Mastering Binary Math

Transcription

1 Cisco certification candidates, from the CCNA to the CCIE, must master binary math. This includes basic conversions, such as binary-to-decimal and decimal-to-binary, as well as more advanced scenarios involving subnetting and VLSM. There s another conversion that might rear its ugly head on your Cisco exam, though, and that involves hexadecimal numbering. Newcomers to hexadecimal numbering are often confused as to how a letter of the alphabet can possibly represent a number. Worse, they may be intimidated after all, there must be some incredibly complicated formula involved with representing the decimal 11 with the letter b, right? Wrong. The numbering system we use every day, decimal, concerns itself with units of ten. Although we rarely stop to think of it this way, if you read a decimal number from right to left, the number indicates how many units of one, ten, and one hundred we have. That is, the number 15 is five units of one and one unit of ten. The number 289 is nine units of one, eight units of ten, and two units of one hundred. Simple enough! Units Of 100 Units Of 10 Units Of 1 The decimal The decimal Hex numbers are read much the same way, except the units here are units of 16. The number 15 in hex is read as having five units of one and one unit of sixteen. The number 289 in hex is nine units of one, eight units of sixteen, and two units of 256 (16 x 16). Units Of 256 Units Of 16 Units Of 1 The hex numeral The hex numeral Since hex uses units of sixteen, how can we possibly represent a value of 10, 11, 12, 13, 14, or 15? We do so with letters. The decimal 10 is represented in hex with the letter a ; the decimal 11 with b ; the decimal 12 with c, 13 with d, 14 with e, and finally, 15 with f. (Remember that a MAC address of ffff.ffff.ffff is a Layer 2 broadcast.) Practice Your Conversions for Exam Success Now that you know where the letters fall into place in the hexadecimal numbering world, you ll have little trouble converting hex to decimal and decimal to hex if you practice. How would you convert the decimal 27 to hex? You can see that there is one unit of 16 in this decimal; that leaves 11 units of one. This is represented in hex with 1b one unit of sixteen, 11 units of one. Work From Left To Right To Perform Decimal Hexadecimal Conversions. Units of 256 Units of 16 Units of 1 Hexadecimal Value Decimal Number B (11) 1b Chris Bryant, CCIE #12933 Page 1

2 Converting the decimal 322 to hex is no problem. There is one unit of 256; that leaves 66. There are four units of 16 in 66; that leaves 2, or two units of one. The hex equivalent of the decimal 322 is the hex figure 142 one unit of 256, four units of 32, and 2 units of 2. Units of 256 Units of 16 Units of 1 Hexadecimal Value Decimal Number Hex-to-decimal conversions are even simpler. Given the hex number 144, what is the decimal equivalent? We have one unit of 256, four units of 16, and four units of 4. This gives us the decimal figure 324. Units of 256 Units of 16 Units of 1 Decimal Value Hexadecimal Number = 324 What about the hex figure c2? We now know that the letter c represents the decimal number 12. This means we have 12 units of 16, and two units of 2. This gives us the decimal figure 194. Units of 256 Units of 16 Units of 1 Decimal Value Hexadecimal Number c = 194 Tips for Exam Day Practice your binary and hexadecimal conversions over and over again before you take your CCNA exams. Binary math questions come in many different forms; make sure you have practiced all of them before exam day. The number one reason CCNA candidates fail their exam is that they re not prepared for the different types of binary math questions they re going to be asked, and that they aren t ready for hexadecimal questions at all. As you can see, hexadecimal conversions are actually simple. You have to practice them, though! You don t have time to learn how to do in on exam day. You ve got to be ready before you go into the exam room, and the only way to be ready is a lot of practice. Finally, make sure you read the question carefully. You ve got hex, decimal, and binary numbers to concern yourself with on your CCNA and CCNP exams. Make sure you give Cisco the answer in the format they re looking for. I have written 20 practice questions that will help you practice your hexadecimal conversion skills. Once you practice with these questions, and know exactly how each answer was arrived at, you ll have no problem with hexadecimal conversions on your Cisco exams. Best of luck! To your success, Chris Bryant, CCIE #12933 Chris Bryant, CCIE #12933 Page 2

