數學知識交流---計算近似值(2)

37 = 36 + 1
sqrt(37) = 6 + X with X < 1
=> 37 = (6 + X)^2
=> 37 = 36 + 12X + X^2

Since X < 1, let X^2 = 0 and we have 12X = 1 and X = 0.083

.083^2 = .007 and this is very small compare to 6

therefore sqrt(37) is about 6.083 and 6.083^2 = 37.003

For sqrt(29), we can use similar technique with
sqrt(29) = (5 + X)^2

=> 29 = 25 + 10X + X^2

and obtain X = 0.4

However, 0.4^2 = 0.16 and may be a bit too big comparing with 5

One more iteration subsituting X^2 with 0.16, we have
29 = 25 + 10X + 0.16
X = 0.384

In this case, X^2 = 0.147 and 5.384^2 = 28.99