If you space looking in ~ approximation methods that you can employ making use of pen, paper and some psychological arithmetic, girlfriend can shot a Binomial Expansion.

You are watching: What is the square root of 43

If you begin at 36, a square number, so the you are looking for #sqrt(36 +7)#, you deserve to now play through that a little:

#sqrt(36 +7) = sqrt 36 sqrt(1 +7/36) = 6 sqrt(1 +7/36)#

You can then usage the Binomial development , ie:

#(1+x)^alpha = 1 + alpha x + (alpha (alpha - 1))/(2!)x^2 + ...#:

In this case:

#6 (1 +7/36)^(1/2)#

#=6 (color(green)(1 +1/2 * 7/36) + 1/2 (-1/2)(1/(2!)) (7/36)^2 + ...)#

Even simply the first two terms offer #79/12 approx 6.583# and #6.583^2 approx 43.34#

We might get a little closer by utilizing a different square number. If you start at 49, one more square number, girlfriend are now looking at:

#sqrt(49 -6) = sqrt 49 sqrt(1 - 6/49) = 7 sqrt(1 - 6/49)#

Using just the an initial 2 terms the the Binomial Expansion:

#= 7 (1 - 1/2 * 6/49 ) = 46/7 approx 6.571#

And #6.571^2 = 43.18# George C.
Feb 17, 2017

#sqrt(43) = 13/2+(3/4)/(13+(3/4)/(13+(3/4)/(13+...)))#

Explanation:

#43# is a prime number, for this reason its square root is irrational.

We can uncover approximations to it as follows...

Note that #43# is roughly fifty percent way between #36=6^2# and #49=7^2#

So a good an initial approximation for #sqrt(43)# would be #13/2#.

We find:

#(13/2)^2 = 169/4 = 42.25#

Given #n > 0# and also #0 we have the right to write #sqrt(n)# together a generalised ongoing fraction:

#sqrt(n) = a+b/(2a+b/(2a+b/(2a+...)))#

where #b = n-a^2#

So in our example:

#n = 43#, #a = 13/2# and #b=43-169/4 = 3/4#

So:

#sqrt(43) = 13/2+(3/4)/(13+(3/4)/(13+(3/4)/(13+...)))#

We can truncate this to get rational approximations.

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For example:

#sqrt(43) ~~ 13/2+(3/4)/13 = 341/52 ~~ 6.5577#

#sqrt(43) ~~ 13/2+(3/4)/(13+(3/4)/13) = 8905/1358 ~~ 6.557437#

A calculator tells me:

#sqrt(43) ~~ 6.5574385243#

See https://yellowcomic.com/s/aCh3Xasm for another example and explanation that this method. 