Let us repeat the problem of the last posting, but solve it exactly using Egyptian division. Egyptian division employs two columns of numbers that form a proportion table. It does not require knowledge of times tables---knowing how to double suffices. Nor does it require guess and check estimation of multiplies usually employed with the standard long division algorithm. The disadvantage is that the calculation is roughly three times as long as for the standard algorithm.
We can start just as we did for approximate division in the last posting :
`\qquad\qquad\qquad[(1543, x),(21, 1)]`.
To this, double the last row and append, double the last row and append, continuing until the numbers at the bottom of the first column bracket `1543`, i.e. `1344 < 1543 < 2688` :
`\qquad\qquad\qquad[(1543, x),(21, 1),(42,2),(84,4),(168,8),(336,16),(672,32),(1344,64),(2688,128)]`.
Now we start add the bottom, succesively adding some rows but not others, trying to make `1543`. `2688` is already too big, so we ignore it. (In practice, we could usually forsee this and do not write that row in the first place.) The next row up, led by `1344` is smaller than 1543, and contributes.
`\qquad\qquad\qquad[(1543, x),(21, 1),(42,2),(84,4),(168,8),(336,16),(672,32),(1344,64)]`.
The next row up, led by `1344` is smaller than `1543`, so that row contributes, as it always does. We still have `1543 - 1344 = 199` to account for. `672` and `336` are both bigger than `199`, so those rows do not contribute, and can be ignored. `168` is smaller, so it contributes, leaving `199 - 168 = 31` to account for. `84` and `42` are larger than `31`, so those rows do not contribute. `21` is smaller, so that row contributes, and leaves a remainder is `31 - 21 = 10`.
On paper, we would strike out the rows to be ignored: here we delete them all.
`\qquad\qquad\qquad[(1543, x),(21, 1),(168,8),(1344,64)]`.
Now, we append the sum of the bottom three rows :
`\qquad\qquad\qquad[(1543, x),(21, 1),(168,8),(1344,64), (21+168+1344,1+8+64)] = [(1543, x),(21, 1),(168,8),(1344,64), (1533,73)]`.
In practice, again, we would not perform the sum `21 + 168 + 1344 =1533`. We do not need it, and even if we did, it is more readily calculated by subtracting the remainder from our original dividend, `1543 - 10 = 1533`.
We do have to perform `1+8+64 = 73`, however, to get the quotient `73`.
So we have, `1543 -: 21 = 73 " remainder "10`.
So far, we have performed divmod division, i.e. found an integer quotient and a remainder. (div and mod are two convenient computer arithmetic terms. div gives only the integer quotient, `1543 " div " 21 = 3`, while mod gives only the remainder,`1543 " mod " 21 = 10`.)
Using wiggle notation, we can write :
`\qquad\qquad1543 -: 21 = 73;_{21}10 = 73 10/21`.
where now we are looking at the result as a mixed fraction.
It is easy to extend Egyptian division so that it gives us a decimal fraction result to any desired degree of accuracy, as we shall see in the next post.
Wednesday, February 24, 2010
Approximate division using a proportion table
Suppose we are trying to get a roughly estimate of how many times `21` goes into `1543`. i.e. we are looking to approximate `1543/21`.
We can set this up as an incomplete proportion table :
`\qquad\qquad\qquad[(1543, x),(21, 1)]`.
Using the proportion table form of the Rule of Three, we can solve for `x` by multiplying the neighbors and dividing by the opposite corner :
`\qquad\qquad\qquad x = \frac{1543 xx 1}{21}`,
i.e., completing this proportion table correctly should indeed give us exactly the quantity we seek. But we are not seeking an exact value, and are instead hoping to extract an approximate answer with less work.
First, we multiply the last row by ten and append, and repeat this until the number at the top left of the table is bracketed by the first values on the two last rows, i.e. `210<1543<2100` :
`\qquad\qquad\qquad[(1543, x),(21, 1),(210, 10),(2100, 100)]`.
Since in the first column `210<1543<2100`, we can conclude from the second column that `10 < x < 100`.
To sharpen this estimate a little, we can round `2100` to `2000` and `1543` to `1500`. Now `1500` is three quarters of `2000`, so `x` is about three quarters of `100`, i.e. `1543/21` is roughly `75`.
We can set this up as an incomplete proportion table :
`\qquad\qquad\qquad[(1543, x),(21, 1)]`.
Using the proportion table form of the Rule of Three, we can solve for `x` by multiplying the neighbors and dividing by the opposite corner :
`\qquad\qquad\qquad x = \frac{1543 xx 1}{21}`,
i.e., completing this proportion table correctly should indeed give us exactly the quantity we seek. But we are not seeking an exact value, and are instead hoping to extract an approximate answer with less work.
First, we multiply the last row by ten and append, and repeat this until the number at the top left of the table is bracketed by the first values on the two last rows, i.e. `210<1543<2100` :
`\qquad\qquad\qquad[(1543, x),(21, 1),(210, 10),(2100, 100)]`.
Since in the first column `210<1543<2100`, we can conclude from the second column that `10 < x < 100`.
To sharpen this estimate a little, we can round `2100` to `2000` and `1543` to `1500`. Now `1500` is three quarters of `2000`, so `x` is about three quarters of `100`, i.e. `1543/21` is roughly `75`.
Extending a proportion table
So far, we have rearranged proportion tables, we have deleted rows and/or columns, and we have cut proportion tables into smaller pieces.
Now we want to make larger tables.
One way to make a larger table is simply to duplicate a row,
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(4, 8, 10, 18),(6, 12, 15, 27)] `.
or a column,
`\qquad\qquad [(2, 4, 5, 9, 5),(4, 8, 10, 18, 10),(6, 12, 15, 27, 15)] \qquad->\qquad [(2, 4, 5, 9, 5),(4, 8, 10, 18, 10),(6, 12, 15, 27, 15)] `.
or both :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9, 5),(4, 8, 10, 18, 10),(4, 8, 10, 18, 10),(6, 12, 15, 27, 15)] `
Of course we can duplicate any row or column as many times as we like. And we can do that to many distinct rows and many distinct columns.
Another way to augment a proportion table is by appending a row that is a multiple of another row. A multiple of one row is necessarily a multiple (usually by different multipliers) of every row :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27),(60, 120, 150, 270)] `.
It should be clear that the result of appending a multiple row will always still be a proportion table, and mutatis mutandis for columns.
One can always append a column that is a sum of two or more other columns (and mutatis mutandis for rows) :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9, 5+9),(4, 8, 10, 18,10+18),(6, 12, 15, 27,15+27)] `,
`\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = \qquad [(2, 4, 5, 9, 14),(4, 8, 10, 18, 28),(6, 12, 15, 27, 42)]`.
More generally, the result of appending a sum or difference of existing rows or of columns is another proportion table.
In the last posting, we dealt with subtables and amalgamation. Amalgamating two rows is equivalent to appending their sum, and deleting the original two rows, and similarly for amalgamating two columns. Any amalgamation whatever of a table (i.e. over a tartanlike decomposition) can be constructed by a suitably chosen succession of amalgamations of pairs of rows and pairs of columns.
Now we want to make larger tables.
One way to make a larger table is simply to duplicate a row,
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(4, 8, 10, 18),(6, 12, 15, 27)] `.
or a column,
`\qquad\qquad [(2, 4, 5, 9, 5),(4, 8, 10, 18, 10),(6, 12, 15, 27, 15)] \qquad->\qquad [(2, 4, 5, 9, 5),(4, 8, 10, 18, 10),(6, 12, 15, 27, 15)] `.
or both :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9, 5),(4, 8, 10, 18, 10),(4, 8, 10, 18, 10),(6, 12, 15, 27, 15)] `
Of course we can duplicate any row or column as many times as we like. And we can do that to many distinct rows and many distinct columns.
Another way to augment a proportion table is by appending a row that is a multiple of another row. A multiple of one row is necessarily a multiple (usually by different multipliers) of every row :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27),(60, 120, 150, 270)] `.
It should be clear that the result of appending a multiple row will always still be a proportion table, and mutatis mutandis for columns.
One can always append a column that is a sum of two or more other columns (and mutatis mutandis for rows) :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9, 5+9),(4, 8, 10, 18,10+18),(6, 12, 15, 27,15+27)] `,
`\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad = \qquad [(2, 4, 5, 9, 14),(4, 8, 10, 18, 28),(6, 12, 15, 27, 42)]`.
More generally, the result of appending a sum or difference of existing rows or of columns is another proportion table.
