Data representation and logic

Write the column values above it — 128 64 32 16 8 4 2 1 — and add the columns that have a 1 under them. 10110101 = 128 + 32 + 16 + 4 + 1 = 181.

Take the biggest column value that fits, subtract it, and repeat with what is left. 181: 128 fits (53 left), 32 fits (21), 16 fits (5), 4 fits (1), 1 fits (0) — so 10110101. Add the columns back to check.

Four bits can hold the values 0 to 15, which is exactly the range of a single hex digit (0 to F). So any four bits swap cleanly for one character.

Group the bits in fours starting from the right, padding the leftmost group with zeros, then turn each group into its digit. 101101 is 0010 1101, which is 2D.

A = 10, B = 11, C = 12, D = 13, E = 14, F = 15. To convert two hex digits to decimal, multiply the left by 16 and add the right: B5 = 11 × 16 + 5 = 181.

Because the meaning is an agreement held by the program reading them, not a property of the bits. 01000001 is 65 read by place value and A read through a character set.

Each character is given a number by an agreed character set — 65 for A in ASCII — and that number is stored in binary. A string is those numbers one after another, so CAT is 67, 65, 84.

ASCII uses one byte per character and covers English. Unicode assigns numbers to every writing system, using more bytes per character where it needs them.

The picture is divided into a grid of pixels, and each pixel's colour is stored as numbers — one byte for greyscale, or three bytes for red, green and blue. More pixels means proportionally more storage.

Audio: the height of the sound wave measured thousands of times a second, each measurement stored as a number. Video: a complete image 24 to 60 times a second, plus the audio — which is why video files are the largest.

A component that takes one or two bits in and gives one bit out, always the same way for the same inputs. Gates wired together are what a CPU is physically built from.

AND: only when both inputs are 1. OR: when at least one input is 1. NOT: when the single input is 0. XOR: when the two inputs are different.

Invert the output. AND with a circle is NAND, OR with a circle is NOR, XOR with a circle is XNOR — each is the original table with every output flipped.

When both inputs are 1, OR gives 1 but XOR gives 0. XOR means the inputs are different, so it rules out the both-on case. A rule saying 'one or the other but not both' is XOR.

AND: a microwave runs only if the door is closed and the timer is running — the rule says 'both'. OR: a car alarm sounds if any one of several doors opens — the rule says 'any'.

Two to the power of the number of inputs: 2 inputs give 4 rows, 3 give 8, 4 give 16. The rows are counted upwards in binary, so no combination is missed.

Give every gate its own column and work left to right. Fill the input columns first, then each gate column using only its own inputs, then combine those for the final output.

Use · for AND, + for OR and NOT for inversion. A circuit where an AND of A and B feeds an OR alongside an inverted C is X = (A·B) + NOT C.

The outputs are arranged so neighbouring squares differ by one variable only. Circle the largest groups of adjacent 1s in sizes 1, 2, 4 or 8; for each circle keep the variables that stay constant and drop those that change.

A shorter expression is a circuit built from fewer gates — less hardware, less power consumption and fewer things to fail. For example A·B + A·(NOT B) simplifies to just A, needing no gates at all.

Write the Boolean expression first, draw the inputs down the left, draw the gates inside the brackets first, then feed their outputs into the final gate and label the output line.

A flat-backed D shape is AND; a curved back with a pointed front is OR; a triangle with a circle is NOT. A circle on any output inverts it; an extra curve at the back makes it exclusive.

Whichever is inside the brackets. Its output feeds the gate outside them. Drawing them the other way round reverses the logic and is the most common mistake.

A + A·B = A, which drops a whole term when A alone is already enough, and A·(B + C) = A·B + A·C, which lets you factor out what is common. A repeated letter is the clue to look for.

Build the truth table for the original expression and for the simplified one. Every row must match. This catches nearly every error, and a shorter expression is never proof on its own.