Part A: how to tell sqrt(140) does not equal 10.7 without using a calculator.
Method 1:
We know that 11*11=121 < 140, so 10.7< sqrt(121) < sqrt(140) because we know that the sqrt() function is monotonic increasing.
Method 2:
If we transform 10.7^2=(10+0.7)^2=10^2+2*10*0.7+0.7^2=100+1.4+0.49 is less than 110, so 10.7^2<110<140, so 10.7 cannot be the answer.
Part B: Find sqrt(140) accurate to 1 place of decimal (without a calculator)
Method 1: use the algorithm of finding square-roots on a piece of paper (to as many places as we wish). The first steps (to 1 decimal) are:
find the greatest integer below the exact square-root. We know this is 11, because 12^2=144.
Take the difference 140-11^2=140-121=19. Add two zeroes.
Divide 1900 by twice 11=22, tag on X to get the number 22X, where X is the quotient 1900/22X=8. 1900/228=8 with remainder 1900-1824=76. Repeat to estimate the second decimal, 7600/456X=1 <5, so the estimate for sqrt(140)=11.81, or 11.8.
It may sound difficult, but is not much more complicated than a long division.
Method 2: (much easier, even mentally)
Use the Newton's method.
As before, estimate the integer closest to the square-root, which is x0=12.
get a better estimate as
x1=x0-(x0^2-140)/(2X0)=12-(144-140)/24=12-4/24=12-1/6=11.83
or sqrt(140)=11.8 to one digit after the decimal.