In the last posting, we dealt with subtables and amalgamation. Amalgamating two rows is equivalent to appending their sum, and deleting the original two rows, and similarly for amalgamating two columns. Any amalgamation whatever of a table (i.e. over a tartanlike decomposition) can be constructed by a suitably chosen succession of amalgamations of pairs of rows and pairs of columns.
Suitable subtle subtables
We can delete, or just simply ignore, arbitrary rows and columns of a given table. The result is a subtable. Any subtable of a proportion table is also a proportion table. Here, we delete the third row, and the first and third columns, of a table to get a subtable :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(4, 9),(8, 18)] `.
In general, making a subtable is a lossy operation---there is less table when we are done.
We can split a table into a complete, nonoverlapping collection of subtables. For instance :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27), (10, 20, 25, 45)] \qquad->\qquad{: ([(2, \qquad4),(4, \qquad8)]\qquad [(5, 9),(10, 18)]), ([(6, 12),(10, 20)]\qquad [(15, 27),(25, 45)]) :} `.
The preceding patchwork of subtables was made by carving up the original table in an obvious way. A nonoverlapping collection does not have to fit together like patchwork, however :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27), (10, 20, 25, 45)] \qquad->\qquad{: ([(\quad2, 9),(10, 45)]\qquad [(4, 5),(20, 25)]), ([(\quad4, 18),(\quad6, 27)]\qquad [(8, 10),(12, 15)]) :} `.
The subtables do not have to be square, or the same size, or deployed in a symmetrical pattern. The most useful decompositions are those that are tartanlike, meaning that any split between cells belonging to distinct subtables goes all the way across the original table.
The two preceding decompositions are tartanlike. This one is not:
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27), (10, 20, 25, 45)] \qquad->\qquad{: ([(2, \qquad4,\qquad5, 9),(4, \qquad8,\qquad10, 18)]), ([(6, 12),(10, 20)]\qquad [(15, 27),(25, 45)]) :} `.
We have arranged the subtables according to the position of its reference element---by convention its top left element---in the original table. Using such a convention is no mere convenience, as we shall soon see.
Tartanlike decompositions can be aggregated, by adding together the elements in each subtable :
`\qquad\qquad\qquad{: ([(2, \qquad4),(4, \qquad8)]\qquad [(5, 9),(10, 18)]), ([(6, 12),(10, 20)]\qquad [(15, 27),(25, 45)]) :}\quad->\quad{: ([(),(\qquad18\qquad),()]\quad [(),(\qquad42\qquad),()]), ([(),(\qquad48\qquad),()]\quad [(),(\quad112\qquad),()]) :}`,
where the result can also be written
`\qquad\qquad\qquad[(18,42),(48,112)]`.
Similarly,
`\qquad\qquad\qquad{: ([(\quad2, 9),(10, 45)]\qquad [(4, 5),(20, 25)]), ([(\quad4, 18),(\quad6, 27)]\qquad [(8, 10),(12, 15)]) :}\quad->\quad[(66,54),(55,45)] `,
where the convention we used before helps us keep the aggregated subtables in proportional relation.
A table filled entirely with ones is a proportion table. By aggregating such a table of suitable size, we can construct any proportion table whatever that has only positive integer elements. For instance :
`\qquad\qquad\qquad [(1,1,1,1,1,1),(1,1,1,1,1,1),(1,1,1,1,1,1),(1,1,1,1,1,1),(1,1,1,1,1,1),(1,1,1,1,1,1)]\quad->\quad[(1,2,3),(2,4,6),(3,6,9)],
where an appropriate tartanlike decomposition that leads from the first of these tables to the second should be obvious.
These three examples illustrate a general truth: aggregating a proportion table always gives another proportion table.
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(4, 9),(8, 18)] `.
In general, making a subtable is a lossy operation---there is less table when we are done.
We can split a table into a complete, nonoverlapping collection of subtables. For instance :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27), (10, 20, 25, 45)] \qquad->\qquad{: ([(2, \qquad4),(4, \qquad8)]\qquad [(5, 9),(10, 18)]), ([(6, 12),(10, 20)]\qquad [(15, 27),(25, 45)]) :} `.
