The minus over the 2 marks it as a _negative digit_ 2. Then to multiply we have: 897 _ 222 ———— 1794 - 1794 1794 —————— 163254 The middle 1794 is subtracted. This process is usually called _short-cut multiplication_. Everybody discovers this trick when he decides that multiplying by 99 is too much work, that it is easier to multiply by 100 and subtract once. BINARY OR TWO NUMBERS We are well accustomed to decimal notation in which we use 10 decimal digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and write them in combinations to designate decimal numbers. In _binary notation_ we use two binary digits 0, 1 and write them in combinations to designate _binary numbers_. For example, the first 17 numbers, from 0 to 16 in the decimal notation, correspond with the following numbers in binary notation: DECIMAL BINARY DECIMAL BINARY 0 0 1 1 9 1001 2 10 10 1010 3 11 11 1011 4 100 12 1100 5 101 13 1101 6 110 14 1110 7 111 15 1111 8 1000 16 10000 In decimal notation, 101 means one times a hundred, no tens, and one. In binary notation, 101 means one times four, no twos, and one. The successive digits in a decimal number from right to left count 1, 10, 100, 1000, 10000, ...—successive _powers_ of 10 (for this term, see the end of this supplement). The successive digits in a binary number from right to left count 1, 2, 4, 8, 16, ...—powers of 2. The decimal notation is convenient when equipment for computing has ten positions, like the fingers of a man, or the positions of a counter wheel. The binary notation is convenient when equipment for computing has just two positions, like “yes” or “no,” or current flowing or no current flowing. Addition, subtraction, multiplication, and division can all be carried out unusually simply in binary notation. The addition table is simple and consists only of four entries. + 0 1 +—————— 0 | 0 1 | 1 | 1 10 The multiplication table is also simple and contains only four entries. × 0 1 +—————— 0 | 0 0 | 1 | 0 1 Suppose that we add in binary notation 101 and 1001: BINARY ADDITION CHECK 101 5 + 1001 9 —————— ——— 1110 14 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.