A NUMERAL is a character or symbol that represents a number. As a result, the picture of the number FOUR '4' is a numeral.

Transcription

1 On this first screen page you will have a simple lesson in Mathematics. In order to fully understand how a computer works, it is necessary to understand what Binary and Hexadecimal Numbers are. In order to understand them, you must first remember some of your earliest recollections of Mathematics. A NUMERAL is a character or symbol that represents a number. As a result, the picture of the number FOUR '4' is a numeral. A NUMBER, on the other hand, MUST have a value. That same 4 that we used in the definition of a numeral could also be a Number. If it actually represents a value, it is a number. 21 are made up of two characters (or numerals) and it represents a value, so it is a Number. Remember, NUMERALs are pictures of numbers, and NUMBERs have a value. The BASE 10 or DECIMAL number system (this is the one that we use every day, and both of these names mean the same thing) uses 10 numerals, or pictures to represent all of the possible numbers that you can make. The NUMERALs are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Our number system is a WEIGHTED number system. This means that each column has a specific value. An example of a non-weighted number system is Roman Numerals. Characters like X, I, V, and L are NUMERALs in the Roman Numeral number system. XXIV represents a NUMBER. The number is 24 in our Decimal Number System. The columns that the numerals are in do not have any specific value. In our number system, the first column to the left of the decimal point is known as the ones column, and the second is the tens, etc.. Our number system is a WEIGHTED number system.

2 Unless you know everything that there is to know about Binary and Hexadecimal Numbers, do NOT skip this page. Read it VERY carefully, it will definitely help you to understand how those number systems work. Nobody actually told you how your number system works when you were young, but I am going to do it now. What I am going to explain to you is how to count (and you thought that you already knew how to do that, WRONG!). You were TOLD what the numbers were, and you memorized them. I am going to teach you the rules behind counting. Then I will use what you know about the Decimal number system so that it can be used to help you to make a little sense out of Binary Numbers. There are a few basic rules that you use when counting. The First one is obviously one that you already know. When counting up from the smallest whole number to???, you always start out using the column immediately to the left of the decimal point. This is the one that you know of as the ONES COLUMN. It has this name because when you figure out the total value of the complete number that you have, one multiplies each number in that column. The second column over is the TENS COLUMN. The next is the HUNDREDS COLUMN, etc.. But, you already know that! When counting UP, you always start with the smallest value and increase by one each time. Visually, we do this by using the ZERO character first, then the ONE character, the TWO characters, etc., until we finally reach the NINE character. I

3 have used the word character each time, but I could also have said NUMERAL. Remember, each of these numerals is used to represent a number. Your parents and teachers TOLD you that the next number after 9 was 10. None of them ever told you why! You took it on blind faith to be gospel truth, because of who it was that told you. I am going to tell you WHY it is true. BEFORE I go on, remember, that we have used up all ten of them, from '0' to '9' as we have counted up from ZERO to NINE. We did this using only the ONES COLUMN. After you have used up all of the numerals that you have in any column, you always follow the same rule (the rule that everyone managed to forget to tell you about). You start over again with the first numeral (the '0') in the column that you are in (in our case, the ones column). You then ADD the value one to the next column to the left (in our case the tens column). Since we had no value before this time, there was actually a zero there that we did not display = 1. You put the character that represents that value, the ONE or '1' in the column. We now have a '1' in the tens column and a '0' in the ones column. Gee, isn't that how we would normally display the value ten? We continue counting by increasing the value in the ones column (and displaying its numeral) until we run out of numerals (at 9). Then, we follow the rule used in the last paragraph again, and voila, we now have a 20. The same rule applies when we go from 99 to 100, except we run out of numerals in two columns at the same time, so we start over in both, and add one to the hundreds column. Now you know WHY we count the way we do. This is really what you have been doing all of your life; it is just that no one has ever thought to tell you this before now. If they had, learning Binary or any other number system would be a whole lot easier

4 Before we move on to Binary and Hexadecimal Numbers, I have a little story to tell you about an alien I once met named Trixi. He was from the planet Trinar. The only picture that I have of him is the one he gave me that was taken when he was visiting the Taj Mahal. You may notice that he only has a total of three fingers. One on his left 'hand' and two on his right 'hand'. He tells me that all of the 'people' from Trinar have only three fingers, just like him. He was explaining to me about how the people on his planet lived, and the one thing that fascinated me the most was the number system that they use. Like us they use a weighted number system that is based on the number of digits that they have on their hands. We would probably call his number system the Trinary number system because it has three digits. These three digits, that are used in all numbers on the planet Trinar, are BOO, BIB, and BOB. In the yellow box at the right I have shown you how Trixi writes these characters. Since I do not have these characters on a human keyboard, I will refer to these characters by the human translation of the name given to the numerals. BOO is similar to our numeral '0', BIB is similar to our '1', and BOB is the same as our '2'.

5 Trixi was very patient with me, and he explained how his number system worked. Once I heard the complete explanation, I told him that his number system follows the EXACT same rules as ours. It seems that even though we come from completely different cultures (and worlds) at least we have something in common. I thought that you might find it interesting, so I decided to include how his number system works. I have shown the numbers used on Trinar and the base ten equivalents at the left. It may come in handy to you some time. You never know when you may find yourself stranded on the planet Trinar! I have labeled the columns used by the people from Trinar with their names (BIB, EE, and IDDY) and with our equivalents 1, 3, and 9. If you think about our number system, the first is also the ones; the second is the base (or number of numerals), which are ten. Since Trinarians use a base three number system, the second column is the 3's column (to us). The next column in our number system is the hundreds, or 10 times 10. Since the Trinarians use base three, their third column has a value of 3 times 3, or 9 (IDDY to them). When you look at the Trinar numbers next to our base ten numbers in the chart at the left, keep in mind that the Trinarians, like us, have decided to NOT show a BOO (a zero to us) if it is at the left side of a number (we would normally write 2 instead of 002). Their system uses exactly the same rules as ours does. Note how they follow the rules by using all of the numerals once in the ones (BIBs) column, then starting

6 over and doing it again. In the next column, with our number system, we have 10 zeros (that are not shown) then we have 10 ONEs, etc. In the Trinarian number system, they have 3 BOOs (that are not shown), then 3 BIBs, and then 3 BOBs. After that, they start over again with 3 BOOs, and continue on forever. The value of their next column is 9 so the first 9 are BOOs (that are not shown), the next 9 are BIBs, then 9 BOBs, then it starts all over again. If you wanted to actually count out loud in Trinarian, it would go something like this... BOO BIB BOB BIBBEE BOO BIBBEE BIB BIBBEE BOB BOBBEE BOO BOBBEE BIB BOBBEE BOB BIBBIDDY BOOEE BOO BIBBIDDY BOOEE BIB BIBBIDDY BOOEE BOB BIBBIDDY BIBBEE BOO BIBBIDDY BIBBEE BIB BIBBIDDY BIBBEE BOB And of course, their version of our number 15 would be pronounced as BIBBIDDY BOBBEE BOO.

7 Computers use the Base Two (Binary) Number System. There are TWO (2) Numerals or Characters used by the Binary Number System. They are ZERO (0) and ONE (1). The reason that this system was originally devised is because computers use electricity. Since electrical impulses are used to represent numbers and it is easy to turn power on and off, some GENIUS decided that ON would represent a 1 and OFF would represent a 0. All values can be represented using only combinations of ONEs and ZEROs. With only 2 choices it seemed logical to call it the Base 2 or Binary number system. The Binary number system is a weighted number system like the Decimal number system that we are all accustomed to using. Following the logic of the weighted number system that meant there had to be only two numerals. Since they both already existed, the characters chosen were '0' and '1'. The same rules as those previously explained in base ten (and Trinary) hold true when you try to count in Binary. When you run out of characters in a column, you add one to the column to the left, and start over with the first character again. Unfortunately, with Binary you run out of numerals a lot quicker (because there are only two of them). With the Binary number system, the column beside the BICIMAL point (that is what it is called) is the ones column, the next one is the twos, the third one is the fours (2 times 2), and the fourth is the eights column (2 times 2 times 2). By looking at the 'blackboard' in the picture, you should be able to see a pattern between Decimal and Binary.