The preceding patchwork of subtables was made by carving up the original table in an obvious way. A nonoverlapping collection does not have to fit together like patchwork, however :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27), (10, 20, 25, 45)] \qquad->\qquad{: ([(\quad2, 9),(10, 45)]\qquad [(4, 5),(20, 25)]), ([(\quad4, 18),(\quad6, 27)]\qquad [(8, 10),(12, 15)]) :} `.
The subtables do not have to be square, or the same size, or deployed in a symmetrical pattern. The most useful decompositions are those that are tartanlike, meaning that any split between cells belonging to distinct subtables goes all the way across the original table.
The two preceding decompositions are tartanlike. This one is not:
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27), (10, 20, 25, 45)] \qquad->\qquad{: ([(2, \qquad4,\qquad5, 9),(4, \qquad8,\qquad10, 18)]), ([(6, 12),(10, 20)]\qquad [(15, 27),(25, 45)]) :} `.
We have arranged the subtables according to the position of its reference element---by convention its top left element---in the original table. Using such a convention is no mere convenience, as we shall soon see.
Tartanlike decompositions can be aggregated, by adding together the elements in each subtable :
`\qquad\qquad\qquad{: ([(2, \qquad4),(4, \qquad8)]\qquad [(5, 9),(10, 18)]), ([(6, 12),(10, 20)]\qquad [(15, 27),(25, 45)]) :}\quad->\quad{: ([(),(\qquad18\qquad),()]\quad [(),(\qquad42\qquad),()]), ([(),(\qquad48\qquad),()]\quad [(),(\quad112\qquad),()]) :}`,
where the result can also be written
`\qquad\qquad\qquad[(18,42),(48,112)]`.
Similarly,
`\qquad\qquad\qquad{: ([(\quad2, 9),(10, 45)]\qquad [(4, 5),(20, 25)]), ([(\quad4, 18),(\quad6, 27)]\qquad [(8, 10),(12, 15)]) :}\quad->\quad[(66,54),(55,45)] `,
where the convention we used before helps us keep the aggregated subtables in proportional relation.
A table filled entirely with ones is a proportion table. By aggregating such a table of suitable size, we can construct any proportion table whatever that has only positive integer elements. For instance :
`\qquad\qquad\qquad [(1,1,1,1,1,1),(1,1,1,1,1,1),(1,1,1,1,1,1),(1,1,1,1,1,1),(1,1,1,1,1,1),(1,1,1,1,1,1)]\quad->\quad[(1,2,3),(2,4,6),(3,6,9)],
where an appropriate tartanlike decomposition that leads from the first of these tables to the second should be obvious.
These three examples illustrate a general truth: aggregating a proportion table always gives another proportion table.
Turning tables
It is useful to have some operations that turn one proportion table into another. In this posting, we deal with some operations that can be applied to many different kinds of table, not just proportion tables.
Let's start with the simple operation of swapping (or transposing) two rows, here the second and the third :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9),(6, 12, 15, 27),(4, 8, 10, 18)] `,
or two columns, here the second and the fourth :
`\qquad\qquad[(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 9, 5, 4),(4, 18, 10, 8),(6, 27, 15, 12)]`.
Repeatedly transposing various pairs of rows can give every possible reordering (or permutation) of the rows :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(6, 12, 15, 27),(2, 4, 5, 9),(4, 8, 10, 18)] `,
and similarly for arbitrary column permutations :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(5, 9, 4, 2),(10, 18, 8, 4),(15, 27, 12, 6)] `.
We can combine arbitrary row permutations and arbitrary column permutations to get arbitrary table permutations. How the rows are reordered is independent of how the columns are reordered. Combining the most recent row and column reorderings, we get :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(15, 27, 12, 6),(5, 9, 4, 2),(10, 18, 8, 4)] `.
All of the foregoing operations simply rearrange the proportion table to make a proportion table that contains exactly the same information. When one can make table from another using only rearrangement, the tables are said to be equivalent up to rearrangment. All of the tables in this posting are equivalent up to permutation---they are in a sense the same `3 xx 4` table, just displayed differently.
Does this account for reliable ways to convert a proportion table to a proportion table? In much later postings, when we deal with group tables and have more restrictions, the answer will be yes.
Here, however, the answer is no---there are other possibilities. One such possibility can be made by giving the table a quarter turn, e.g.
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(6, 4, 2),(12,8,4),(15,10,5),(27,18,9)] `.
