Binary and hexadecimal

Before we talk about the code, a bit of background knowledge is in order. When programming at a low level, understanding of binary and hexadecimal is mandatory. Since you may already know about both of these, a summary of the RGBDS-specific information is available at the end of this lesson.

So, what’s binary? It’s a different way to represent numbers, in what’s called base 2. We’re used to counting in base 10, so we have 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Here’s how digits work:

  42 =                       4 × 10   + 2
     =                       4 × 10^1 + 2 × 10^0
                                  ↑          ↑
    These tens come from us counting in base 10!

1024 = 1 × 1000 + 0 × 100  + 2 × 10   + 4
     = 1 × 10^3 + 0 × 10^2 + 2 × 10^1 + 4 × 10^0
       ↑          ↑          ↑          ↑
And here we can see the digits that make up the number!

ℹ️

^ here means “to the power of”, where X^N is equal to multiplying X with itself N times, and X ^ 0 = 1.

Decimal digits form a unique decomposition of numbers in powers of 10 (decimal is base 10, remember?). But why stop at powers of 10? We could use other bases instead, such as base 2. (Why base 2 specifically will be explained later.)

Binary is base 2, so there are only two digits, called bits: 0 and 1. Thus, we can generalize the principle outlined above, and write these two numbers in a similar way:

  42 =                                                    1 × 32  + 0 × 16  + 1 × 8   + 0 × 4   + 1 × 2   + 0
     =                                                    1 × 2^5 + 0 × 2^4 + 1 × 2^3 + 0 × 2^2 + 1 × 2^1 + 0 × 2^0
                                                              ↑         ↑         ↑         ↑         ↑         ↑
                                          And since now we're counting in base 2, we're seeing twos instead of tens!

1024 = 1 × 1024 + 0 × 512 + 0 × 256 + 0 × 128 + 0 × 64  + 0 × 32  + 0 × 16  + 0 × 8   + 0 × 4   + 0 × 2   + 0
     = 1 × 2^10 + 0 × 2^9 + 0 × 2^8 + 0 × 2^7 + 0 × 2^6 + 0 × 2^5 + 0 × 2^4 + 0 × 2^3 + 0 × 2^2 + 0 × 2^1 + 0 × 2^0
       ↑          ↑         ↑         ↑         ↑         ↑         ↑         ↑         ↑         ↑         ↑

So, by applying the same principle, we can say that in base 2, 42 is written as 101010, and 2048 as 1000000000. Since you can’t tell ten (decimal 10) and two (binary 10) apart, RGBDS assembly has binary numbers prefixed by a percent sign: 10 is ten, and %10 is two.

Okay, but why base 2 specifically? Rather conveniently, a bit can only be 0 or 1, which are easy to represent as “ON” or “OFF”, empty or full, etc! If you want, at home, to create a one-bit memory, just take a box. If it’s empty, it stores a 0; if it contains something, it stores a 1. Computers thus primarily manipulate binary numbers, and this has a slew of implications, as we will see throughout this entire tutorial.

Hexadecimal

To recap, decimal isn’t practical for a computer to work with, instead relying on binary (base 2) numbers. Okay, but binary is really impractical to work with. Take %10000000000, aka 2048; when in decimal only 4 digits are required, binary instead needs 10! And, did you notice that I actually wrote one zero too many? Fortunately, hexadecimal is here to save the day! 🦸

Base 16 works just the same as every other base, but with 16 digits, called nibbles: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.

  42 =            2 × 16   + 10
     =            2 × 16^1 + A × 16^0

1024 = 4 × 256  + 0 × 16   + 0
     = 4 × 16^2 + 2 × 16^1 + 4 × 16^0

Like binary, we will use a prefix to denote hexadecimal, namely $. So, 42 = $2A, and 1024 = $400. This is much more compact than binary, and slightly more than decimal, too; but what makes hexadecimal very interesting is that one nibble corresponds exactly to 4 bits!

NibbleBits
$0%0000
$1%0001
$2%0010
$3%0011
$4%0100
$5%0101
$6%0110
$7%0111
$8%1000
$9%1001
$A%1010
$B%1010
$C%1100
$D%1101
$E%1110
$F%1111

This makes it very easy to convert between binary and hexadecimal, while retaining a compact enough notation. Thus, hexadecimal is used a lot more than binary. And, don’t worry, decimal can still be used 😜

(Side note: one could point that octal, i.e. base 8, would also work for this; however, we will primarily deal with units of 8 bits, for which hexadecimal works much better than octal. RGBDS supports octal via the & prefix, but I have yet to see it used.)

💡

If you’re having trouble converting between decimal and binary/hexadecimal, check if your favorite calculator program doesn’t have a “programmer” mode, or a way to convert between bases.

Summary

  • In RGBDS assembly, the hexadecimal prefix is $, and the binary prefix is %.
  • Hexadecimal can be used as a “compact binary” notation.
  • Using binary or hexadecimal is useful when individual bits matter; otherwise, decimal works just as well.
  • For when numbers get a bit too long, RGBASM allows underscores between digits (123_465, %10_1010, $DE_AD_BE_EF, etc.)