旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � Matt Johnson's Z80 Programming Tutorial � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � Programming the TI-86 � PART 02 � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � Binary / Hex / Logical Operators / Bit Masks � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 If you catch any mistakes or have any additions, comments, anything, e-mail me immediately! My e-mail address is Matt514@gte.net My Home Page is at http://home1.gte.net/matt514/matt.htm My TI-86 Web Page is at http://www.dogtech.com/cybop/ti86/ 컴컴컴컴컴컴컴� BINARY/HEX/ETC 컴컴컴컴컴컴컴컴컴 This lesson is very powerful. I do not go into detail about Z80 instructions because this lesson can be applied to almost every CPU around! Very important, binary, hex, bit manipulation, and logical operators IS when you KNOW your programming Assembly!! This lesson can be very boring. But it explains a lot. Assembly language requires a lot of math, especially hex and binary. Also you will learn logical operators such as AND, OR, XOR. Bit masks will also be explained in this lesson. A few people requested my this tutorial be on binary/hex/etc, which was my original plans anyway. Zshell school and others don't go into much detail on these subjects, which are extremely important.. Enjoy! 컴컴컴컴컴컴컴� NUMBERING SYSTEMS 컴컴컴컴컴컴컴컴컴 All numbering systems are based on any value of digits. The decimal system has 10 digits, the binary system has 2 digits, and the hexadecimal system has 16 digits. Take the number 315 for example. To convert this number into a decimal number, simply multiply each part (3, 1, 5) by its place value, and add them. Every Place Value are in multiples of 10. In this example, 5 is in the one's place, 1 is in the ten's place, and 3 is in the hundred's place. You could also say 5 is in the 10 ^ 0 place, 1 is in the 10 ^ 1 place, and 3 is in the 10 ^ 2 place. 旼컴컴컴컴� Value � 旼컴컴� Base 3 1 5 � � 旼 Place � � � � � � � � 읕컴 5 * 10 ^ 0 = 5 * 1 = 5 � 읕컴컴 1 * 10 ^ 1 = 1 * 10 = 10 읕컴컴컴 3 * 10 ^ 2 = 3 * 100 = 300 컴컴� 315 This shouldn't be difficult to understand. Decimal has the 1's place, 10's place, 100's place, etc. So if in the 1's place, there is a value 5, then the total amount of the 1's place is 5. 3 5 | | | |___ One's Place | |______ Ten's Place What if 35 is a decimal number? Then: Decimal Value = One's Place + Ten's Place = (3 * 10 ^ 1) + (5 * 10 ^ 0) = (3 * 10) + (5 * 1) = 30 + 5 = 35 What if 35 is a hex number? Then: [more on this later] = One's Place + Ten's Place = (3 * 16 ^ 1) + (5 * 10 ^ 0) = (3 * 16) + (3 * 1) = 48 + 5 = 53 35 HEX is 53 decimal. 컴컴컴컴컴컴컴� BINARY 컴컴컴컴컴컴컴컴컴 The Binary system is a Base 2 numbering system composed of two digits, 0 and 1. Every Place Value are in multiples of 2. Most computer systems operate on binary logic. Binary is used right down to the circuits, where electronic switches can be either in the ON state or the OFF state. Electronics which take use of binary, bits, and bytes, are called digital circuits. BIT 0 NIBBLE 0000 BYTE 00000000 A binary 0 or 1 is a bit. A pattern of of 4 bits is a nibble. A pattern of 8 bits is a BYTE. A pattern of 16 bits is a WORD. A pattern of 32 bits is a DOUBLE WORD. A bit represents ON or OFF, 0 or 1, SET or RESET. A bit is 1-bit (duh!), a nibble is 4-bits, and a byte is 8-bits. One hex digit represents a 4-bit number. More on that later. In Computers and electronic circuits decimal numbers are represented by binary numbers. Binary Numbers also serve as codes that represent letters of the alphabet, the color of a pixel, etc.. Binary numbers are often set up as patterns to represent letters. The common ASCII system uses an 7-bit value to code for 128 values. When 8-bit values are used, a maximum of 256 characters can be coded for. This allows the use of both standard ASCII and extended ASCII. 