It should be clear that in the rotated version, rows are proportional to rows and columns to columns, as before, i.e. that the rotated version of the original proportion table is also a proportion table. We could not have made the rotated version from permutations alone or the original, either, because permutations only ever turn a `3 xx 4` table into a `3 xx 4` table, never into a `4 xx 3` table, as this quarter turn has just done.
Of course, this quarter-turned table can now be permuted to give many `4 xx 3` tables, all of which are equivalent up to permutation. Indeed, there will be as many distinct `4 xx 3` tables are there are distinct `3 xx 4` tables.
Does this exhaust all the possibilities? What if one makes a quarter turn in the other direction? It turns out that the two different ways of quarter-turning the original table lead to two tables that differ by a half turn,
`\qquad\qquad\qquad [(6, 4, 2),(12,8,4),(15,10,5),(27,18,9)] `,
and
`\qquad\qquad\qquad [(9,18,27),(5,10,15),(4,8,12),(2, 4, 6)] `.
Unlike a quarter turn, a half turn can be made just from permutations: reverse the order of the rows, and reverse the order of the columns. In this way, we get
`\qquad\qquad\qquad [(6, 4, 2),(12,8,4),(15,10,5),(27,18,9)] \qquad->\qquad [(9,18,27),(5,10,15),(4,8,12),(2, 4, 6)] `,
and we also get
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(27,15,12,6),(18,10,8,4),(9,5,4,2)] `.
It should also be mentioned here that rotating by a quarter turn clockwise and reversing the order of the columns (or, alternatively, rotating a quarter of a turn anticlockwise and reversing the order of the rows) is often called transposing the table.
Transposing a table is like flipping a rectangular piece of paper while keeping the original top left corner at the top and left, going from portrait orientation to landscape orientation, or vice versa :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 6),(4,8,12),(5,10,15),(9,18,27)] `.
Transposing in this sense is a little different from transposition as introduced above. Transposition swaps a pair of rows, or else a pair of columns. Transposing, on the other hand turns all the columns to rows and all the rows to columns, in particular way.
We shall actually have little use either for quarter turns or for transposing, while we are dealing with proportion tables. Transposing become important when dealing with matrices, which come much later.
Let's start with the simple operation of swapping (or transposing) two rows, here the second and the third :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9),(6, 12, 15, 27),(4, 8, 10, 18)] `,
or two columns, here the second and the fourth :
`\qquad\qquad[(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 9, 5, 4),(4, 18, 10, 8),(6, 27, 15, 12)]`.
Repeatedly transposing various pairs of rows can give every possible reordering (or permutation) of the rows :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(6, 12, 15, 27),(2, 4, 5, 9),(4, 8, 10, 18)] `,
and similarly for arbitrary column permutations :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(5, 9, 4, 2),(10, 18, 8, 4),(15, 27, 12, 6)] `.
We can combine arbitrary row permutations and arbitrary column permutations to get arbitrary table permutations. How the rows are reordered is independent of how the columns are reordered. Combining the most recent row and column reorderings, we get :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(15, 27, 12, 6),(5, 9, 4, 2),(10, 18, 8, 4)] `.
All of the foregoing operations simply rearrange the proportion table to make a proportion table that contains exactly the same information. When one can make table from another using only rearrangement, the tables are said to be equivalent up to rearrangment. All of the tables in this posting are equivalent up to permutation---they are in a sense the same `3 xx 4` table, just displayed differently.
Does this account for reliable ways to convert a proportion table to a proportion table? In much later postings, when we deal with group tables and have more restrictions, the answer will be yes.
Here, however, the answer is no---there are other possibilities. One such possibility can be made by giving the table a quarter turn, e.g.
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(6, 4, 2),(12,8,4),(15,10,5),(27,18,9)] `.
It should be clear that in the rotated version, rows are proportional to rows and columns to columns, as before, i.e. that the rotated version of the original proportion table is also a proportion table. We could not have made the rotated version from permutations alone or the original, either, because permutations only ever turn a `3 xx 4` table into a `3 xx 4` table, never into a `4 xx 3` table, as this quarter turn has just done.
Of course, this quarter-turned table can now be permuted to give many `4 xx 3` tables, all of which are equivalent up to permutation. Indeed, there will be as many distinct `4 xx 3` tables are there are distinct `3 xx 4` tables.
