Method of complements

Complement numbers on an adding machine c. 1910. The smaller numbers, for use when subtracting, are the nines' complement of the larger numbers, which are used when adding.

In mathematics and computing, the method of complements is a technique used to subtract one number from another using only addition of positive numbers. This method was commonly used in mechanical calculators and is still used in modern computers.

The nines' complement of a number is formed by replacing each digit with nine minus that digit. To subtract a decimal number y (the subtrahend) from another number x (the minuend) two methods may be used:

In the first method the nines' complement of x is added to y. Then the nines' complement of the result obtained is formed to produce the desired result.

In the second method the nines' complement of y is added to x and one is added to the sum. The leading digit '1' of the result is then discarded. Discarding the initial '1' is especially convenient on calculators or computers that use a fixed number of digits: there is nowhere for it to go so it is simply lost during the calculation. The nines' complement plus one is known as the tens' complement.

The method of complements can be extended to other number bases (radixes); in particular, it is used on most digital computers to perform subtraction, represent negative numbers in base 2 or binary arithmetic and test underflow and overflow in calculation. [1]

Numeric complements

The radix complement of an n digit number y in radix b is, by definition, b^n-y. The radix complement is most easily obtained by adding 1 to the diminished radix complement, which is (b^n-1)-y. Since (b^n-1) is the digit b-1 repeated n times (because b^n-1 = b^n-1^n = (b-1)(b^{n-1}+b^{n-2}+...+b+1)=(b-1)b^{n-1}+...+(b-1); see also binomial numbers). The diminished radix complement of a number is found by complementing each digit with respect to b-1 (that is, subtracting each digit in y from b-1).

The subtraction of y from x may be performed as follows. Adding the diminished radix complement of x to y results in the value b^n-1-x + y or b^n -1-(x-y) which is the diminished radix complement of x-y, except for possible padding digits b-1. The diminished radix complement of this is the value x-y. Alternatively, adding the radix complement of y to x results in the value x+b^n-y or x-y+b^n. Assuming y ≤ x , the result will always be greater or equal to b^n and dropping the initial '1' is the same as subtracting b^n, making the result x-y+b^n-b^n or just x-y, the desired result.

In the decimal numbering system, the radix complement is called the ten's complement and the diminished radix complement the nines' complement. In binary, the radix complement is called the two's complement and the diminished radix complement the ones' complement. The naming of complements in other bases is similar. Some people, notably Donald Knuth, recommend using the placement of the apostrophe to distinguish between the radix complement and the diminished radix complement. In this usage, the four's complement refers to the radix complement of a number in base four while fours' complement is the diminished radix complement of a number in base 5. However, the distinction is not important when the radix is apparent (nearly always), and the subtle difference in apostrophe placement is not common practice. Most writers use one's and nine's complement, and many style manuals leave out the apostrophe, recommending ones and nines complement.

Decimal example

Digit Nines'
complement
0 9
1 8
2 7
3 6
4 5
5 4
6 3
7 2
8 1
9 0

The nines' complement of a decimal digit is the number that must be added to it to produce 9; the complement of 3 is 6, the complement of 7 is 2, and so on, see table. To form the nines' complement of a larger number, each digit is replaced by its nines' complement.

Consider the following subtraction problem:

  873  (x, the minuend)
- 218  (y, the subtrahend)

First method

We compute the nines' complement of the minuend, 873. Add that to the subtrahend 218, then calculate the nines' complement of the result.

  126  (nines' complement of x)
+ 218  (y, the subtrahend)
=====
  344

Now calculate the nines' complement of the result

  344 (result)
  655 (nine's complement of result, the correct answer)

Second method

We compute the nines' complement of 218, which is 781. Because 218 is three digits long, this is the same as subtracting 218 from 999.

Next, the sum of x and the nines' complement of y is taken:

  873  (x)
+ 781  (nines' complement of y = 999-y)
=====
 1654  (999 + x-y)

The leading "1" digit is then dropped, giving 654.

