By Robert M (adapted by Duane Alan Hahn)
Table of Contents
Original Lesson
In previous lessons, I have hinted that there is a standard method for counting using bits. In this lesson, I will introduce that standard method. First, I want to reiterate that this is NOT the only way to count using bits. It is simply the most common method, and therefore has considerable support for it built into most microprocessors including the 650X processor Family used in most classic gaming consoles and computers from the late 70s and 80s.
I must assume that if you can read the lessons in this class and understand them, then you must be familiar with decimal numbers and basic arithmetic like adding and subtracting. All decimal numbers are made using the ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. When you write a decimal number larger than 9 you must use two or more digits like this: 123 which is one hundred and twenty three. The 1 in 123 does not represent 1 it represents 100. The 2 in 123 is not 2 it is 20. The 3 in 123 does represent 3. Thus we call the 1 the hundreds digit, the 2 the tens digit, and 3 the one's digit. Each digit in a large decimal number represents that digit multiplied by a power of 10 based on its position and summed together.
Thus (1 * 10^2) + (2 * 10^1) + (3 * 10^0) = 100 + 20 + 3 = 123 Tada!
We count numbers inside computers exactly the same way except that the numbering system in computers does not have 10 digits, it only has 2 digits: 0 and 1 (bits!). The numbering system that uses only 0 and 1 is called the binary numbering system or simply binary. Since it only has 2 digits the value of each binary digit in a large binary number is multiplied by a power of 2 instead of a power of 10 as we do for decimal numbers.
The following table shows the positional value of the first 16 bits in the binary number system, as a power of 2. Note that the first position is numbered zero and not one.
Bit position.   Bit value   2^0 = 1 => bit 2^1 = 2 2^2 = 4 => octet 2^3 = 8 => nybble 2^4 = 16 2^5 = 32 2^6 = 64 2^7 = 128 => byte 2^8 = 256 2^9 = 512 2^10 = 1024 2^11 = 2048 2^12 = 4096 2^13 = 8192 2^14 = 16384 2^15 = 32768 => word
Every computer in the world has a limited number of bits. When you use binary numbers in a computer program, you must decide how many bits you will use to store each number that your program must keep track of. For 650X processors, the most common numbers of bits to use will be 8 (byte) and 16 (word) because the processor works with data an entire byte at a time for each instruction (see opcodes in lesson 3). You can use any number of bits, but since 8 and 16 are so common, we will focus on them for the remainder of this lesson. The same rules will apply to binary numbers made of any number of bits.
This should not seem strange, as you can use any number of digits to represent a decimal number. If you don't need all the digits, then your number will have 1 or more leading zeros.
0000123 is the 7digit decimal number 123, by convention leading zeros are not shown for decimal numbers, but they are always present nonetheless. The same is true for binary numbers.
In the real world you can write as big of decimal number as your writing surface will allow. In a computer, your writing surface is bits. If you do not have enough bits allocated/available to store a binary number then your program can not store that number and the information will be lost. Such an event is called overflow if the number is too big, or underflow if the number is too small. We will examine overflow and underflow in detail in the next lesson.
You may be curious to know what is the biggest binary number you can store in a byte or a word. You already learned the formula in lesson 2—enumeration. Given N bits, you can store 2^N different values. Counting positive integers starts at zero not one, so N bits can store binary numbers from 0 to (2^N)1
Example:
A byte has 8 bits so N=8. Therefore, a byte can hold unsigned positive integers from 0 to (2^8)1=2561=255. From 0 to 255.
Algorithm to Convert Decimal to Binary
To convert a decimal number to binary, you must repeatedly divide that number by 2. The remainder of each division is next binary digit in the binary representation of that decimal number. when there is nothing left to divide, the conversion is done.
Example: Convert 123 decimal to its binary equivalent.
123 / 2 = 61 with a remainder of 1 + 61 / 2 = 30 with a remainder of 1 + 30 / 2 = 15 with a remainder of 0 + 15 / 2 = 7 with a remainder of 1 + 7 / 2 = 3 with a remainder of 1 + 3 / 2 = 1 with a remainder of 1 + 1 / 2 = 0 with a remainder of 1 +  123 Decimal = 1111011 binary
Algorithm to Convert Binary to Decimal
Any binary number can be converted to its decimal equivalent, which is often easier for humans to read and understand. To convert a binary number to decimal you must multiply each digit of the binary number by its positional value (see the first table above) and add the digit values together to form the decimal value.
Example: Convert 1111011 binary to decimal
1111011 binary  + 1 * 2^0 = 1 * 1 = 1 + 1 * 2^1 = 1 * 2 = 2 + 0 * 2^2 = 0 * 4 = 0 + 1 * 2^3 = 1 * 8 = 8 + 1 * 2^4 = 1 * 16 = 16 + 1 * 2^5 = 1 * 32 = 32 + 1 * 2^6 = 1 * 64 = 64  123 decimal
Up until now I have been explicitly stating whether a written number is binary or decimal. As you can see it can become tedious to write "binary" or "decimal" after every number to clearly indicate the numbering system being used. Luckily, a handy shorthand notation exists for us to use to save time. For the remainder of this class I will always make use of this notation to indicate whether a number is binary or decimal.
