This new first part is in response to a query.
#color(blue)("Additional explanation: demonstration of rounding by example")#
I chose to use the number #23.421# as the starting point for my example
Round to 2 decimal places
#23.42color(red)(1) ->#round down as #color(red)( 1) <5 ->23.42#
#23.42color(red)(2) ->#round down as #color(red)( 2) <5 ->23.42#
#23.42color(red)(3) ->#round down as #color(red)( 3) <5 ->23.42#
#23.42color(red)(4) ->#round down as #color(red)( 4) <5 ->23.42#
#23.42color(red)(5) ->#round up as #color(red)( 5) >=5 ->23.43#
#23.42color(red)(6) ->#round up as #color(red)( 6) >=5 ->23.43#
#23.42color(red)(7) ->#round up as #color(red)( 7) >=5 ->23.43#
#23.42color(red)(8) ->#round up as #color(red)( 8) >=5 ->23.43#
#23.42color(red)(9) ->#round up as #color(red)( 9) >=5 ->23.43#
#23.43color(red)(0) ->#round down as #color(red)( 0) <5 ->23.43#
#23.43color(red)(1) ->#round down as #color(red)( 1) <5 ->23.43#
#23.43color(red)(2) ->#round down as #color(red)( 2) <5 ->23.43#
#23.43color(red)(3) ->#round down as #color(red)( 3) <5 ->23.43#
#23.43color(red)(4) ->#round down as #color(red)( 4) <5 ->23.43#
#23.43color(red)(5) ->#round up as #color(red)( 5) >=5 ->23.44#
#23.43color(red)(6) ->#round up as #color(red)( 6) >=5 ->23.44#
And so on:
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#color(blue)("Answering the question")#
By calculator #2sqrt(3) = 3.46410....#
The dots indicate that the digits go on for a very long time.
As we need the last digit to be the #10^("ths")#, the cut off point is:
#color(green)(3.4color(red)(|)6410....)#
#color(white)(3.4)color(red)(uarr)#
#color(red)("Cut off point")#
We now consider the 3 units#+4/10+6/100#
The digit to the right is 6. As this satisfies the condition of 5 or more we round up the 4 by 1 and consider all the digits to the right of the cut off point as 0's:
#color(green)(3.5000..) = 3.5#
