We proceed: 1 and 1 is 10; write down 0 and carry 1; 0 and 0 is 0, and 1 to carry is 1; and 1 and 0 is 1; and then we just copy the last 1. To check this we can convert to decimal and see that 101 is 5, 1001 is 9, and 1110 is 14, and we can verify that 5 and 9 is 14.
One of the easiest ways to subtract in binary notation is to add a ones complement (that is, the analogue of the nines complement) and use end-around-carry (for these two terms, see the end of this supplement). A ones complement can be written down at sight by just putting 1 for 0 and 0 for 1. For example, suppose that we subtract 101 from 1110:
| Direct Subtraction | Check | Subtraction by Adding Ones Complement |
|---|---|---|
| 1110 | 14 | 1110 |
| - 101 | - 5 | + 1010 |
| 1001 | 9 | (1)1000 |
| ↓ | ||
| ⎯→ 1 | ||
| 1001 |
Multiplication in the binary notation is simple. It amounts to (1) adding if the multiplier digit is 1 and not adding if the multiplier digit is 0, and (2) moving over or shifting. For example, let us multiply 111 by 101:
| Binary Multiplication | Check |
|---|---|
| 111 | 7 |
| × 101 | × 5 |
| 111 | |
| 111 | |
| 100011 | 35 |
The digit 1 in the 6th (or nth) binary place from the right in 100011 stands for 1 times 2 to the 5th (or n-1 th) power, 2 × 2 × 2 × 2 × 2 = 32. The result 100011 is translated into 32 plus 2 plus 1, which equals 35 and verifies.
Division in the binary notation is also simple. It amounts to (1) subtracting (yielding a quotient digit 1) or not subtracting (yielding a quotient digit 0), and (2) shifting. We never need to try multiples of the divisor to find the largest that can be subtracted yet leave a positive remainder. For example, let us divide 1010 (10 in decimal) into 10001110 (142 in decimal):
- 1110 (14 in decimal)
- ————
- 1010)10001110
- 1010
- 1111
- 1010
- 1011
- 1010
- 10 (remainder, 2 in decimal)
In decimal notation, digits to the right of the decimal point count powers of ⅒. In binary notation, digits to the right of the binary point count powers of ½: ½, ¼, ⅛, ¹/₁₆.... For example, 0.1011 equals ½ + ⅛ + ¹/₁₆, or ¹¹/₁₆.
If we were accustomed to using binary numbers, all our arithmetic would be very simple. Furthermore, binary numbers are in many ways much better for calculating machinery than any other numbers. The main problem is converting numbers from decimal notation to binary. One method depends on storing the powers of 2 in decimal notation. The rule is: subtract successively smaller powers of 2; start with the largest that can be subtracted, and count 1 for each power that goes and 0 for each power that does not. For example, 86 in decimal becomes 1010110 in binary: