Converting single unit expressions into mixed unit expressions in standard form

So far, we have looked at converting mixed units to a single unit, and also at performing addition while remaining in mixed units.  It is time to learn how to go from single units expressions to given mixed units.

Suppose, for instance, we wish to convert `208` inches into yards, feet, and inches.  One can of course say trivially that

`\qquad\qquad\qquad 208  =  0_{3}0_{12}208,`

i.e. 0 yard, 0 feet, 208 inches.  That probably isn't what we wanted, so we need to be more specific---we want to convert into yards, feet, and inches in standard form.

The way to to this is by converting smaller units into complete larger units, i.e. by carrying; the way to do that is by successive division.  How many whole feet, i.e. how many complete lots of `12` inches, go into `208` inches?  Dividing `12` into `208`, i.e. `12 \\ 208` or `208 -: 12`, we find that `12` goes `17` times, with a remainder of `4`, often written `17  "r"  4`.  In  `17  "r"  4`, "r" stands for "remainder".  Instead of this, we shall write `17_{12}4`, because if we did the division correctly, then `17_{12]4` has to be a wiggle expression equivalent to `208`.

In any case, `208` inches is equivalent to `17` feet and `4` inches, and this would be our final answer if we were using only feet an inches as units.  But since we are also using yards, we divide `17` by `3`, whence `17  = 5_{3}2`, i.e. `17` feet is `5` yards and `2` feet.

So in the end, `208` inches is `5` yards, `2` feet, `4` inches.  In wiggle without units,

`\qquad\qquad\qquad 208  =  5_{3}2_{12}4.`

The calculation proceeds from right to left, dividing by successive interbases, writing only the remainder, and passing the quotient to the next interbase for division.  The final quotient is then written before the leftmost interbase.

The work of successive division can be set forth as follows :

`\qquad\qquad\qquad{:"       "\ul{"      5"}_{"   3 "}2`
`\qquad\qquad\qquad{:"   3 "\ul{")   17"}_{" 12 "}4`
`\qquad\qquad\qquad{:" 12 )  208."`

Brave souls may even elect to drop the divisors on the left, and write them just on the right, at levels between the lines, from the beginning.


`\qquad\qquad\qquad{:"       "\ul{"      5"}_{"   3 "}2`
`\qquad\qquad\qquad{:"      "\ul{"    17"}_{" 12 "}4`
`\qquad\qquad\qquad{:"         208."`