8 It is now time to learn how to convert a Base ten number to a Binary number. There are two easy ways to do this. The first one uses the values of the Binary columns as its basis. I have explained how this method works in the area to the left of this text box. Use this simple 5-step method and follow along with me as I explain exactly how this works. The example used is changing the base ten number 99 to its Binary equivalent. To do so, start with the Binary 1's column and keep moving over to the next column on the left until you reach a value that is larger than the base ten number that you are trying to convert to Binary. That column will be the 128's column. Back up to the right ONE column, and put a ONE in the chart in that column (I put a red 1 there on the chart). Next, subtract that value from the original decimal value (99-64 = 35). The new decimal value is 35. Back up to the right ONE column. If the column value is smaller than the new decimal value (if 32 is smaller that 35, which it is) put a ONE in the chart in that column (I put a red 1 there on the chart) and subtract the column value from the decimal value (35-32 = 3). If the column value is larger than the decimal value (if 32 is larger than 35, which it is NOT) put a ZERO in the column on the chart. Now you just keep backing up one column to the right and following the rules that you used in the paragraph above (rule 3).

9 16 is greater than 3 so a ZERO is put in the 16's column. 8 is greater than 3 so a ZERO is put in the 8's column. 4 is greater than 3 so a ZERO is put in the 4's column. 2 is SMALLER than 3, so a ONE is placed in the 2's column and 2 is subtracted from 3 (3-2 = 1). 1 is equal to 1 so a ONE is placed in the 1's column and the conversion is now finished. The Binary equivalent of 99 is We normally would write that as leaving a space every four characters counting from the right. This makes it easier to read the value, and it also has a use when converting to Hexadecimal numbers. Now, I am going to ask you to try to convert several base 10 numbers to Binary. You can use the rules at the left to make it easier. Create a chart like the one at the left on a piece of paper. Write each of the following base ten numbers on the paper, and use your chart to convert them to Binary numbers Now that you have completed the task, you may want to try using a calculator to convert base ten numbers to Binary numbers. Assuming that you are using a Windows-based computer, you have a calculator already built-in. To access it, you click the START button (usually at the bottom left of your screen on the task bar). Then go up it to PROGRAMS. Then over to ACCESSORIES. Finally, go down the accessories column until you come to CALCULATOR. Not all people will have it loaded on their computer. If you don't, then you will have to use an alternate means of doing this. First, Click the pull-down menu on your calculator, and change its VIEW to SCIENTIFIC. Then, click the radio button that says BYTE, and the one that says DEC (for Decimal). Now, enter one of the numbers from above (11). Click the radio button for Binary (BIN) and see what happens. Now you are on your own. Try entering other numbers in Binary and Decimal, and check using the calculator to see if you come up with the correct answer. If you want to try entering numbers larger than the decimal number 255, you will have to click the radio button WORD or DWORD. Don't worry about what they

10 mean, just be happy with the fact that they will help you to work with larger numbers. The number that you enter MUST be a POSITIVE number (or strange, but true, things will happen to the binary value). Try thinking up some more base ten numbers and converting them on paper. Then use the Calculator to check your answers. Good luck!! Some people find the method to convert Decimal numbers to Binary numbers that you have been shown to be difficult to understand. As a result, I will show you a second method to do the same thing. I call this method the 'Ladder Method'. To use this method you first have to draw a ladder (gee, I wonder where I got the name of this method from?). In the top rung of the ladder you place the Decimal number to be converted. To the right of this number I always place the value of the base that I am converting to (not necessary, I just do it because I use this method for other base conversions.). Next you divide the Decimal number by the base (99 / 2 = 49 and 1 remainder). The new decimal value (49) is placed inside the rung of the ladder immediately below the original Decimal number. The Remainder (1) is placed outside the ladder on the left side beside the new decimal value.