Does this exhaust all the possibilities? What if one makes a quarter turn in the other direction? It turns out that the two different ways of quarter-turning the original table lead to two tables that differ by a half turn,
`\qquad\qquad\qquad [(6, 4, 2),(12,8,4),(15,10,5),(27,18,9)] `,
and
`\qquad\qquad\qquad [(9,18,27),(5,10,15),(4,8,12),(2, 4, 6)] `.
Unlike a quarter turn, a half turn can be made just from permutations: reverse the order of the rows, and reverse the order of the columns. In this way, we get
`\qquad\qquad\qquad [(6, 4, 2),(12,8,4),(15,10,5),(27,18,9)] \qquad->\qquad [(9,18,27),(5,10,15),(4,8,12),(2, 4, 6)] `,
and we also get
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(27,15,12,6),(18,10,8,4),(9,5,4,2)] `.
It should also be mentioned here that rotating by a quarter turn clockwise and reversing the order of the columns (or, alternatively, rotating a quarter of a turn anticlockwise and reversing the order of the rows) is often called transposing the table.
Transposing a table is like flipping a rectangular piece of paper while keeping the original top left corner at the top and left, going from portrait orientation to landscape orientation, or vice versa :
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 6),(4,8,12),(5,10,15),(9,18,27)] `.
Transposing in this sense is a little different from transposition as introduced above. Transposition swaps a pair of rows, or else a pair of columns. Transposing, on the other hand turns all the columns to rows and all the rows to columns, in particular way.
We shall actually have little use either for quarter turns or for transposing, while we are dealing with proportion tables. Transposing become important when dealing with matrices, which come much later.
Tuesday, February 23, 2010
More about proportion tables
Look again at the standard multiplication table :
By simply changing the '`xx`' to a `1` we have a proportion table. The stubs no longer deserve special coloring because they have no special role. This is generally true. Any multiplication table can be converted to a proportion table in this way.
Conversely, any '`1`' in a proportion table can be treated as if it is the '`xx`' symbol in a multiplication table. So, for instance, Hutton gives his multiplication table, which he also says is called Pythagoras's table, in the following form, although his is `12 xx 12`.
The '`1`' at the top left can serve as a multiplication sign. This remains true for any '`1`' entry found in an arbitrary proportion table.
It is readily apparent that the quantities directly below and directly to the right of the '`1`' serve as as stubs of a multiplication table, with the body of the table between these two arms, both below and to the right of the '`1`'. But the numbers directly above, and also directly to the left of the '`1`' can also serve as stubs. So, for instance, `8 xx 2 1/2 = 20`. On this view, here one has table bodies in all four quadrants.
It is better, however, to think of entire row containing the '`1`' as a row stub, and the entire column containing the '`1`' as a column stub. Any value in the table has a marker value on the same row in the column stub, and a marker value in the same column in the row stub. The '`1`', the arbitrary value in the table, and the two marker values lie at the corners of a rectangular box.
The four values
`\qquad\qquad\qquad{:(2 1/2\qquad, 1),(20\qquad, 8):}`
can be thought of as a `2 xx 2` subtable, with any value in this subtable equivalent to the product of its adjacent values divided by its diagonally opposite value :
`\qquad\qquad\qquad 20 = \frac{8 xx 2 1/2}{1}`.
In this form, we are using the proportion table version of the Rule of Three. Any `2 xx 2` subtable can serve in this way, and no '`1`' is required in the subtable, or indeed, anywhere in the proportion table :
so that, e.g.
`\qquad\qquad\qquad 4" mph" = \frac{28" miles " xx ""8}{56" hours"}.`
Here, we have used the top left entry of the subtable of the subtable as the target element, and we have continued to follow the convention of writing the same-row neighbor before the same column neighbor.
But we could use of these three same elements to find the fourth. e.g. if the top right element is our target element, we can write :
`\qquad\qquad\qquad 28" miles" = \frac{4" mph " xx" "56" hours"}{8}.`
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 |
| 10 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
By simply changing the '`xx`' to a `1` we have a proportion table. The stubs no longer deserve special coloring because they have no special role. This is generally true. Any multiplication table can be converted to a proportion table in this way.