컴컴컴컴컴컴컴� MAKE-UP OF BINARY 컴컴컴컴컴컴컴컴컴 Converting Binary into Decimal: 旼컴컴컴컴� Value � 旼컴컴� Base 1 0 1 � � 旼 Place � � � � � � � � 읕컴 1 * 2 ^ 0 = 1 * 1 = 1 � 읕컴컴 0 * 2 ^ 1 = 0 * 2 = 0 읕컴컴컴 1 * 2 ^ 2 = 1 * 4 = 4 컴컴� 5 8 4 2 � � � (Combinations) 1 0 1 � � � (Place Value) 4 2 1 So 101 is 1*1+0*2+1*4, or simply 4+1+0, which is 5! Understand!?!? Also 101 has 8 combinations, 000,001,010,011,100,101,110,111. Cool Ehh? Converting Decimal into Binary: 旼컴컴컴컴컴컴 Value � 旼컴컴컴컴 Base � � 旼컴컴 Quotinent � � � � � � 旼 Remainder 5 / 2 = 2 + 1 2 / 2 = 1 + 0� 1납(Final Quotient is Final Remainder) 납� Answer: 101 So 5 is BINARY 100 + BINARY 00 + BINARY 1 = 101 So 5 is DECIMAL 4 + DECIMAL 0 + DECIMAL 1 = 5 컴컴컴컴컴컴컴� Bits, Bytes, & Words in Detail 컴컴컴컴컴� _________________________________________________________ 1 BIT � 2 Combinations � 0 to 1 � 2 BIT � 4 Combinations � 0 to 3 � 3 BIT � 8 Combinations � 0 to 7 � 4 BIT � 16 Combinations � 0 to 15 � 5 BIT � 32 Combinations � 0 to 31 � 6 BIT � 64 Combinations � 0 to 63 � 7 BIT � 128 Combinations � 0 to 127 � 8 BIT � 256 Combinations � 0 to 255 � 9 BIT � 512 Combinations � 0 to 511 � 10 BIT � 1024 Combinations � 0 to 102 � 12 BIT � 4096 Combinations � 0 to 4095 � 15 BIT � 32768 Combinations � 0 to 32767 � 16 BIT � 65536 Combinations � 0 to 65535 � 24 BIT � 16,777,216 Combinations � 0 to 16,777,215 � 32 BIT � 4,294,967,296 Combinations � 0 to 4,294,967,295� ---------------------------------------------------------- If you look at this chart, you see that, for example, 4 bits can represent up to 16 different things. 4-bits is enough to count up to 15 (from 0 to 15 = 16 total numbers) The number of combinations in 2 ^ n, where n is the number of digits. This means that in a binary number, every digit to the left is two times the place value of its right neighbor. Lets take a 4-bit number (called a NIBBLE), for example. 1 0 1 1 - Binary Number -------- 8 4 2 1 - One's place, Two's place, Four's Place, Eight's Place. Binary Number---| | Place Value-| | 8 * 1 = 8 4 * 0 = 0 2 * 1 = 2 + 1 * 1 = 1 ----- ---- 11 컴컴컴컴컴컴컴� NIBBLES 컴컴컴컴컴컴컴컴컴 Nibbles are a pattern of four bits. It is also a good little snake game too :-). The reason nibbles are important is for two things: Hexadecimal and BCD (Binary Coded Decimal). In BCD, a NIBBLE is used to represent on decimal number. This is because a nibble can represent from 0 to 15. 3-bits would not be enough since it only goes from 0 to 7. In Hex, a NIBBLE represents one hex digit. Hex is base 16 meaning that there are a total of 16 numbers per digit. So 1 HEX digit is 1 NIBBLE. Since the range of a 4-bit number (0 to 15) is the same as the range of a hex number (0 to 15), it makes these numbers EXTREMELY EASY to convert between the two. These TOPICS are explained in righteous DETAIL after the section on BYTES. 컴컴컴컴컴컴컴� BYTES 컴컴컴컴컴컴컴컴컴 Lets take an 8-bit number (called a BYTE) for example: 10101111 Some people find it easier to seperate the number in groups of 4, just like seperating decimal numbers by commas. 0110 1111 - Binary Number --------- 7654 3210 - Bit Number Bit 0 (the far right bit), is 1 in this example. Bit 0 is called the LSB (Least Significant Bit) Bit 7 (the far left bit), is 0 in this example. Bit 7 is called the MSB (Least Significant Bit) Bit 0 to bit 3 is called the L.O. (Low Order Nibble) Bit 4 to bit 7 is called the H.0. (High Order Nibble) Lets find the place value for each of these bits 0 1 1 0 1 1 1 1 ---------------------- 128 64 32 16 8 4 2 1 Bit 0: 1 * 1 = 1 Bit 1: 1 * 2 = 2 Bit 2: 1 * 4 = 4 Bit 3: 1 * 8 = 8 Bit 4: 0 * 16 = 0 Bit 5: 1 * 32 = 32 Bit 6: 1 * 64 = 64 + Bit 7: 0 * 128 = 0 --------------------- This byte is 111 in decimal. 