11 You then continue with the last step until your decimal value is less than your base value. 49 / 2 = 24 with 1 remainder. Put the 24 inside the next rung down, and the 1 outside it to the left. 24 / 2 = 12 with 0 remainder. The 12 go inside the next rung and the zero goes outside to the left. The biggest mistake that most people make when using this method is they forget to put the zeros where they belong. 12 / 2 = 6 with 0 remainder. Then 6 / 2 = 3 with 0 remainder, and finally 3 / 2 = 1 with 1 remainder. Once all of the numbers have been placed, your answer is there; all you have to do is be able to read it. Start reading it with the 1 inside the rung at the very bottom of the ladder. It is your first numeral (the one at the LEFT) of your answer. Then go left to the remainder column to find the next numeral (also a 1 in this case). Then read the rest of the numerals in the remainder column FROM THE BOTTOM TO THE TOP. 0 then 0, then another 0, a 1, and finally another 1. The correct answer is that 99 in decimal is equal to in Binary (just like it was on the last screen page). It is easier to read Binary values if you group them in fours from the bicimal point, that is why I wrote the number down as Try using this same method to convert each of the following Decimal values to Binary. When you have finished, check out your answers by using the method that you were given on the last screen page. After you are sure that they are right, check them again by using a calculator Good Luck!!

12 Next, I will show you how to convert from a Binary Number to a Decimal Number. The simplest method is to add up the values of the columns that have a ONE in them. Look at the example at the left. The Binary value to be converted to Decimal is Notice again that it is easier to read a binary value if it is broken up into nibble sized pieces (groups of 4 organized from the right side). Place the column values (in base ten or Decimal) above each numeral in the Binary number. Each column that has a ONE in it gets added to the final value. The reason that I did it the way that I did (showing 1 X? and 0 X?) is because I will use the same method when explaining how to change from Hexadecimal to Decimal later in the lesson. Get used to doing it this way now, and it will be easier later. Try figuring out the decimal equivalent of the following Binary values using this method. The answers are at the top of the next screen page

13 Good Luck!! Do NOT try to understand what is happening in the picture at the left until I explain it to you. You will probably just confuse yourself. Here are the answers to the questions at the bottom of the previous page. If you had any trouble, try using some of the numbers that you know from previous pages for which you already know both the binary and the decimal values = = = = = 115 Now we will try to perform a little miracle. I am going to teach you how to add using binary numbers. Remember when I showed you how easy it was to count in binary by pointing out to you that it was the same way you have always counted (except for the slight difference of having only two numerals)? Well, now

14 I am going to show you that you already know how to add binary numbers, you just don't know it. When you were in grade one, your teacher made you learn those boring number facts, like = 2, = 4, and so on. What you didn't realize at the time was that unless you memorized those facts, you would never be able to add (without a calculator, like real people do). Unfortunately, you have to memorize the Binary number facts (or is it FACT as in Free Academy of Career Training) to be able to add binary numbers. So-o-o-o-o, get your thinking caps on, and start memorizing. Here they are = = = 1 Not too hard yet, is it? By the way, there is really only one number fact left. Now it gets really hard though because = 0 with a carry of 1. That may seem ridiculous, but remember that 1 is the highest valued single number that we have in Binary. If you took the highest base ten number 9 and added 1 to it, you would end up with 0 (zero) in the ones column and you would end up carrying a 1 (one) into the next column. It seems that we do the same thing in Binary. To prove that, how do you write a decimal 2 in Binary? 10 (one zero). That is a zero in the ones column and a 1 in the twos column. Now I know I told you that you only needed those four number facts, but I think that you will find it easier to do if you learn just one more = 1 with a carry of 1 (which is the same thing as 11 base two and you know that 11 in binary is equal to 3 in base ten). It is now time to look at the simple addition question at the left. The decimal equivalent of each of the two Binary numbers to be added is given. The top binary number 111 is equal to 7 in base ten. The bottom number 110 base two is equal to 6 in the decimal number system. If you add 6 and 7 in base ten you get an answer of in base ten is equal to 1101 in Binary. Now let's use those number facts and do the binary addition properly = 1 That looks after the ones column = 0 with a carry over to the next column (the 4s column) of 1. Now we have three 1s to add together, and you know that = 1 with a carry over to the next column (the 8s column) of 1. Finally, there is only a 1 in the fourth column, so bring it down and you have your answer. All you have to do is add the way you always have, and remember that you only have five basic number facts to deal with and you will get the right answer every time. Try the following questions. Check your answers (after you do the binary addition) by putting the base ten equivalents out to the right like I have, and using