Decimal Notation: A decimal number is written with no special notation at all. It is written the same way you have always written decimal numbers.
Binary Notation: All binary numbers shall be preceded by the % symbol.
Examples:
123 is decimal and %1111011 is its binary equivalent. 101 is decimal = One hundred and one. %101 is binary = 4 + 1 = Five
Another aspect of binary notation is MostSignificant and LeastSignificant Bits (MSB and LSB). By convention, a binary number is written from left to right, the same as a decimal number. The leftmost digit of a number represents the greatest portion of the overall value of the whole number. It is the MostSignificant digit in the number. The rightmost digit of any number represents the least portion of the whole value. It is the LeastSignificant digit. Each digit in a binary number is a bit. Therefore, the MostSignificant Bit (MSB) of a binary number is the leftmost bit, and the LeastSignificant Bit (LSB) is the right most bit.
Examples:
%100010010 ^ ^   MSB LSB %10111010010010 ^ ^   MSB LSB %100110 ^ ^   MSB LSB
Please get comfortable with this notation for binary numbers because you will be using it extensively when writing assembly language programs!
You may have noticed that so far our discussion of binary has been restricted to positive integers. Computers can count negative numbers, so now we will begin to explore methods for representing negative numbers in binary. We will complete this discussion in Lesson 5.
Sign Magnitude Format
The first format for negative numbers we will explore is the signmagnitude format. The format is easy to understand. For any given binary number add an additional bit to represent the sign of the number. If the sign bit is set, then the number is negative. If the sign bit is clear then the number is positive. The sign bit is the MSB of a signmagnitude binary number.
Examples:
%0101 = %0 sign and %101 magnitude = 5 %1101 = %1 sign and %101 magnitude = 5  +> The MSB is the Sign bit
There is a problem with this method of representing negative numbers. Can you see what that problem is?
The problem is with the number zero. Using this system there are 2 instances of zero, a negative zero and a positive zero. There is no such thing as negative zero, so this may not be the best method for representing negative numbers.
Two's Complement Format (Introduction)
An alternative to the signmagnitude format is the two's complement format. Two's complement is superior to signmagnitude (usually, but not always) because there is only one way to represent zero. Another advantage of two's complement is that when you add or subtract positive and negative numbers, the result is the correct number in two's complement format.
To really understand two's complement, you must know how to add and subtract binary numbers. We won't learn how to add and subtract binary numbers until lesson 5, so we will suspend further discussion of negative binary numbers until lesson 5.
Yes, you can represent REAL numbers in binary. REAL numbers can also be thought of as fractions (1.23 for example). This is an advanced topic and I don't want to go into it now because I believe it will confuse more people than it will help them to understand. You can write many, many programs without every needing to use REAL numbers. So just put them out of your mind for now.
%00110101
%01000101
%11100001
%10100110
%01001011
%01001110
%11001001
Bonus Questions:
213 / 2 = 106 with a remainder of 1 + 106 / 2 = 53 with a remainder of 0 + 53 / 2 = 26 with a remainder of 1 + 26 / 2 = 13 with a remainder of 0 + 13 / 2 = 6 with a remainder of 1 + 6 / 2 = 3 with a remainder of 0 + 3 / 2 = 1 with a remainder of 1 + 1 / 2 = 0 with a remainder of 1 + %11010101 binary = 213 decimal.
%00101100  + 0 * 2^0 = 0 * 1 = 0 + 0 * 2^1 = 0 * 2 = 0 + 1 * 2^2 = 1 * 4 = 4 + 1 * 2^3 = 1 * 8 = 8 + 0 * 2^4 = 0 * 16 = 0 + 1 * 2^5 = 1 * 32 = 32 + 0 + 0 0 + 0 + 4 + 8 + 0 + 32 + 0 + 0 = 44 decimal.
1087 / 2 = 543 with a remainder of 1 543 / 2 = 270 with a remainder of 1
270 / 2 = 135 with a remainder of 0
135 / 2 = 67 with a remainder of 1
67 / 2 = 33 with a remainder of 1
33 / 2 = 16 with a remainder of 1
16 / 2 = 8 with a remainder of 0
8 / 2 = 4 with a remainder of 0
4 / 2 = 2 with a remainder of 0
2 / 2 = 1 with a remainder of 0
1 / 2 = 0 with a remainder of 1
So 1087 decimal = %10000111011 binary.