Conversely, any '`1`' in a proportion table can be treated as if it is the '`xx`' symbol in a multiplication table. So, for instance, Hutton gives his multiplication table, which he also says is called Pythagoras's table, in the following form, although his is `12 xx 12`.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
| 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 |
| 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 |
| 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
| 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 |
| 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 |
| 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 |
| 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 |
| 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
The '`1`' at the top left can serve as a multiplication sign. This remains true for any '`1`' entry found in an arbitrary proportion table.
| `5` | `8` | `2` | `1200` | `14" hours"` |
| `10" mph"` | `16" mph"` | `4" mph"` | `2400" mph"` | `28" miles"` |
| `2 1/2` | `4` | `1` | `600` | `7" hours"` |
| `22 1/2` | `36` | `9` | `5400` | `63" hours"` |
| `250` | `400` | `100` | `60000` | `700" hours"` |
| `20` | `32` | `8` | `4800` | `56" hours"` |
| `5/6` | `1 1/3` | `1/3` | `200` | `2 1/3" hours"` |
It is readily apparent that the quantities directly below and directly to the right of the '`1`' serve as as stubs of a multiplication table, with the body of the table between these two arms, both below and to the right of the '`1`'. But the numbers directly above, and also directly to the left of the '`1`' can also serve as stubs. So, for instance, `8 xx 2 1/2 = 20`. On this view, here one has table bodies in all four quadrants.
It is better, however, to think of entire row containing the '`1`' as a row stub, and the entire column containing the '`1`' as a column stub. Any value in the table has a marker value on the same row in the column stub, and a marker value in the same column in the row stub. The '`1`', the arbitrary value in the table, and the two marker values lie at the corners of a rectangular box.
The four values
`\qquad\qquad\qquad{:(2 1/2\qquad, 1),(20\qquad, 8):}`
can be thought of as a `2 xx 2` subtable, with any value in this subtable equivalent to the product of its adjacent values divided by its diagonally opposite value :
`\qquad\qquad\qquad 20 = \frac{8 xx 2 1/2}{1}`.
In this form, we are using the proportion table version of the Rule of Three. Any `2 xx 2` subtable can serve in this way, and no '`1`' is required in the subtable, or indeed, anywhere in the proportion table :
| `5` | `8` | `2` | `1200` | `14" hours"` |
| `10" mph"` | `16" mph"` | `4" mph"` | `2400" mph"` | `28" miles"` |
| `2 1/2` | `4` | `1` | `600` | `7" hours"` |
| `22 1/2` | `36` | `9` | `5400` | `63" hours"` |
| `250` | `400` | `100` | `60000` | `700" hours"` |
| `20` | `32` | `8` | `4800` | `56" hours"` |
| `5/6` | `1 1/3` | `1/3` | `200` | `2 1/3" hours"` |
so that, e.g.
`\qquad\qquad\qquad 4" mph" = \frac{28" miles " xx ""8}{56" hours"}.`
Here, we have used the top left entry of the subtable of the subtable as the target element, and we have continued to follow the convention of writing the same-row neighbor before the same column neighbor.
But we could use of these three same elements to find the fourth. e.g. if the top right element is our target element, we can write :
`\qquad\qquad\qquad 28" miles" = \frac{4" mph " xx" "56" hours"}{8}.`
Saturday, February 20, 2010
Multiplication tables and proportion tables
Here is a typical modern multiplication table. (Older tables usually go up to `12 xx 12`.)
At the left and along the top are the stubs---here, in this example, the stubs are colored. The rest is the body of the table. One looks up the numbers to be multiplied in the stubs, and then finds the product in the body, at the intersection of an appropriate row and column. So, for instance, `"4 "xx" 5 = 20"`:
The standard multiplication table and how to use it are presumably familiar. We can put arbitrary quantities in the stubs, and fill out the body of the corresponding multiplication table :
One can build a multiplication table with any number of rows and any number of columns.
In the body of a multiplication table :
Given a single nonzero row, one can reconstruct any other row from the value at a single position in the row to be reconstructed, and mutatis mutandis for columns.
Any table of numbers that can be the body multiplication table is called a proportion table. To be a proportion table, then, a table of numbers has only :
The following are equivalent:
Old proportion notation:
`\qquad\qquad 750" men : "22,500" rations of bread :: "1200" men : What?"`
Algebraic notation:
`\qquad\qquad\qquad\frac{750" men"}{"22,500 rations of bread"}=\frac{1200 " men"}{x}.`
Proportion table form:
To find a missing entry in a `2 \times 2` proportion table, one multiplies the neighboring entries, and divides by the opposite entry:
`\qquad\qquad\qquad x = \frac{"22,500 rations of bread" \times "1200 men"}{"750 men"},`
`\qquad\qquad\qquad = "36,000 rations of bread".`
This three argument operation---multiplying two numbers and dividing by a third---is remarkably useful. One computer language, FORTH, even defines a single operation, '*/' or starslash, for doing this.