컴컴컴컴컴컴컴� BINARY CODED DECIMAL 컴컴컴컴컴컴컴컴컴 BINARY CODED DECIMAL (BCD): A system in which each decimal digit is assigned its binary equivalent. For example, 1 9 � � � 읕컴 9 = 1001 � 읕컴컴 1 = 0001 So 19=0001 1001 and 23=0010 0011 and 123 = 0001 0010 0011 ...and 69 = 0110 1001 ... and you thought that you couldn't express your feelings toward that one girl you saw in school in Binary! :) Understand? It is pretty easy! 컴컴컴컴컴컴컴� HEXADECIMAL 컴컴컴컴컴컴컴컴컴 Let's Face it. Binary Numbers are difficult to use. They take up a lot of space, look ugly, and smell bad too! Plus it's a pain in the neck using 4, 8, and 16 bit values with decimal, since you can't easily convert binary numbers into decimal numbers. That's where HexaDecimal comes in! A hexadecimal number is a numbering system based on 16 digits. This means that every place value is in multiples of 16. ..But then how do we show 16 values in one digit! There's only 10 numbers on the keyboard 0..9. Well after the number 9, letters of the alphabet are used instead. So to count in hexadecimal, it would be 0,1,2,3,4,5,6,7,8,9,A, B,C,D,E,F. Lets look at a sample hexadecimal number. 4CF What the hell is that? Well its like any other old number, except that it just looks different (and is hexadecimal!) Let's take this number and show its place values. 4 C F � � � � � 읕컴 1's Place � 읕컴컴 16's Place 읕컴컴컴 256's Place Let's See it's Binary equivalents: 4 C F � � � � � 읕컴 1111 � 읕컴컴 1100 읕컴컴컴 0100 The neat thing about hexadecimal is any 4-bit value can be stored as one hex digit. This is because 4-bits equal 16, 1-Hex equals 16. So in Binary 4=0100 C=1100 F=1111 4CF= 0100 1100 1111, 4CF= 010011001111 Hmm.. 010011001111 is a 12-bit number. A 12-bit value has 2 ^ 12 combinations. A 3 digit hex number has 16 ^ 3 combinations. Well that means that a 12-bit binary number is just like a 3-digit hex number! A 8-bit binary number is like a 2-digit hex number, and a 4-bit binary number is like a 1-digit hex number. Each Hex digit has a 4-bit equivalent! That's why people use Hexadecimal! Because 4CF is a whole lot easier to handle then freaking 010011001111! 4 C F � � � � � 읕컴 15 * 10 ^ 0 = 15 * 1 = 15 � 읕컴컴 12 * 10 ^ 1 = 12 * 16 = 192 읕컴컴컴 9 * 10 ^ 2 = 4 * 256 = 1024 컴컴컴 1231 컴컴컴컴컴컴컴� Logical Operators 컴컴컴컴컴컴컴컴컴 The Z80 has many instructions that change a byte in some form or fashion. I will not go into the detail behind the instructions themselves (that will be another lesson), however you will learn the concept of bit manipulation. The three most important logical operators are AND, OR, and XOR. If you ever read anything about digital electronics, this should all sound familiar. AND is a digital logic gate in electronics. It actually uses transisters to compare two bits. AND is used to logically "and" two numbers together. When you "AND" two bits for example, the result is 1 (true) only when both bits are 1 (true). Lets look at a "truth table". "AND" A B RESULT ---------------- 0 0 0 0 1 0 1 0 0 1 1 1 Basically, if A and B are 1 (TRUE) then the RESULT is 1 (TRUE) What if the numbers are more then 1-bit? Well then you are "AND"ing a number bit by bit. "AND" A & B A 11100101 B 00110101 --------------- 00100101 In Byte A & B, bit 0, bit 2, and bit 5 are BOTH 1 (TRUE). If you look at the truth table, it takes two TRUE (1) bits to make a TRUE result So in the answer, bit 0, bit 2, and bit 5 are ALL 1 but the rest of the