Originally Posted by

**Soroban** Hello, Vicky!

I *think* I have an answer to your question.

In the Old Days, say 100 B.C. (Before Calculators), we had only our brains and a pencil.

If we want to evaluate, for example, $\displaystyle \frac{1}{\sqrt{2}}$, we would:

. . [1] Find an approximation for $\displaystyle \sqrt{2}$ on a square-root table (1.4142)

. . [2] Perform the long division: .$\displaystyle 1 \div 1.414212$

The divison looks like this: . $\displaystyle \begin{array}{cccc}& - & - - - - - - \\ 1.4142& ) & 1.0000000 \end{array}$

. . $\displaystyle \begin{array}{ccccccccccccc}$$\displaystyle

& & & & & & 0. & 7 & 0 & 7 & 1 & \hdots\\

& & -- & -- & -- & -- & -- & -- & -- & --& -- & --\\

1\;4\;1\;4\;2 & ) & 1 & 0 & 0 & 0 & 0. & 0 & 0 & 0 & 0 & \hdots\\

&&& 9 & 8 & 9 & 9 & 4 \\

&& -- & -- & -- & -- & -- \\

&&&& 1 & 0 & 0 & 6 & 0 \\

&&&&&&&& 0 \\

&&&& -- & -- & -- & -- & -- \\

&&&& 1 & 0 & 0 & 6 & 0 & 0 \\

&&&&& 9 & 8 & 9 & 9 & 4 \\

&&&&& -- & -- & -- & -- & -- \\ \end{array}$

. - - . . . . . . . . . . . . . . . $\displaystyle \begin{array}{cccccccccccc}$$\displaystyle

1 & 6 & 0 & 6 & 0 \\

1 & 4 & 1 & 4 & 2 \\

-- & -- & -- & -- & -- \\

& 1 & 9 & 1 & 8 & 0 & \hdots \end{array}$

Pretty tedious, dividing by a 5-digit number.

. . (Worse, if we had more decimal places.)

Rationalizing, we would have: .$\displaystyle \frac{\sqrt{2}}{2}$

And I would much prefer to do: .$\displaystyle 1.414213562... \div 2$

. . Wouldn't you?

And Quacky's view on combining fractions is excellent!