Binary and Hexadecimal

Binary is simply a way of writing numbers with only two digits, 0 and 1, and hexadecimal is a way of writing them with sixteen. The names just count digits: bi- means two, and hexadecimal glues Greek hexa (six) to decimal (ten), so sixteen. Binary is the natural language of hardware, where a wire is either on or off, and hex is the shorthand engineers use to read it quickly. The formula $n = \sum b_i\,2^i$ looks scary, but it's just three pieces: $b_i$ (each bit, 0 or 1), $2^i$ (what that position is worth: 1, 2, 4, 8, …) and $\sum$ (add them all up). It's the same thing you already do with a number like 407 in decimal; bear with it.

Place value with 2s. You've used place value for years: the decimal numeral 407 means $4 \cdot 10^2 + 0 \cdot 10 + 7$. Binary is the same idea with powers of 2 in place of powers of 10, so each digit, a bit, only needs to be 0 or 1:

Hardware likes base 2 because a wire only has to tell two voltages apart, high and low, which stays reliable even with noise.

Decimal to binary. Divide by 2 and keep the remainders. Think about why that works. Write $n = 2q + b_0$. Every bit except the last one is worth an even amount, so the remainder $b_0$ has to be the last bit, and $q$ is all the other bits shifted down one place. So you just repeat on $q$. Notice the remainders come out last bit first.

How many patterns. Each of $k$ bits has 2 choices, so by the product rule there are $2^k$ patterns, standing for $0$ through $2^k - 1$. A byte (8 bits) holds 0 to 255. Going the other way, $n \ge 1$ needs $\lfloor \log_2 n \rfloor + 1$ bits. The largest $k$-bit number is the geometric series $1 + 2 + 4 + \cdots + 2^{k-1} = 2^k - 1$, which is why 0xFF + 1 = 0x100.

Hexadecimal. Long rows of bits are hard to read, so engineers write them in base 16, with digits 0 to 9 and A to F (A = 10, …, F = 15). Since $16 = 2^4$, each hex digit is exactly four bits, and converting is just regrouping:

Decimal digits don't line up with bits like this, because 10 isn't a power of 2. That's why datasheets give register addresses and bit masks in hex: you can read the bits right off. Code marks the base with a prefix, so 0xB5, 0b10110101 and 181 are the same number in C, Python and JavaScript. Octal (base 8, three bits per digit) survives in Unix file permissions like 0755.

In this unit you can type answers the way code writes them: 0b1011 for binary and 0x2D for hex.

Counting in binary. Count 0, 1, 10, 11, 100, … and keep an eye on each bit: bit 0 flips every step, bit 1 every 2 steps, bit $k$ every $2^k$ steps. So each bit of a counter is a square wave at half the frequency of the bit below it. In other words, a binary counter is a frequency divider, and that's how a chip makes slower clocks from a fast one. There's an animation further down that shows a 4-bit counter doing exactly this.

Every register map in a microcontroller datasheet is written in hex, because each hex digit lines up with four bits. And watch crystals run at 32,768 Hz because a 15-bit binary counter divides that down to exactly one tick per second.