Lester Hill’s “The checking of the accuracy …”, part 11

The field F_{101}

All essential points connected with the checking of telegraphic sequences by the methods proposed in this paper may be fully illustrated in one finite field. For our purposes, perhaps the most useful field is F_{101}, to which we shall confine our attention in the following sections. The elements of the field F_{101} are the one hundred and one marks\footnote{Hill actually uses the symbol X in place of 100.} 0, 1, 2, \dots, 100. The operations of addition and multiplication are effected as explained in a previous example; and are abbreviated as suggested. To determine sums and products, we regard the marks of the field momentarily as integers of elementary arithmetic. Thus we have

\sum_1^n f_i = f_h,\ \ \ \ \ \ (\prod_1^n f_i = f_k),

the f_i being n marks of F_{101}, distinct or not, if, when the f_i are momentarily regarded as integers of elementary arithmetic, the congruence

\sum_1^n f_i \equiv f_h \pmod{101},\ \ \ \ \ \ (\prod_1^n f_i \equiv f_k \pmod{101}),

holds. It will not be possible to provide a full multiplication table for the field F_{101}. But the following special table will be found convenient.

\begin{array}{r|rrr} x & x^2 & 1/x & -x\\ 1 & 1 & 1 & 100 \\ 2 & 4 & 51 & 99 \\ 3 & 9 & 34 & 98 \\ 4 & 16 & 76 & 97 \\ 5 & 25 & 81 & 96 \\ 6 & 36 & 17 & 95 \\ 7 & 49 & 29 & 94 \\ 8 & 64 & 38 & 93 \\ 9 & 81 & 45 & 92 \\ 10 & 100 & 91 & 91 \\ 11 & 20 & 46 & 90 \\ 12 & 43 & 59 & 89 \\ 13 & 68 & 70 & 88 \\ 14 & 95 & 65 & 87 \\ 15 & 23 & 27 & 86 \\ 16 & 54 & 19 & 85 \\ 17 & 87 & 6 & 84 \\ 18 & 21 & 73 & 83 \\ 19 & 58 & 16 & 82 \\ 20 & 97 & 96 & 81 \\ 21 & 37 & 77 & 80 \\ 22 & 80 & 23 & 79 \\ 23 & 24 & 22 & 78 \\ 24 & 71 & 80 & 77 \\ 25 & 19 & 97 & 76 \\ 26 & 70 & 35 & 75 \\ 27 & 22 & 15 & 74 \\ 28 & 77 & 83 & 73 \\ 29 & 33 & 7 & 72 \\ 30 & 92 & 64 & 71 \\ 31 & 52 & 88 & 70 \\ 32 & 14 & 60 & 69 \\ 33 & 79 & 49 & 68 \\ \end{array}

\begin{array}{r|rrr} x & x^2 & 1/x & -x\\ 34 & 45 & 3 & 67 \\ 35 & 13 & 26 & 66 \\ 36 & 84 & 87 & 65 \\ 37 & 56 & 71 & 64 \\ 38 & 30 & 8 & 63 \\ 39 & 6 & 57 & 62 \\ 40 & 85 & 48 & 61 \\ 41 & 65 & 69 & 60 \\ 42 & 47 & 89 & 59 \\ 43 & 31 & 47 & 58 \\ 44 & 17 & 62 & 57 \\ 45 & 5 & 9 & 56 \\ 46 & 96 & 11 & 55 \\ 47 & 88 & 43 & 54 \\ 48 & 82 & 40 & 53 \\ 49 & 78 & 33 & 52 \\ 50 & 76 & 99 & 51 \\ 51 & 76 & 2 & 50 \\ 52 & 78 & 68 & 49 \\ 53 & 82 & 61 & 48 \\ 54 & 88 & 58 & 47 \\ 55 & 96 & 90 & 46 \\ 56 & 5 & 92 & 45 \\ 57 & 17 & 39 & 44 \\ 58 & 31 & 54 & 43 \\ 59 & 47 & 12 & 42 \\ 60 & 65 & 32 & 41 \\ 61 & 85 & 53 & 40 \\ 62 & 6 & 44 & 39 \\ 63 & 30 & 93 & 38 \\ 64 & 56 & 30 & 37 \\ 65 & 84 & 14 & 36 \\ 66 & 13 & 75 & 35 \\ \end{array}

\begin{array}{r|rrr} x & x^2 & 1/x & -x\\ 67 & 45 & 98 & 34 \\ 68 & 79 & 52 & 33 \\ 69 & 14 & 41 & 32 \\ 70 & 52 & 13 & 31 \\ 71 & 92 & 37 & 30 \\ 72 & 33 & 94 & 29 \\ 73 & 77 & 18 & 28 \\ 74 & 22 & 86 & 27 \\ 75 & 70 & 66 & 26 \\ 76 & 19 & 4 & 25 \\ 77 & 71 & 21 & 24 \\ 78 & 24 & 79 & 23 \\ 79 & 80 & 78 & 22 \\ 80 & 37 & 24 & 21 \\ 81 & 97 & 5 & 20 \\ 82 & 58 & 85 & 19 \\ 83 & 21 & 28 & 18 \\ 84 & 87 & 95 & 17 \\ 85 & 54 & 82 & 16 \\ 86 & 23 & 74 & 15 \\ 87 & 95 & 36 & 14 \\ 88 & 68 & 31 & 13 \\ 89 & 43 & 42 & 12 \\ 90 & 20 & 55 & 11 \\ 91 & 100 & 10 & 10 \\ 92 & 81 & 56 & 9 \\ 93 & 64 & 63 & 8 \\ 94 & 49 & 72 & 7 \\ 95 & 36 & 84 & 6 \\ 96 & 25 & 20 & 5 \\ 97 & 16 & 25 & 4 \\ 98 & 9 & 67 & 3 \\ 99 & 4 & 50 & 2 \\ 100 & 1 & 100 & 1 \end{array}

Squares, reciprocals, negatives

Using the scheme of reciprocals shown in this Table, we may easily perform an rational operations in F_{101}.

Example:

Suppose, for example, that we wish to solve the system of equations:

36x-79y=52,\ \ \ 90x+85y = 98.

They may be written

x-79y/36 = 13/9,\ \ \ \ x+17y/18 = 49/45

and the fractions are quickly evaluated. Thus: -79/36 = 96. Determining the fractions in this manner, we write the two equations in the form:

x+96y=80,\ \ \ \ x+29y=37,

whence y=73 and x=41

The modulus 101 is very convenient to work with. The residue, modulo 101, of any integer is immediately obvious, at sight of the integer, and is therefore obtained without computation.

One thought on “Lester Hill’s “The checking of the accuracy …”, part 11

  1. Pingback: Lester Hill’s “The checking of the accuracy …”, part 12 | Yet Another Mathblog

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s