How to subtract one positive integer (the *subtrahend*) from another, larger positive integer (the *minuend*) (Gardner 1973):

**Step 1.** Place the larger number on the bottom horizontal row.

**Step 2.** Place the binary complement of the number to be subtracted in the margin below the bottom horizontal row. That is, where there would be a counter in the number itself, leave the square empty; and where there would not be a counter, place one on the square. Do this for all entries from right to left through the highest power of 2 occurring in the decomposition of the larger number.

**Step 3.** Combine all the counters on the bottom horizontal row.

**Step 4.** *Abbreviate* the board from right to left; that is, remove every two counters in a square and place one counter in the square to its left. This process often causes a "chain reaction" that "carries" into other squares, requiring further *abbreviation*.

**Step 5.** Add one more counter in the 1's place (\(2^0\)-place) square in the bottom row. This may cause a "chain reaction" that "carries" into squares to the left, requiring further *abbreviation*.

**Step 6.** The desired answer can now be obtained by converting the *abbreviated* difference back to a decimal number, except that you must ignore the farthest left counter (which may already have "fallen" off the left side of the board).

**Example:** That 116 – 84 = 32 is illustrated below.

**Steps 1 and 2.** The larger integer, 116, decomposed as \[116 = 2^6 + 2^5 + 2^4 +{\phantom{2^3}}+2^2+{\phantom{2^1}}+{\phantom{2^0}} = 64 + 32 + 16 +{\phantom{8}}+4+{\phantom{2}}+{\phantom{1}},\] is represented by counters in the bottom horizontal row.** **Now we decompose 84 as \[\phantom{0}84 = 2^6 + {\phantom{2^5}}+ 2^4 +{\phantom{2^3}}+2^2+{\phantom{2^1}}+{\phantom{2^0}} = 64 +{\phantom{32}}+ 16 +{\phantom{8}}+4+{\phantom{2}}+{\phantom{1}},\] and write its complement \[{\phantom{84=2^6++}} {2^5} + {\phantom{2^4}} +{2^3}+{\phantom{2^2}} +{2^1}+{2^0}={\phantom{x2^6+}} 32 + {\phantom{16}} +8+{\phantom{4}} +2+1.\] The complement is represented in the margin below the bottom horizontal row.

The binary complement of a positive integer is sometimes called the "1's complement" because one can compute it by interchanging 0's and 1's in the binary-digit representation of the positive integer. For instance:

64 32 16 8 4 2 1

1 0 1 0 1 0 0 \(\leftarrow\) Binary-digit representation for 84

0 1 0 1 0 1 1 \(\leftarrow\) 1's complement of 84

The binary complement of 84 is therefore 32 + 8 + 2 + 1.

**Step 3.** Combine all of the counters in the bottom horizontal row:

**Step 4.** Abbreviate the row from right to left. Note that there is now a counter in the 128 or \(2^7\) place:

**Step 5.** Add +1; that is, add one more counter to the 1's place, and ...

... abbreviate the row from right to left:

**Step 6.** Now, ignoring the farthest left counter, the difference can be read from this row as 116 – 84 = 32.

Next: multiplication on the chessboard calculator