4.1 Logic

The pack’s temperature sensor answers with two bytes on the I²C bus: 0xFF then 0x80. Read one way, that is −0.5 °C, the pack outside in November. Read another way, it is 255.5 °C. The bits are the same; what changes is the convention applied to them, and the sensor does not send it.

The oldest convention is binary. Leibniz publishes in 1703 his "Explanation of Binary Arithmetic", which counts with the characters 0 and 1 alone. He has no machine in mind; through the letters of the Jesuit Bouvet, he recognizes in it the hexagrams of the I Ching. In 1854, George Boole publishes The Laws of Thought, an algebra where variables are only 0 or 1; Jevons, ten years later, gives it the rule we use, x + x = x, the "or" of two truths is a truth. Peirce sees as early as 1886 that an electrical circuit could compute it. Nobody does for fifty years.

Then a master’s student, Claude Shannon, at MIT in 1937, and Nakashima in Japan and Shestakov in the USSR each on their own, notice that the relays of a telephone exchange obey this algebra: two contacts in series make an AND, two in parallel make an OR, a contact at rest makes a NOT. The gates of this RAS are those contacts, etched in silicon.

Many computers of the 1950s have words of 36 bits, which cut into threes, hence octal. In 1964, IBM announces the System/360 and fixes addressing on the eight-bit byte. Eight does not cut into threes; it cuts into two nibbles, and each nibble is written with a single digit, 0 to F. Hexadecimal is binary’s shorthand, four bits at a time: 0xFF, two full nibbles, 255.

An odometer running backwards goes from 000000 to 999999; a byte that backs up from zero shows 0xFF. That is two’s complement, which von Neumann proposes in 1945: the most significant bit, on the left, weighs −128 instead of +128. So 0xFF is 255 read unsigned and −1 read signed; 0x80 is 128 or −128. On 12 signed bits, the count runs from −2048 to 2047. The TMP102 delivers its temperature on 12 signed bits, 0.0625 °C per step: −0.5 °C is −8 steps, 0xFF8, left-justified in two bytes, 0xFF 0x80. Read unsigned, 4088 steps, 255.5 °C.

Two other conventions live in the same eight bits. BCD, where each nibble is a decimal digit: the seconds register of a DS3231 clock says 0x59 for 59 seconds, because the display reads in decimal; in binary, 0x59 would be 89. ASCII, standardized in 1963, where 0x41 is the letter A and 0x30 the character "0", which is not zero. Three readings of 0x59: 89, 59, or the letter Y.

In code, bits are touched one at a time with a mask. PORTB |= (1 << 3) sets bit 3 to 1 and leaves the other seven; PORTB &= ~(1 << 3) clears it, the other seven untouched; (x >> 4) & 0x0F brings the high nibble, bits 7 to 4, down to the place of bits 3 to 0, and cuts the rest. A mask says which bits it touches; it is enough to write it in binary.

The classic mistake has two faces. The sign misread: −0.5 °C becomes 255.5 °C. And the two bytes in the wrong order: the TMP102 sends the most significant byte first, a BMS that speaks SMBus sends the least significant first, and a pack at 25 °C, 0x19 0x00, read backwards gives 0x0019, that is 0.06 °C. Danny Cohen named this quarrel in 1980, Big-Endians and Little-Endians, after the war of the egg in Swift.