bits are 0. OR is also used as a digital logical gate thing-a-mahgig. It also uses transisters. OR is used to logically "OR" two numbers together. When you "OR" to bits for example, the result it 1 (true) if 1 of the bits are 1 (true). Let's look at a "truth table" A B RESULT ---------------- 0 0 0 0 1 1 1 0 1 1 1 1 Basically, if A or B is 1 (TRUE), then the RESULT is 1 (TRUE) What if the numbers are more then 1-bit? Well then you are "OR"ing a number bit by bit. "OR" A & B A 11100101 B 00110101 --------------- 11110101 As you can see, whenever a bit in "A" or a bit in "B" are 1 (true), then the result of that bit is 1 (true). In Byte A & B, bit 1, and bit 3 are BOTH 0 (FALSE). If you look at the truth table, it takes two FALSE (0) bits to make a FALSE result Likewise, it takes One TRUE bit to make a result of 1 (TRUE) So in the answer, bit 1, and bit 3 are ALL 0 but the rest of the bits are 1. XOR is a more complicated digital logic gate. However the rules are still simple. XOR is used to logically "Exclusive OR" two numbers together. When you XOR two bits, for example, the RESULT is 0 (FALSE) if both bits are THE SAME. Let's look at a TRUTH table, A B RESULT ---------------- 0 0 0 0 1 1 1 0 1 1 1 0 As you can see, whenever both A & B are the SAME, then the RESULT is 0. Or whenever ONLY ONE bit is 1 (TRUE), the RESULT is 1 (TRUE.) What if the numbers are more then 1-bit? Well then you are "Exclusive OR"ing a number bit by bit. "XOR" A & B A 11100101 B 00110101 --------------- 11010000 As you can see, whenever a bit in "A" or a bit in "B" are EQUAL, then the result of that bit is 0 (FALSE). In Byte A & B, bit 0,1,2,3, and bit 5 are EQUAL. If you look at the truth table, it takes two EQUAL bits to make a FALSE result NOT is a really simple gate. If A=1 then A=0. If A=0 then A=1. It is like, A is not A. Let's look at a truth table: A RESULT ----------- 0 1 1 0 What if the numbers are more then 1-bit? Well then you are "NOT"ing (inverting) a number bit by bit. "NOT" A A 11100101 --------------- 00011010 Byte A becomes inverted What if you "NOT" A, and byte A is the binary number 1111? A 00001111 ---------- 11110000 As you might have not expected, byte A is not zero. NOT inverts all bits in a byte. So any bits that are zero are now 1. 컴컴컴컴컴컴컴� The POWER OF BINARY - Bit MASKS! 컴컴컴컴컴컴컴� Don't get too excited now! :).. Heheh, by now, you probably are half asleep drooling on your keyboard! Anyway, the use of bit masks are very powerful, yet least talked about. With bit masks, you can change bits, set bits, reset bits, ignore bits, and manipulate bits like the gullible electronic entities they are! How the hel....lo? Yes, Bit masks and the such allow you to do all these. Bit masks are named because you "MASK" (Hide) some of the bits you don't need to mess with and only mess with what you need. We use LOGICAL OPERATORS and a SERIES of BITS (like a byte) for the mask. So basically, a mask is a series of bits, usually a BYTE, that is setup up so when it is applied with a logical operator, one or more bits are ignored. 1.) How to SET a bit. (Set means BIT = 1) To set a bit, we can use the logical operator OR. Lets take byte "A" for example: A = 1110101 ||__ Bit 0 |____ Bit 1 Ok, suppose we want to set the bit 1, which equals 0 right now. First we need to create a MASK that IGNORES all the other bits and we then can CONCENTRATE on BIT 1. We need a mask, where, when the OR operation is applied, does not allow the any bits to change EXCEPT bit 1. Do you see why? Mask "B" : B = 00000010 Now when you "OR" A & B, you get A = 1110101 B = 0000010 ----------- 1110111 |____ Bit 1 now SET!! SO