Finding the square root of 19.3 by bisection

We will try to find it with the precision of one decimal.

We know by guessing that 4² < 19.3 < 5². Thus the square root of 19.3 lies between x₀ = 4 and y₀ = 5. Now we start the procedure

Step 1.
m₀ = 4.5.
Since m₀² = 20.25 > 19.3, the root must be between x₀ = 4 and m₀ = 4.5.
We set x₁ = x₀ = 4 and y₁ = m₀ = 4.5.

Step 2.
m₁ = 4.25.
Since m₁² = 18.0625 < 19.3, the root must be between m₁ = 4.25 and y₁ = 4.5.
We set x₂ = m₁ = 4.25 and y₂ = y₁ = 4.5.

Step 3.
m₂ = 4.375.
Since m₂² = 19.140625 < 19.3, the root must be between m₂ = 4.375 and y₂ = 4.5.
We set x₃ = m₂ = 4.375 and y₃ = y₂ = 4.5.

Step 4.
m₃ = 4.4375.
Since m₃² = 19.691... > 19.3, the root must be between x₃ = 4.375 and m₃ = 4.4375.
We set x₄ = x₃ = 4.375 and y₄ = m₃ = 4.4375.

Step 5.
m₄ = 4.40625.
Since m₄² = 19.415... > 19.3, the root must be between x₄ = 4.375 and m₄ = 4.40625.
We set x₅ = x₄ = 4.375 and y₅ = m₄ = 4.40625.

Note that the difference between x₅ and y₅ is less then 0.03125. So we can now say that the square root of 19.3 is roughly 4.4 with possible error 0.032, in particular the first decimal digit is definitely correct (actually, we could have already stopped after Step 4 for this conclusion). Since 4.4² = 19.36, we are really close to the root of 19.3.
Note that another round would decrease the error to 0.016, the next one to 0.008 and so on.