Bytes, hex and bit width

Bits come in groups

Nobody handles bits one at a time. A single bit answers a single yes-or-no question, and almost everything a circuit does needs more than that, so bits are dealt with in fixed-size groups. Two sizes come up so often that they have names.

The byte is the group that matters most. Almost every computer built since the 1970s stores its memory as a long row of bytes, and bigger values are made of several bytes side by side. The nibble matters for a different reason: it is exactly the size of one hexadecimal digit, which is the subject of the next section.

Hex: one digit per nibble

Binary is easy for a circuit and tiresome for a person. Try reading 1011011010011100 out loud and you will lose your place halfway through. Hexadecimal, or hex for short, is a shorthand for writing bit patterns. It uses sixteen digits: 0 to 9 as usual, then the letters A to F for ten to fifteen. Sixteen is 2⁴, so one hex digit stands for exactly one nibble, and a byte is always exactly two hex digits. Here is the whole table.

To turn a byte into hex, split it into its two nibbles and look each one up. To go back, replace each hex digit with its nibble. There is no arithmetic in either direction, only the lookup, and after a little practice the table is in your head. Try it below: each nibble has its own hex digit, and flipping a bit only ever changes the digit it belongs to.

Width: how many bits there are

The previous lesson introduced the width of a circuit: the fixed number of bits it holds. A width is nearly always a whole number of bytes, and four widths do most of the work in the world: 8 bits, 16 bits, 32 bits and 64 bits. An eight bit width holds one byte, and a sixty-four bit width holds eight of them.

The width sets the biggest number that fits. With n bits there are 2ⁿ patterns, one of which is zero, so the biggest value is 2ⁿ − 1. For a byte that is 255. For sixteen bits it is 65,535, and for thirty-two bits it is a little over four billion. Click every bit on below and read the value.

Now imagine adding one to all-ones. In decimal, 999 plus one needs a fourth column. In binary, 11111111 plus one needs a ninth bit, and an eight bit circuit does not have one. The 1 that should have gone into that ninth column has nowhere to go, so the eight bits that remain read 00000000. The value has wrapped round to zero. Nothing broke; the circuit did exactly what its wires allow. When you build an adder in stage 5 you will see that lost 1 come out on a wire of its own, called the carry out, so that a circuit which cares can catch it.

What to remember

• A nibble is 4 bits and a byte is 8. A byte holds 256 patterns, so 0 to 255 as an unsigned number.
• Hex uses the digits 0 to 9 and A to F. One hex digit is one nibble, so a byte is two hex digits.
• Converting between hex and binary is a lookup, one nibble at a time, with no arithmetic.
• Common widths are 8, 16, 32 and 64 bits. With n bits the biggest value is 2ⁿ − 1.
• Adding one to all-ones wraps round to zero, and the lost carry comes out on its own wire in an adder.