15 the simulator to get the correct answer. I have separated the binary numbers in groups of fours from the 1s column to make them easier to read. The correct answers are on the next screen page The correct answers are = = = = If you want to, try some more and make sure that you are correct. Make up the questions in base ten first, then convert each number to binary, then add them up.

16 If you have been around computers for a while, you have probably heard terms like BIT, BYTE, and NIBBLE. Since these words are directly related to Binary numbers, now would probably be a good time to remind you what they mean. A BIT is the smallest amount of information that a computer can access and/or process. This term refers to any individual ONE or ZERO that makes up a Binary Number. Each column in a Binary number is ONE BIT of information. It comes from the shortening of the phrase 'BInary digit' (or is it 'Binary digit'?). If you have eight bits of information, you have a BYTE. With one Byte of information you can store values from to These Binary values represent the Decimal equivalent of 0 to 255. One Byte is the also the number of Bits needed to represent any one character (the number of bits in an EXTENDED ASCII character). A NIBBLE (or nybble as I have also seen it spelled) is 4 Bits of information, or half of a Byte. It can be used to represent numbers between 0 (0000) and 15 (1111). If you know anything about Hexadecimal Numbers, you will know that one Hex character is a representation of four Bits (or one Nibble). I think of Dracula when I can't remember whether a byte or a nibble is bigger. A nibble on the neck from The Count is obviously not as bad as a bite would be Now is also a good time to remind you about the definition of the word WORD, as it relates to computers. The data bus of a computer is merely a group of parallel conductors (thin pieces of metal that allow electricity to easily pass through them). If a computer has an eight bit data bus (it has eight paths for the data to travel along AT THE SAME TIME), it uses eight bit WORDs. A computer with a sixteen bit data bus uses sixteen bit WORDs. The word WORD refers to the size of the data bus (number of parallel conductors) of a computer.

17 You may have also heard about the Hexadecimal Number System. Though some people make it out to be some deep dark secretive way of communicating, this is actually only a method developed by some early programmers to make it easier to work with large Binary Numbers. Hexadecimal numbers made programming easier to understand and machine language programs easier to enter. 'Hex' is actually just a shorthand method for using Binary Numbers. Hexadecimal is a weighted number system like Binary and Base 10. Hex is a prefix that stands for 6 and of course decimal stands for 10. There are 16 numerals in the Hexadecimal number system. Since there are only ten numerals (or characters) used in the base 10 number system, someone got the brilliant idea of using the first few letters of the alphabet to represent the other 6 characters. As a result, by looking at the 'blackboard' at the left you can see that the alphabetic characters A to F represent the values 10 through 15. The blackboard at the left shows you how to write all of the Hexadecimal values from 0 to 20. If you look closely, you will notice that this number system follows the exact same rules as Decimal, Binary, and even Trinary. You write down the numerals in the first column until you run out (in this case 0 to F). Then, you add one to the next column, start over at zero again, and cycle through the numerals again. Try using this new knowledge that you have and write down all of the decimal numbers from 0 to 40, then put the Hex equivalents beside each number. Obviously, do not look at this screen while you are doing it. Here are the answers.

