THE HEXADECIMAL NUMBERING SYSTEM
So far, all the work we’ve done with numbers has been with
the decimal numbering system. But we’re getting
close to the point where we need to look at binary numbers, meaning decimal
numbers are difficult, or even impossible, for what we want to do.
However, a one-byte number requires eight 1s and 0s in
binary, and a two-byte number requires 16 1s and 0s in binary. In addition, it’s hard to instantly tell the
equivalent decimal number by looking at a binary number. Needless to say, sometimes binary numbers can
be inconvenient to work with.
Which is why we have another numbering system that solves
these problems: hexadecimal. Binary has
a maximum “ones”, “tens”, etc. digit of 1, and Decimal has a maximum ones,
tens, etc. digit of 9. Hexadecimal has a
maximum of 15.
But how do we represent this?
We can’t, for example, count 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, etc. because then by trying to have a maximum of 15 in the “ones”
place, we end up with a tens place!
Thus, to make sure we have 10, 11, 12, 13, 14, 15 in the ones place, we
represent these numbers with letters. 10
= A, 11 = B, 12 = C, 13 = D, 14 = E, and 15 = F.
So, to count from 0 to 32 in hexadecimal, we have the following: 0, 1,
2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 1A, 1B, 1C, 1D, 1E, 1F, 20.
Okay, you’re about to kill me for not explaining why hexadecimal is,
in some situations, easier to work with than binary numbers. One digit of hexadecimal is equal to four
digits of binary, which means you can store an 8-digit binary number in just
two digits of hexadecimal. 1111 is equal
to the hexadecimal digit F, so you can easily write 1111111111111111 as FFFF.
In addition, notice that ten of the sixteen possible values for a
digit of hexadecimal—0 to 9—have a decimal equivalent! So it’s far easier to translate decimal to
hexadecimal and hexadecimal to decimal quickly.
BUT, but but but but, you sadly have to discover the special,
opportune moments for hexadecimal by yourself.
Most of the time when you write a regular program, hexadecimal is used
more for tradition than for being a “reasonable numbering system” to use. For instance, hexadecimal is the numbering
system traditionally used for numbering RAM addresses, and hexadecimal is the
number system traditionally used to list the numerical equivalents of
characters in a string. Yet these are
only traditional, not convenient. As you
program in ASM, you’ll discover when it’s appropriate to use hexadecimal to
save you time.
263 =
1 + 7?
In lesson 4, I left you with the thought: if H = 0, and L = 6, HL = 6. But if H = 1 and L = 7, HL does not equal 17.
So why is it that when you put H = 1 and
L = 7 together that you don’t get 17?
You have to think of it in terms of binary numbers. H = 00000001 in binary, and 7 = 00001111 in binary. If you put them together, and come out with
0000000100001111, that number is indeed equal to 263.
This is also why H = 0
and L = 6 means HL = 6. H = 00000000 and
L = 00001110, so HL = 0000000000001110.
Similarly, if we have a
given value for HL, we can find out what H and L are equal to. Remember that 1 byte is equal to 8 bits, so
by splitting the 16 bits of HL in half, we get H in the first half and L in the
second. We can tell from the first eight
bits of 0000000100001111 that H = 00000001 = 1 and L = 00001111, that last
eight bits, which equals 7.
Let’s look at the last
paragraph in hexadecimal for practice.
Remember that 1 byte is equal to 2 digits in hexadecimal, so by splitting
the 4 digits of HL (hexadecimal) in half, we have H = 01 and L = 07 in
hexadecimal. (Easier to visualize than
binary, right?)
TWO-BYTE
REGISTERS
As a review from lesson
7, there are several 1-byte registers that can be combined into two-byte
registers for important functions. H and
L can be combined, D and E can be combined, and B and C can be combined. A and F can also be combined, but for very
little purpose, which we will look at later.
Avoid AF for right now.
You
already know that HL is used to store RAM addresses. But it is also used for 2-byte math. If you need to solve a problem involving
numbers too big to fit inside of register A, use HL to do your math.
DE is
also used to store RAM addresses. It can
be used to access RAM if need be, but you should avoid using DE for data
storage and access until necessary.
Rather, DE is kind of like HL’s “partner.” It is the register most often used to hold a
RAM address when it needs to be saved for only a few instructions, and it is
used to tell HL where to store data when large amounts need to be copied. It is also the register most often used when
HL needs to do math. Why is DE so
important in all of this? Because HL and
DE can exchange their values. You can tell DE to hold whatever is inside of
HL, and HL to hold whatever is inside of DE.
No other registers, not even one-byte registers, allow you to do this so
easily. So strictly speaking, don’t use
DE to hold and store data that HL can handle, but it should be the first
register you use when HL needs a helping hand.
BC is
another 2-byte register. It can do
almost anything that DE can do, but it cannot exchange with HL. You should use BC in conjunction with HL only
when DE is tied up. However, BC does
have a special purpose, in that when you need to copy large amounts of data, BC
is a byte-counter. It means just that: it holds how many bytes
you need to copy, or how many bytes you need when a function you use has a parameter
of a number of bytes.
There
are other two-byte registers, we’ll look at those later.
NEW
CODING INSTRUCTIONS
The next couple few
pages contain some instructions you can use with 2-Byte registers as
parameters. 2-Byte value means a number
up to 65535. HL BC DE means you can use either HL, DE or BC.
Rather than doing any
coding exercises for this lesson, we’ll work with these new functions next lesson,
when we work some more on ASM Gorillas.