Complement and negative number

1. About complement

In order to perform arithmetic operations, it is necessary to be able to represent negative numbers even in the range of integers.

  The complement is a number that makes the sum of the digits of each digit a constant number. If it is 1's complement, it is a number that makes the sum of each digit 1, and if it is 10's complement, it is a number that makes the sum of the digits 10. The 10's complement for 4 is 7, the 10's complement for 4 is 6, and the 10's complement for 5 is 5. In decimal, there are 1 to 10's complement, in binary there are 1 and 2's complement.

 

2. Conversion to complement

In the case of the decimal system, calculation by complement is rarely performed, but only 10's complement is used in mental arithmetic or when using the abacus. However, since it is different from the writing method, practice is required.

  In the case of the binary system, 2's complement is required for calculation, and for this, 1's complement is obtained. Since 1's complement is expressed only as 1 or 0, if it is 1's complement, 0 is converted to 1 and 1 is converted to 0 so that the sum is 1, but it is the same as the number of digits before conversion. For example, 1's complement for 1001101₂ becomes 0110010₂, and 1's complement for 01001101₂ becomes 10110010₂.
  Next, 2's complement is converted so that the sum becomes 2, and it is difficult to convert immediately because rounding occurs at every digit, but it is easily solved by adding 1 to 1's complement. That is, 1001101₂ is converted to 0110010₂ (1's complement) → 0110011₂ (2's complement), and 01001101₂ is converted to 10110010₂ (1's complement) → 10110011₂ (2's complement).

 

Before conversion	1	0	0	1	1	0	1
1's complement	0	1	1	0	0	1	0
2's complement	0	1	1	0	0	1	1

 

Before conversion	0	1	0	0	1	1	0	1
1's complement	1	0	1	1	0	0	1	0
2's complement	1	0	1	1	0	0	1	1

 

To check for 2's complement, add the number before conversion and 2's complement and check that rounding occurs at every digit. That is, the sum of each digit becomes 2.

 

3. Handling negative numbers

Even if 0 and 1 recognized by the computer are repeatedly used, only positive numbers, which are numbers greater than or equal to 0, can be expressed, to express negative numbers, the largest digit is designated as the sign digit. Accordingly, even if the same number of digits is used, the range of the number to be expressed is halved, and even negative numbers can be written.
  If you write it based on the 8-digit binary number, it is as follows.

Positive (8 binary digits)		Negative (8 binary digits)
Decimal	Binary	Decimal	Binary
＋12710	011111112	－110	111111112
＋12610	011111102	－210	111111102
·	·	－310	111111012
·	·	·	·
·	·	·	·
＋310	000000112	·	·
＋210	000000102	－12610	100000102
＋110	000000012	－12710	100000012
010	000000002	－12810	100000002

 

With the 8th digit as the sign, 0 is a positive number, and 1 is a negative number. In the case of the next 7 digits, if the decimal number is converted to a binary number according to the existing method in the positive number section, it matches. In the case of a negative number interval, the decimal number is converted to binary number and then converted to 2's complement, it is match.
  For example, when converting decimal －126₁₀ to binary, 1 is placed in the 8th digit, and the binary number of 126₁₀ is 1111110₂, it converted to 0000001₂ (1's complement) → 00000102 (2's complement), combined with results to match 10000010₂.
  If it is decided to use both positive and negative numbers for 8 binary digits in this way, it is possible to convert decimal numbers in the range of -128₁₀ to +127₁₀, if it is decided to use only positive numbers, it is possible to convert in the range of 0₁₀ to ＋255₁₀.

 

4. Conclusion

Just learn how to calculate this yourself, and usually use a calculator on your computer or a converter on the Internet.