 1654
-1000  -(999 + 1) 
=====
  654  (x-y-1)

This is not yet correct. We have essentially added 999 to the equation in the first step. Then we removed 1000 when we dropped the leading 1 in the result 1654 above. This will thus make the answer we get (654) one less than the correct answer (x-y). To fix this, we must add 1 to our answer:

 654
  +1
====
 655  (x-y)

Adding a 1 gives 655, the correct answer to our original subtraction problem.

Magnitude of numbers

In the following example the result of the subtraction has fewer digits than x:

   123410 (x, the minuend)
 - 123401 (y, the subtrahend)

Using the first method the sum of the nines' complement of x and y is

   876589 (nines' complement of x)
+  123401 (y)
=========
   999990

The nines' complement of 999990 is 000009. Removing the leading zeros gives 9 the desired result.

If the subtrahend, y, has fewer digits than the minuend, x, leading zeros must be added in the second method. These zeros become leading nines when the complement is taken. For example:

  48032  (x)
-   391  (y)

can be rewritten

  48032  (x)
- 00391  (y with leading zeros)

Replacing 00391 with its nines' complement and adding 1 produces the sum:

  48032  (x)
+ 99608  (nines' complement of y)
+     1
=======
 147641

Dropping the leading "1" gives the correct answer: 47641.

Binary example

Binary
digit
Ones'
complement
0 1
1 0

The method of complements is especially useful in binary (radix 2) since the ones' complement is very easily obtained by inverting each bit (changing '0' to '1' and vice versa). And adding 1 to get the two's complement can be done by simulating a carry into the least significant bit. For example:

  01100100  (x, equals decimal 100)
- 00010110  (y, equals decimal 22)

becomes the sum:

  01100100  (x)
+ 11101001  (ones' complement of y)
+        1  (to get the two's complement)
==========
 101001110

Dropping the initial "1" gives the answer: 01001110 (equals decimal 78)

Negative number representations

The method of complements normally assumes that the operands are positive and that yx, logical constraints given that adding and subtracting arbitrary integers is normally done by comparing signs, adding the two or subtracting the smaller from the larger, and giving the result the correct sign.

Let's see what happens if x < y. In that case, there will not be a "1" digit to cross out after the addition since x-y+b^n will be less than b^n. For example (in decimal):

  185  (x)
- 329  (y)

Complementing y and adding gives:

  185  (x)
+ 670  (nines' complement of y)
+   1
=====
  856

This is obviously the wrong answer; the expected answer is -144. But it isn't as far off as it seems; 856 happens to be the ten's complement of 144. This issue can be addressed in three ways:

Practical uses

Comptometer from the 1920s, with nines' complements marked on each key

The method of complements was used in many mechanical calculators as an alternative to running the gears backwards. For example:

In computers

Use of the method of complements is ubiquitous in digital computers, regardless of the representation used for signed numbers. However, the circuitry required depends on the representation:

Manual uses

The method of complements was used to correct errors when accounting books were written by hand. To remove an entry from a column of numbers, the accountant could add a new entry with the ten's complement of the number to subtract. A bar was added over the digits of this entry to denote its special status. It was then possible to add the whole column of figures to obtain the corrected result.

Complementing the sum is handy for cashiers making change for a purchase from currency in a single denomination of 1 raised to an integer power of the currency's base. For decimal currencies that would be 10, 100, 1,000, etc., e.g. a $10.00 bill.

In grade school education

In grade schools, students are sometimes taught the method of complements as a shortcut useful in mental arithmetic.[3] Subtraction is done by adding the ten's complement of the subtrahend, which is the nines' complement plus 1. The result of this addition used when it is clear that the difference will be positive, otherwise the ten's complement of the addition's result is used with it marked as negative. The same technique works for subtracting on an adding machine.

References

  1. Florida Tech
  2. Easy Instructions for Operation the Controlled Key Comptometer, Comptometer Division, Felt and Tarrant Mfg. Co., Chicago, 1917, p. 12
  3. Carl Barnett Allendoerfer (1971). Principles of Arithmetic and Geometry for Elementary School Teachers. Macmillan.
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