We can always find a solution as long as we have three appropriately positioned values, and as long as we don't have to divide by zero. A zero in the top stub of the multiplication table leads to a column of zeroes in the body of the table; a zero in the side stub leads to a row of zeroes in the body. A zero only appears in the body of the table as part of a row stub or column stub. In short, a zero appears somewhere in the stubs exactly when a zero appears in the body of the table. A table with no zeroes is called zero free. We shall prefer to work with zero free proportion tables.
In writing this proportion table calculation, we have followed the convention of writing the row neighbor before the column neighbor in the numerator, with the opposite centered below them in the denominator. Long ahead, when we get to group tables, this convention will prove useful.
| `\times` | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 |
| 10 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
At the left and along the top are the stubs---here, in this example, the stubs are colored. The rest is the body of the table. One looks up the numbers to be multiplied in the stubs, and then finds the product in the body, at the intersection of an appropriate row and column. So, for instance, `"4 "xx" 5 = 20"`:
| `\times` | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 |
| 10 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
The standard multiplication table and how to use it are presumably familiar. We can put arbitrary quantities in the stubs, and fill out the body of the corresponding multiplication table :
| `\times` | `2 1/2` | `4` | `1` | `600` | `7" hours"` |
| `2` | `5` | `8` | `2` | `1200` | `14" hours"` |
| `4" mph"` | `10" mph"` | `16" mph"` | `4" mph"` | `2400" mph"` | `28" miles"` |
| `1` | `2 1/2` | `4` | `1` | `600` | `7" hours"` |
| `9` | `22 1/2` | `36` | `9` | `5400` | `63" hours"` |
| `100` | `250` | `400` | `100` | `60000` | `700" hours"` |
| `8` | `20` | `32` | `8` | `4800` | `56" hours"` |
| `1/3` | `5/6` | `1 1/3` | `1/3` | `200` | `2 1/3" hours"` |
One can build a multiplication table with any number of rows and any number of columns.
In the body of a multiplication table :
- every row is proportional to every other row, and
- every column is proportional to every other column.
Given a single nonzero row, one can reconstruct any other row from the value at a single position in the row to be reconstructed, and mutatis mutandis for columns.
Any table of numbers that can be the body multiplication table is called a proportion table. To be a proportion table, then, a table of numbers has only :
- satisfy the proportionality condition on its rows/columns.
The following are equivalent:
Old proportion notation:
`\qquad\qquad 750" men : "22,500" rations of bread :: "1200" men : What?"`
Algebraic notation:
`\qquad\qquad\qquad\frac{750" men"}{"22,500 rations of bread"}=\frac{1200 " men"}{x}.`
Proportion table form:
| `"750 men"` | `"1200 men"` |
| `"22,500 rations of bread "` | `"What?"` |
To find a missing entry in a `2 \times 2` proportion table, one multiplies the neighboring entries, and divides by the opposite entry:
`\qquad\qquad\qquad x = \frac{"22,500 rations of bread" \times "1200 men"}{"750 men"},`
`\qquad\qquad\qquad = "36,000 rations of bread".`
This three argument operation---multiplying two numbers and dividing by a third---is remarkably useful. One computer language, FORTH, even defines a single operation, '*/' or starslash, for doing this.
We can always find a solution as long as we have three appropriately positioned values, and as long as we don't have to divide by zero. A zero in the top stub of the multiplication table leads to a column of zeroes in the body of the table; a zero in the side stub leads to a row of zeroes in the body. A zero only appears in the body of the table as part of a row stub or column stub. In short, a zero appears somewhere in the stubs exactly when a zero appears in the body of the table. A table with no zeroes is called zero free. We shall prefer to work with zero free proportion tables.
In writing this proportion table calculation, we have followed the convention of writing the row neighbor before the column neighbor in the numerator, with the opposite centered below them in the denominator. Long ahead, when we get to group tables, this convention will prove useful.
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