Converting hexadecimal numbers to other systems, like octal, often involves a middle step: binary. Think of binary as a universal translator. The key is knowing how to represent each hexadecimal digit as its own unique binary string. Since a single hex digit can represent values from 0 to 15, we need exactly four binary digits, or bits, to match that range. This four-bit group has a special name: a nibbles.

Why four bits? It comes down to powers of two. With one bit, you can represent two values (0 or 1). With two bits, you get four values (00, 01, 10, 11). Three bits give you eight values. It's not until you have four bits that you can represent $2^4$, or 16, distinct values—enough for every hex digit from 0 through F.

Each position in a four-bit binary number has a specific value, a concept known as bit weightings. From right to left, the positions represent $2^0$ (the 1s place), $2^1$ (the 2s place), $2^2$ (the 4s place), and $2^3$ (the 8s place). To find the binary equivalent of a hex digit, you simply figure out which of these values—8, 4, 2, and 1—add up to it.

For example, let's take the hex digit C. We know C represents the decimal value 12. Which combination of 8, 4, 2, and 1 equals 12? That would be 8 + 4. So, we place a 1 in the 8s and 4s positions and a 0 in the others. This gives us the binary string 1100.

What about a simpler digit, like 3? We only need a 2 and a 1. So, we put a 1 in those positions. To keep it a four-bit string, we must add leading zeros. The hex digit 3 becomes 0011.

Here is the complete mapping of hexadecimal digits 0 through F to their 4-bit binary equivalents. Committing this to memory is the fastest way to perform conversions.

With this one-to-one mapping, you can take any hexadecimal number, no matter how long, and convert it to binary by simply replacing each hex digit with its corresponding nibble. For instance, the hex number A3 becomes 10100011. You just look up A (1010) and 3 (0011) and place them side-by-side.

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