BASICALLY REMEMBER THIS: When you are using "OR" to SET a BIT (or bits), then in the MASK, use 0 to ignore that bit and 1 for the bit(s) you want set. 2.) How to CLEAR a bit (Clear means BIT = 0) To clear a byte, we can use the logical operator AND. Lets take byte "A" for example: A = 1110101 ||__ Bit 0 |____ Bit 1 Ok, suppose we want to clear Bit 0, which equals 1 right now. So we need to create a MASK that ignores ALL BITS EXCEPT BIT 0. We need a MASK, that, when the AND operation is applied, does not allow any bits to change EXCEPT BIT 0. Can you see why? Pretty Easy! Mask "B" : B = 11111101 So when you AND A & B, you get A = 1110101 B = 1111101 ----------- 1110100 |___ BIT 0 is NOW CLEAR!! SO BASICALLY REMEMBER THIS: When you are using "AND" to CLEAR a BIT (or bits), then, in the MASK, use 1 to ignore that BIT and 0 for the BIT(s) you want to CLEAR. 3.) How to CHANGE a BIT. (Change means invert BIT, or BIT = NOT BIT) To change (invert) a BIT, we use the logical operator XOR. We do not use NOT because NOT inverts EVERY bit. We only want to invert one or more bits. Lets take byte "A" for example: A = 1110101 | |____ Bit 7 Ok, suppose we want to CHANGE bit 7, which equals 1 right now. We need to create a MASK that ignores all bits except bit 7. Now this should be simple by now! Mask "B" : 10000000 So when you XOR A & B, you get this: A = 1110101 B = 1000000 ----------- 0110101 |___ BIT 7 has now been CHANGED (inverted)!! SO BASICALLY REMEMBER THIS: When you are using "XOR" to CHANGE a BIT (or bits), then, in the MASK, use 0 to ignore that BIT and 1 for the BIT(s) you want to CHANGE. 컴컴컴컴컴컴� Setting/Clearing/Changing Multiple Bits 컴컴컴컴컴� Setting, clearing, or changing multiple bits are really easy! You just change the masks. For example, if you wanted to SET bit 6 and bit 2, just make sure that when you are using the "OR" operation, that the MASK has a 1 for the the bits you want set (bit 6 and bit 2) and 0 for the bits you want to ignore (the other 6 in that byte).. Just read the little paragraph after each section labled "SO BASICALLY REMEMBER THIS:" in the previous section. 컴컴컴컴컴컴� The POWER of AND - Ignoring bits 컴컴컴컴컴� AND is very useful when you want to ignore (or mask away) some bits. For example, if you are only interested in the lower 4-bits of a byte, because the upper four bits are junk, then just simply ignore (erase), the upper 4-bits. You do this by using the mask 00001111. Now only the lower 4-bit remains. 컴컴컴컴컴컴� The POWER of OR - Setting bits 컴컴컴컴컴� You can use OR to set bits, as you learned before. What if you wanted an easy way to add 128 to a number? If you knew a number was always 63 or below, you could easily add 128 by setting bit 7. Never though about that, Ehh? 컴 More Binary Power using XOR - Simple Encryption methods 컴컴컴컴� XOR can be used to encrypt and decrypt things suprising simple! Suppose you wanted to encrypt a letter. You simply XOR that letter by any 8-bit number, like 17, and then it is encrypted. When you XOR the encrypted letter again it then gets decrypted! I will use a number as an example. Lets ENCRYPT the number 15 and the "password" is 43. This means that we are going to XOR 13 by 43. XOR (to encrypt) 13 = 001101 (13 is the letter to encrypt) 43 = 101011 (43 is the password) ----------- 100110 (which is 38) Encrypted number is 38. Now lets do that again. XOR (to decrypt) 38 = 100110 (38 was previously encrypted, now here to decrypt) 43 = 101011 (43 is the password) ----------- 001101 (which is 13, successfully decrypted!) Letters work exactly the same! All letters are are a series of bits, Usually 7 to 8-bits. On the IBM/PC, 7-bits are used to represent 128 ASCII characters, and 8-bits represents 128 standard ASCII characters and 128 Extended ASCII characters. REMEMBER, LETTERS ARE JUST BINARY NUMBERS! XOR is the most simple form of encrypting available, so don't thing the XOR technique is going to make you a top notch security program. 