abundant number''' is an [[odd number]] <math>n</math> that its [[sum-of-divisors|sum-of divisors]] greater than the [[double|twice]] of itself.
==Examples==
*The first example is 945  (''[[3 (number)|3]]<sup>3</sup>× [[5 (number)|5]]× [[7 (number)|7]]''). Its [[prime number|prime]] [[factor|factors]] are [[3 (number)|3]], [[5 (number)|5]], and [[7 (number)|7]]. The next following eleven odd abundant numbers are
1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615.
*Odd abundant numbers below 500000 are in On-Line Encyclopedia of Integer Sequences [http://oeis.org/A005231/b005231.txt, A005231].


The mathematical operation of finding a root is the opposite operation of [[exponentiation]], and therefore involves a similar but reverse thought process.
==Formulas==
The following [[formula]]


Firstly, one needs to know how [[Arithmetic precision|precise]] the result is expected to be.  This is because often [[square root]]s are [[Irrational numbers|irrational]]. For example, square root of a nice round [[whole number]] '''28''' is a [[Fraction (mathematics)|fraction]] which in its [[Decimal places|decimal notation]] has [[Infinity|infinite]] length, and therefore it is impossible to express it exactly:<math display="block">\sqrt{28} \approx 5.291502622129181....</math>
<math>945+630n</math><ref>{{Cite web|url=http://ms.appliedprobability.org/data/files/Articles%2038/38-1-2.pdf|title=More Odd Abundant Sequences|first=|last=|date=2005|website=|publisher=JAY. SCHIFFMAN|accessdate=2017-01-2}}</ref>
presents 62 [[abundant number]]s, but it fails if


Moreover, for some [[real number]]s the square root is a [[complex number]].  For example, square root of '''-4''' is a complex number '''2''[[imaginary unit|i]]''''' :<math display="block">\sqrt{-4} = 2 i</math>
<math>n\le62</math>.


In many cases there may be multiple valid answers.  For example, square root of '''4''' is '''2''', but '''-2''' is also a valid answer.  One can verify that they are both valid answers by squaring each candidate answer and checking if you obtain '''4''' as the result of verification:<math display="block">2^2 = 2 \times 2 = 4</math>
The second formula


<math display="block">(-2)^2=(-2) \times (-2) = 4</math>
<math>3465+2310n</math><ref>{{Cite web|url=http://ms.appliedprobability.org/data/files/Articles%2038/38-1-2.pdf|title=More Odd Abundant  Sequences|first=|last=|date=2005|website=|publisher=JAY. SCHIFFMAN|accessdate=2017-01-26}}</ref>
presents 192 [[abundant number]]s, but fails if


Please note that calculating a square root is a special case of the problem of [[Nth root|calculating N<sup>th</sup> root]].
<math>n\le192</math>


== Calculating ==
The [[third]] formula
Most [[calculator]]s provide a function for calculation of a square root.


{| class="wikitable"
<math>2446903305+1631268870n</math> <ref>{{Cite web|url=http://ms.appliedprobability.org/data/files/Articles%2038/38-1-2.pdf|title=More Odd Abundant  Sequences|first=|last=|date=2005|website=|publisher=JAY. SCHIFFMAN|accessdate=2017-01-26}}</ref>
!
!General Steps
!Example
|-
|width="20%" valign="top"|How to calculate a square root using a simple [[calculator]].
|width="40%" valign="top"|
* ''First, make sure the operating space is clear. This is usually accomplished by clicking the '''C''' button a couple of times.''
* Then type the number whose root you are trying to calculate.
* Then press the square root button (<math>\sqrt{}</math>).
* The number you see on the screen is one of the answers.  Remember, that often there are multiple valid answers, as explained above.
|width="40%" valign="top"|
* Press '''C''' button a couple of times.
* Type '''16'''
* Press <math>\sqrt{}</math> button.
* The answer is '''4.'''  ''Keep in mind that '''-4''' is also a valid answer.''
|}


== Estimating ==
fails if <math>n\le135939073</math>.
If the result does not have to be very precise, the following estimation techniques could be helpful:
{| class="wikitable"
!Methodology
!Example
|-
|width="49%" valign="top"|Suppose you need to find [[square root]] of some number <math>N</math>. 
Find some number <math>A</math> such that <math>A^2</math> (that is <math>A</math> squared, or <math>A</math> times <math>A</math>) is [[Approximation|approximately]] [[Equality (mathematics)|equal]] to <math>N</math> ''(but how close? This needs to be expanded)''.


Then we can think of <math>A</math> as being [[Approximation|approximately]] a [[square root]] of <math>N</math>.
==Properties==
|width="49%" valign="top"|Suppose we need to estimate the square root of 2.
We know that <math>1^2 = 1</math>, and <math>2^2 = 4</math>.
* An calculation was given by Iannucci shows how to find the smallest abundant number not divisible by the first '''''n''''' primes.
 
*An abundant number with abundance 1 is called a [[quasiperfect number]], although none have yet been found. A quasiperfect number must be an odd square number having a value above 10<sup>30<sup>.
Therefore, one of the answers to <math>\sqrt{2}</math> is somewhere between 1 and 2.  
|}


==References==
==References==
*{{Cite web|url = http://mathworld.wolfram.com/SquareRoot.html | title = Square Root}}


{{math-stub}}
{{Reflist}}
 
{{Math-stub}}


[[Category:Mathematics]]
[[Category:Integer sequences]]

Latest revision as of 01:09, 25 April 2017

An odd abundant number is an odd number [math]\displaystyle{ n }[/math] that its sum-of divisors greater than the twice of itself.

Examples

  • The first example is 945 (33× 5× 7). Its prime factors are 3, 5, and 7. The next following eleven odd abundant numbers are

1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615.

  • Odd abundant numbers below 500000 are in On-Line Encyclopedia of Integer Sequences A005231.

Formulas

The following formula

[math]\displaystyle{ 945+630n }[/math][1] presents 62 abundant numbers, but it fails if

[math]\displaystyle{ n\le62 }[/math].

The second formula

[math]\displaystyle{ 3465+2310n }[/math][2] presents 192 abundant numbers, but fails if

[math]\displaystyle{ n\le192 }[/math]

The third formula

[math]\displaystyle{ 2446903305+1631268870n }[/math] [3]

fails if [math]\displaystyle{ n\le135939073 }[/math].

Properties

  • An calculation was given by Iannucci shows how to find the smallest abundant number not divisible by the first n primes.
  • An abundant number with abundance 1 is called a quasiperfect number, although none have yet been found. A quasiperfect number must be an odd square number having a value above 1030.

References

Template:Reflist

Template:Math-stub

  1. ↑ Template:Cite web
  2. ↑ Template:Cite web
  3. ↑ Template:Cite web
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