Application of the Binomial Theorum using fractional values to reduce terms.
Application of the Binomial Theorum.
The Binomial Series
In various newsgroups and on-line math sites we find students seeking help having been given the task of finding square roots by other methods than by calculator etc.
Here is one example:
Calculate 26� to six decimal places
I was about to write a revision of the Binomial Series, but saved myself from aching fingers by finding this excellent tutorial, this gives all the information required to perform the above task.
http://www.krysstal.com/binomial.html
Here is my small addition:
26� = (25 + 1)� 25 = the largest perfect square below 26
Bringing 25 out of the brackets gives (25(1 + 1/25))� or 25�(1 + 1/25)� = 5(1 + 1/25)�
Applying the Series then multiplying by 5 gives the result.
The success of the method depends upon expressing 26 in the form u2 + v where v/u2 is small. It is not always possible to express an integer in this form if u is confined to integral values, for example, in the calculation of 2� the best we can do is to write:
2� = (4 - 2)� = 4(1 - 1/2)�
The terms in the binomial expansion of (1 - 1/2)� do not rapidly become small and it is necessary to to take more than 10 terms in order to obtain a result correct to 6 decimal places.
In such cases we choose a suitable fractional value for u in the following way:
The squares of the first few positive integers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121……… We notice that 102/72 is very nearly equal to 2.
In fact: 2 = (102-2)/72 or 102/72 - 2/72
therefore 2½ = (102/72 - 2/72 )½ or 10/7(1 - 1/50)½
4 terms of the binomial expansion give a result correct to 7 decimal places.
I have written a Qbasic program to calculate binomial expansions ( do the donkey work!) and show the terms required. binom_rt.zip download now
Adapted from ALGEBRA by A.PAGE M.Sc., Ph.D University of London Press 1947
We shall try to find the square root of 121, which we presume we do not know the answer to.
We need to find a number close to, and below, 121 that we know the square root of. In this case it will be 100, the square root being 10.
Let x = 100
The difference between 121 and 100 = 21
Let delta x = 21
Formula:
Sqrt( x + delta x ) approx = sqrt(x) + (delta x / (2 * sqrt(x))
Example: sqrt(100 + 21) approx = sqrt(100) + (21/(2 * sqrt(100)))
approx = 10 + 21/20
approx = 10 + 1
approx = 11
Regards to 'Dan' who posted the method in www.allexperts.com
This method is similar to long division, some practice will make the
process a little easier.
Let's try to find the square root of 545.6 correct to three decimal places...
Starting at the decimal point and working left and right we partition
the number into periods of two digits each.
Each partition will give a one digit portion of the root.
545.6 will result in this format:
05 45.60 00 00
The first period is 5. The largest square less than or equal
2 05 45 .60 00 00 4
Subtract 4 from 5 and bring down the next period.
2 05 45.60 00 00 4 1 45
We now double the extraction, which at this point is 2 and place it to
the left of the subtraction.
We must now find the largest digit n so that 4n times n is as close
as possible to 145.
In this case it is 43 times 3 (123).
2 05 45.60 00 00 4 1 45 4n x n 1 23
Place the value of n above the second period then subtract 123 from 145 and bring down the third period.
2 3 05 45.60 00 00 4 1 45 4n x n 1 23 22 60
Now we repeat the process, double the extraction (2 times 23 = 46),
find the largest digit so that 46n times n is as close as possible to 2260.
In this case it is 464 times 4.
2 3. 4 05 45.60 00 00 4 1 45 4n x n 1 23 22 60 46n x n 18 56
I'll allow you to complete the process.....
We pick, let's say, 1515 to find the square root of.
Now take a rough guess at the root, say 50.
Square = 1515
Guess = 50
Square/Guess = 30.3
Average = ( 50 + 30.3 )/2 = 40.15
This result becomes the new guess.
Square = 1515
Guess = 40.15
Square/Guess = 37.733 (3 dp)
Average = ( 40.15 + 37.733 )/2 = 38.942
Square = 1515
Guess = 38.904
Square/Average = 38.942
Average = ( 38.942 + 38.904 )/2 = 38.923
Square = 1515
Guess = 38.923
Square/Guess = 38.923
Your calculator will confirm this is correct to three decimal places.