컴컴컴컴컴컴� The POWER of my FIST 컴컴컴컴컴� Just joking around! Trying to make this binary tutorial interesting because it is so long! 컴컴컴컴컴컴� How are binary/hex numbers represented then? 컴컴컴� How does the TASM (Table Assembler) know if a number is a hex number, decimal number, or binary? If a number begins with a $ or ends with a "h" then it is known to be a hex number. If a number begins with a % or ends with a "b" then it is known to be a binary number. If a number includes neither of these, then it is decimal. It is best to use $ or % since it appears to be more clear and easier to read the source then. 컴컴컴 Table of Binary/Hex/Decimal Numbers or Conversions 컴컴컴 DEC BIN BCD HEX ------------------------------------------------------------- 0 0 0000 0000 0 1 1 0000 0001 1 2 10 0000 0010 2 3 11 0000 0011 3 4 100 0000 0100 4 5 101 0000 0101 5 6 110 0000 0110 6 7 111 0000 0111 7 8 1000 0000 1000 8 9 1001 0000 1001 9 10 1010 0001 0000 A 11 1011 0001 0001 B 12 1100 0001 0010 C 13 1101 0001 0011 D 14 1110 0001 0100 E 15 1111 0001 0101 F 16 10000 0001 0110 10 17 10001 0001 0111 11 18 10010 0001 1000 12 19 10011 0001 1001 13 20 10100 0010 0000 14 21 10101 0010 0001 15 22 10110 0010 0010 16 23 10111 0010 0011 17 24 11000 0010 0100 18 25 11001 0010 0101 19 26 11010 0010 0110 1A 27 11011 0010 0111 1B 28 11100 0010 1000 1C 29 11101 0010 1001 1D 30 11110 0011 0000 1E 31 11111 0011 0001 1F 32 100000 0011 0010 20 64 1000000 0110 0100 40 96 1100000 1001 0110 60 99 1100011 1001 1001 63 컴컴컴컴컴컴컴� SUMMARY (Conversions) 컴컴컴컴컴컴컴컴컴 Has your mind exploded yet? Here's a "quick" summary. Numbers are written as successive powers of the base of the numbering system. Thus 15 = (1 * 10 ^ 1)+(5 * 10 ^ 0) = (1 * 10) + (5 * 1) = 10 + 5 = 15 Binary is a numbering system composed of 0's and 1's. A BIT is a binary 0 or 1 A NIBBLE is a combination of 4-bits, with a maximum value of 16 A BYTE is a combination of 8-bits, with a maximum value of 256. 컴 Converting Binary to Hex 컴 F 3 F=1111 3=0011 F3=1111 0011 =11110011 컴 Converting Hex to Binary 컴 1 C 1=0001 C=1100 1C=0001 1100 = 1 1100 =11100 컴 Converting Hex to Decimal 컴 143 1 * 16 ^ 2 = 256 4 * 16 ^ 1 = 64 3 * 16 ^ 0 = 3 컴컴컴� 323 So 143 in hex is 323 in decimal. 컴 Converting Decimal to Hex 컴 320 320/16=20 + 0 20/16= 1 + 4_ 1__ (Final Quotient is Final Remainder) 컴컴 140 So 320 in decimal is 140 in hex. Each time you divide you move on to the next left place value you need to convert! So Simple! 컴 Converting Binary to Decimal 컴 10011 1 * 2 ^ 5 = 1 * 16 = 16 0 * 2 ^ 4 = 0 * 8 = 0 0 * 2 ^ 3 = 0 * 4 = 0 1 * 2 ^ 2 = 1 * 2 = 2 1 * 2 ^ 1 = 1 * 1 = 1 컴컴컴� 19 컴 Converting Decimal to Binary컴 230 230/2 = 115 + 0 115/2 = 57 + 1� 57 /2 = 28 + 1납 28 /2 = 14 + 0납� 14 /2 = 7 + 0납납 7 /2 = 3 + 1납납� 3 /2 = 1 + 1납납납 1납납납� (Final Quotient is Final Remainder) 컴컴컴컴컴컴컴� 11100110 컴컴컴컴컴컴컴� CONCLUSION 컴컴컴컴컴컴컴컴컴 This concludes to what is probably the largest BINARY and HEX tutorial on the face of this Earth! There should be no question in mind about how binary, hex, bit masks, etc, work after you read this tutorial! Thanks for Reading!! -Cyber Optic If you catch any mistakes or have any additions, comments, anything, e-mail me immediately! My e-mail address is Matt514@gte.net My Home Page is at http://home1.gte.net/matt514/matt.htm My TI-86 Web Page is at http://www.dogtech.com/cybop/ti86/