%1000010100011110        + 1 * 2^1 = 2    + 1 * 2^2 = 4    + 1 * 2^3 = 8    + 1 * 2^4 = 16   + 1 * 2^8 = 256  + 1 * 2^10 = 1024 + 1 * 2^15 = 32768 2+4+8+16+256+1024+32768 = 34078 decimal
%00110101
%01000101
%11100001
%10100110
%01001011
%01001110
%11001001
The LSB is the rightmost digit of each number, so the new binary number is: %1010111 which in decimal is: 64+0+16+0+4+2+1 = 87
unsigned = 64 + 0 + 0 + 0 + 4 + 0 + 1 = 69 decimal
signed = (1) * (0 + 0 + 0 + 4 + 0 + 1 ) = 5 decimal
unsigned = 0 + 32 + 0 + 16 + 0 + 2 + 0 = 50
signed = (+1) * ( 32 + 0 + 16 + 0 + 2 + 0 ) = 50
unsigned = 64 +32 + 0 + 0 + 4 + 2 + 0 = 102
signed = (1) * ( 32 + 0 + 0 +4 +2 +0 ) = 38
unsigned = 2
signed = 2
There are two basic ways to solve this problem. I will demonstrate both because I think it's important to see the relationships that make both methods arrive at the same answer.
If we convert that to decimal we get:
2^15 + 2^14 + 2^13 + 2^12 + 2^11 + 2^10 + 2^9 + 2^8 + 2^7 + 2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2^1 + 2^0 =
32768 + 16384 + 8192 + 4096 + 2048 + 1024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 65535.
combinations = 2 ^ 16 = 65536 combinations. Recall however that the enumeration begins at 0, so that leaves 65536  1 = 65535 as the largest possible unsigned integer in a word.
Bonus Questions:
This is a trick question. The correct answer is both. As we learned in the first lesson bits represent whatever you the programmer say that they represent. They can represent both at the exact same time. It doesn't matter, they are only bits. Meaning is given by you the programmer with the logic of your code.
It is a code because enumerations represent only positive values from 0 to N.
We are rapidly descending from broad concepts to specific details in assembly programming. Most often, subsequent lessons will require an understanding of previous lessons. If you have questions after studying this material, do not hesitate to ask.
Other Assembly Language Tutorials
Be sure to check out the other assembly language tutorials and the general programming pages on this web site.


Lesson 4: Binary Counting
How to get started writing 6502 assembly language. Includes a JavaScript 6502 assembler and simulator.
Atari Roots by Mark Andrews (Online Book)
This book was written in English, not computerese. It's written for Atari users, not for professional programmers (though they might find it useful).
Machine Language For Beginners by Richard Mansfield (Online Book)
This book only assumes a working knowledge of BASIC. It was designed to speak directly to the amateur programmer, the parttime computerist. It should help you make the transition from BASIC to machine language with relative ease.
The 6502 Instruction Set broken down into 6 groups.
Nice, simple instruction set in little boxes (not made out of tickytacky).
The Second Book Of Machine Language by Richard Mansfield (Online Book)
This book shows how to put together a large machine language program. All of the fundamentals were covered in Machine Language for Beginners. What remains is to put the rules to use by constructing a working program, to take the theory into the field and show how machine language is done.
An easytoread page from The Second Book Of Machine Language.
6502 Instruction Set with Examples
A useful page from Assembly Language Programming for the Atari Computers.
Continually strives to remain the largest and most complete source for 6502related information in the world.
By John Pickens. Updated by Bruce Clark.
Guide to 6502 Assembly Language Programming by Andrew Jacobs
Below are direct links to the most important pages.
Goes over each of the internal registers and their use.
Gives a summary of whole instruction set.
Describes each of the 6502 memory addressing modes.
Describes the complete instruction set in detail.
HTMLified version.
Nick Bensema's Guide to Cycle Counting on the Atari 2600
Cycle counting is an important aspect of Atari 2600 programming. It makes possible the positioning of sprites, the drawing of sixdigit scores, nonmirrored playfield graphics and many other cool TIA tricks that keep every game from looking like Combat.
How to Draw A Playfield by Nick Bensema
Atari 2600 programming is different from any other kind of programming in many ways. Just one of these ways is the flow of the program.
Cart Sizes and Bankswitching Methods by Kevin Horton
The "bankswitching bible." Also check out the Atari 2600 Fun Facts and Information Guide and this post about bankswitching by SeaGtGruff at AtariAge.
Atari 2600 programming specs (HTML version).
Atari 2600 Programming Page (AtariAge)
Links to useful information, tools, source code, and documentation.
Atari 2600 programming site based on Garon's "The Dig," which is now dead.
Includes interactive color charts, an NTSC/PAL color conversion tool, and Atari 2600 color compatibility tools that can help you quickly find colors that go great together.
The Atari 2600 Music and Sound Page
Adapted information and charts related to Atari 2600 music and sound.
A guide and a check list for finished carts.
A multiplatform Atari 2600 VCS emulator. It has a builtin debugger to help you with your works in progress or you can use it to study classic games.
A very good emulator that can also be embedded on your own web site so people can play the games you make online. It's much better than JStella.
If assembly language seems a little too hard, don't worry. You can always try to make Atari 2600 games the faster, easier way with batari Basic.
Disclaimer
View this page and any external web sites at your own risk. I am not responsible for any possible spiritual, emotional, physical, financial or any other damage to you, your friends, family, ancestors, or descendants in the past, present, or future, living or dead, in this dimension or any other.
Use any example programs at your own risk. I am not responsible if they blow up your computer or melt your Atari 2600. Use assembly language at your own risk. I am not responsible if assembly language makes you cry or gives you brain damage.