18 B F C D E F A B C D E A

19 Now that you have the basic idea of how Hex works, let's see if I can make a light bulb or two come on in your head. We were at this stage before, just a few screen pages ago when you were learning about Binary. It is now time to learn how to convert a Base ten number to a Hexadecimal number. There are two ways to do this. The first one uses the values of the Hex columns as its basis. I have explained how this method works in the area to the left of this text box. Use this simple 4-step method and follow along with me as I explain exactly how this works. The example used is changing the base ten number 327 to its Hexadecimal equivalent. To do so, start with the Hex 1's column and keep moving over to the next column on the left until you reach a value that is larger than the base ten number that you are trying to convert to Hex. That column will be the 4096's column. Back up to the right ONE column, and divide the decimal value by the column value. 327 / 256 = 1 with a remainder of 71. Put the answer, 1 in the 256's column, and use the remainder as the decimal value for the next step. Back up one column to the right to the 16's column and divide our decimal value by / 16 = 4 with a remainder of 7. Put the answer, the 4, in the 16's column and since the next column is the last one, put the 7 in that column. That wasn't so hard was it? You are finished already. 327 in base ten is equal to 147 in Hexadecimal. By the way, with non-decimal values, we usually read the values as one four seven, NOT one hundred and forty seven. The latter insinuates that the value is a base ten number. Now, I and going to ask you to try to convert several base 10 numbers to Hex. You can use the rules at the left to make it easier. Create a chart like the one at the left on a piece of paper. Write each of the following base ten numbers on the paper, and use your chart to convert them to base sixteen numbers

20 Now that you have completed the task, you may use your calculator again. You can use it to check to see if you correctly converted the decimal values to Hex. Try thinking up some more base ten numbers and converting them on paper. Then use the Calculator to check your answers. Good luck!! Now let's use the Ladder Method to convert a number from decimal to hexadecimal. You've already used it converting from decimal to binary, so this should almost be like a review of the method for you. To use this method, you first have to draw a ladder (remember that?). In the top rung of the ladder you place the Decimal number to be converted. To the right of this number I always place the value of the base that I am converting to (not necessary, but it helps me to remember what value I am dividing by). Next you divide the Decimal number by the base (327 / 16 = 20 and 7 remainder). The new decimal value (20) is placed inside the rung of the ladder immediately below the original Decimal number. The Remainder (7) is placed outside the ladder on the left side beside the new decimal value. You then continue with the last step until your decimal value is less than your base value. 20 / 16 = 1 with 4 remainder. The biggest mistake that most people make when using this method is they forget to put the zeros where they belong, especially if there is no remainder.

21 Once all of the numbers have been placed, your answer is there; all you have to do is be able to read it. Start reading it with the 1 inside the rung at the very bottom of the ladder. It is your first numeral (the one at the LEFT) of your answer. Then go left to the remainder column to find the next numeral (a 4 in this case). Then read the rest of the numerals in the remainder column FROM THE BOTTOM TO THE TOP. All we have left is a 7. The correct answer is that 327 in decimal are equal to 147 in Hex. If you have a number that is greater than 9 and less than 16, I suggest that you circle the value (or change it immediately to the hex equivalent) so that you will not read the converted value incorrectly (put a 14 in your number when it should really be an E. Try using this method to convert each of the following Decimal values to Hexadecimal. When you have finished, you will find the answers are at the top of the next page. After you are sure that they are right, check them out Good Luck!!

22 Here are last page's answers. 109 = 6D 217 = D9 451 = 1C3 178 = B2 279 = = CE 256 = 100 Next, I will show you how to convert from a Hexadecimal Number to a Decimal Number. With this method each number in the hex value is taken one at a time and multiplied by its column value. Start by placing the column values (in base ten or Decimal) above each numeral in the Hexadecimal number. Take each column one at a time and multiply the hex value (e.g. C (or 12) times the column value 1 and extend it out to the right. Do this for all of the columns, then add up the answers that you got ( ) and place the answer (6764) underneath. This is very similar to the Binary to Decimal version that you have used previously.

