Using the duplex method to calculate square roots

I have been assigned to find out how a calculator figures out square roots, so far the shortest thing I can see is "the duplex method". But the thing is that the explanation on Wikipedia makes no sense for me:

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Find the square root of 2,080,180,881. Solution by the duplex method: this ten-digit square has five digit-pairs, so it will have a five-digit square root. The first digit-pair is 20. Put the colon to the right. The nearest square below 20 is 16, whose root is 4, the first root digit. So, use 2·4=8 for the divisor. Now proceed with the duplex division, one digit column at a time. Prefix the remainder to the next dividend digit.

divisor; gross dividend: 8) 20:  8   0   1   8    0   8   8   1
read the dividend diagonally up: 4   8   7  11   10  10   0   8
minus the duplex:    16: xx  25  60  36   90 108  00  81
actual dividend:      : 48  55  11  82   10  00  08  00
minus the product:      : 40  48  00  72   00  00   0  00
remainder:     4:  8   7  11  10   10   0   8  00
quotient:     4:  5,  6   0   9.   0   0   0   0
Duplex calculations:
Quotient-digits ==> Duplex deduction.
5       ==> 52= 25
5 and 6 ==> 2(5·6) = 60
5,6,0   ==> 2(5·0)+62 = 36
5,6,0,9 ==> 2(5·9)+2(6·0) = 90
5,6,0,9,0 ==> 2(5·0)+2(6·9)+ 0 = 108
5,6,0,9,0,0 ==> 2(5·0)+2(6·0)+2(0·9) = 0
5,6,0,9,0,0,0 ==> 2(5·0)+2(6·0)+2(0·0)+92 = 81


Hence the square root of 2,080,180,881 is exactly 45,609.

" What does it actually mean by dividend diagonally up, where do the rows connect, how does the answer come out to 45609?

Please help me understand, sorry if this seams like a pointless question but it's my first time here.

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I am still wondering how you concluded that this method is best from Methods of Computing Square Roots. For example, did you look at Newton's method? Also, did you look at the convergence properties of the algorithms? For your immediate question, please review math.stackexchange.com/questions/48695/… – Amzoti Dec 21 '12 at 2:46
@Amzoti, the Newton's method doesn't make too much sense to me but I will try to look at it again. And I'll also check out the link. – user115422 Dec 21 '12 at 20:37