23 Try using this method to convert each of the following HexaDecimal values to Decimal. When you have finished, you will find the answers at the top of this page because I used the same values on the last page. After you are sure that they are right, check them out. 6D D9 1C3 B2 117 CE 100 Good Luck!! After you finish this screen page, it is time to try the lesson test. This test will obviously include a lot of Binary, Hex, and Decimal conversion questions. You will even have some Binary addition to do. If you feel that you need to go over

24 ANY of the screen pages in this lesson, before attempting the test, do it NOW! (Or at least before ending this screen page) Please do NOT use a calculator. The whole idea of the test is to see what you have learned, not whether or not you can punch some numbers into a calculator. Only YOU will really know how you got the mark that you got, but if you do not do it on your own, you will only hurt yourself. The Hexadecimal number system is NOT really used by computers. Huh? Then why the H#%% are we learning about it? Hex, as it is affectionately called (by whom?), is actually a short form method of writing Binary numbers. It came into existence because programmers back in the 'Stone Ages' were making too many stupid mistakes. By the way, the 'Stone Ages' is the years between the beginning of time (around the Second World War when the first electronic computers were built), and the beginning of the "New Age' (when the IBM PC was released). It seemed that a lot of the programming at that time had to be done totally using binary (because, it is, after all the only thing that a computer can understand). Programmers were having trouble getting all those ones and zeros correct so someone came up with something to make it easier for the programmer. Here is an example of what the problem was. Which of the following two numbers would you be less likely to make a mistake copying from one place to another, the Binary number or the Hex number? EB37DA8 If you said the Binary number, the men in white coats are probably knocking on your door as you read this. You can see that it is much easier to read the Hex number, and it would obviously be easier to transpose it from one place to another successfully. As a result, someone created a software interface that allowed programmers to enter numbers in Hex. The software then converted the Hex to Binary before it was actually entered into machine code. You have probably noticed in the drawing at the left that I have grouped all of my Binary numerals in groups of four starting at the ones column. Just doing this, makes Binary easier to comprehend, but that is not the only reason that it was done. If you look at the column values that are assigned to the first four binary columns you will see that they are 8, 4, 2, and 1. The smallest number that can be made with a four bit (BInary digit) value is 0000 or zero. The largest is 1111 or 15. Surprisingly, the smallest digit in Hex is 0, and the largest is F (or 15). To convert between Binary and Hex, you simply separate ALL of the ones and zeros into groups of four (as I have) and place the 8, 4, 2, 1 value above EACH GROUP OF FOUR! Then pretend that the real Binary number does not exist.

25 Instead, treat it as a bunch of four bit Binary values (that will always have a Decimal equivalent of between 0 and 15). Convert each of the four bit values to decimal (because our minds find this easy to do, not because you have to), and then finally change the Decimal value to a Hex digit. For example the first group from the right is This converts to 15 in Decimal. 15 in Decimal are equal to F in Hex. The next group is This is 12 in Decimal, and C in Hex. You just continue this way until you are finished the number. If you want to convert the other way, from Hex to Binary, you do it in reverse. Convert each Hex digit to Decimal first. Next convert each Decimal value to a FOUR bit Binary value. Remember, you MUST ALWAYS use four bits. If the Hex value is 3, the Binary is 0011, not 11. Finally put all of the ones and zeros together, and voila, you have finished. Remember, zeros are placeholders and can NOT be left out. That's it; you are now an official Binary-Hex Ghuru! If you can NOT convert from Binary to Hex, and from Hex to Binary in your head, without using a calculator or a piece of paper, please take a five minute break to let what you have read sink in, then come back and read this over. It is actually VERY SIMPLE! If you are dealing with any number greater than 15 at any time, you are doing it the hard way! Good Luck on this test. And on the exam for this module.

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