3 1. Convert the following hexadecimal number to decimal: 1c 2. Convert the following hexadecimal number to decimal: f1 3. Convert the following hexadecimal number to decimal: 2a9 4. Convert the following hexadecimal number to decimal: 14b 5. Convert the following hexadecimal number to decimal: 3e4 6. Convert the following decimal number to hexadecimal: Convert the following decimal number to hexadecimal: Convert the following decimal number to hexadecimal: Convert the following decimal number to hexadecimal: Convert the following decimal number to hexadecimal: Convert the following hex number to binary: Convert the following hex number to binary: Convert the following hex number to binary: a9 14. Convert the following hex number to binary: 3c 15. Convert the following hex number to binary: Convert the following binary string to hex: Convert the following binary string to hex: Convert the following binary string to hex: Convert the following binary string to hex: Convert the following binary string to hex: Answers begin on the next page. No peeking! Chris Bryant, CCIE #12933 Page 3

4 Before we go through the answers and how they were achieved, let's review the meaning of letters in hexadecimal numbering: A = 10, B = 11, C = 12, D = 13, E = 14, F = 15. (And remember that ffff.ffff.ffff is a Layer 2 broadcast!) Conversions involving hexadecimal numbers will use this chart: 1. Convert the following hexadecimal number to decimal: 1c 1 c There is one unit of 16 and twelve units of = Convert the following hexadecimal number to decimal: f1 There are fifteen units of 16 and 1 unit of = 241 f 1 3. Convert the following hexadecimal number to decimal: 2a9 2 a 9 There are two units of 256, ten units of 16, and nine units of = 681 Chris Bryant, CCIE #12933 Page 4

5 4. Convert the following hexadecimal number to decimal: 14b 1 4 b There is one unit of 256, four units of 16, and 11 units of = Convert the following hexadecimal number to decimal: 3e4 3 e 4 There are three units of 256, fourteen units of 16, and four units of = Convert the following decimal to hexadecimal: 13 When converting decimal to hex, work with the same chart from left to right. Are there any units of 256 in the decimal 13? No. 0 Are there any units of 16 in the decimal 13? No. 0 0 Are there any units of 1 in the decimal 13? Sure. Thirteen of them. Remember how we express the number "13" with a single hex character? 0 0 d The answer is "d". It's not necessary to have any leading zeroes when expressing hex value. 7. Convert the following decimal to hexadecimal: 784 Are there any units of 256 in the decimal 784? Yes, three of them, for a total of 768. Place a "3" in the 256 slot, and subtract 768 from Chris Bryant, CCIE #12933 Page 5

6 = 16 Obviously, there's one unit of 16 in 16. Since there is no remainder, we can place a "0" in the remaining slots The final result is the hex number "310". 8. Convert the following decimal to hexadecimal: 419 Are there any units of 256 in the decimal 419? Yes, one, with a remainder of Are there any units of 16 in the decimal 163? Yes, ten of them, with a remainder of three. 1 a Three units of one take care of the remainder, and the hex number "1a3" is the answer. 1 a 3 9. Convert the following decimal to hexadecimal: 1903 Are there any units of 256 in the decimal 1903? Yes, seven of them, totaling This leaves a remainder of Are there any units of 16 in the decimal 111? Yes, six of them, with a remainder of By using the letter "f" to represent 15 units of 1, the final answer "76f" is achieved. 7 6 f Chris Bryant, CCIE #12933 Page 6

7 10. Convert the following decimal to hexadecimal: 345 Are there any units of 256 in 345? Sure, one, with a remainder of Are there any units of 16 in 89? Yes, five of them, with a remainder of Nine units of nine give us the hex number "159" Convert the following hex number to binary: 42 First, convert the hex number to decimal. We know "42" in hex means we have four units of 16 and two units of 1. Since = 66, we have our decimal. Now we've got to convert that decimal into binary. Here's our chart showing how to convert the decimal 66 into binary: The correct answer: Convert the following hex number to binary: 12 First, convert the hex number to decimal. The hex number "12" indicates one unit of sixteen and two units of one; in decimal, this is 18. Now to convert that decimal into binary. Use the same chart we used in Question 11: The correct answer: Chris Bryant, CCIE #12933 Page 7

8 13. Convert the following hex number to binary: a9 First, convert the hex number to decimal. Since "a" equals 10 in hex, we have 10 units of 16 and nine units of = 169 Now convert the decimal 169 to binary: The correct answer: Convert the following hex number to binary: 3c First, convert the hex number to decimal. We have three units of 16 and 12 units of 1 (c = 12), giving us a total of 60 ( ). Convert the decimal 60 into binary: The correct answer: Convert the following hex number to binary: 74 First, convert the hex number to decimal. We have seven units of 16 and four units of 1, resulting in the decimal 116 ( ). Convert the decimal 116 into binary: The correct answer: The next five questions dealt with converting binary to hex. We're going to use much the same method in solving these questions, but this point bears repeating: Make sure to answer the question in the format that Cisco is asking for on your exams. Chris Bryant, CCIE #12933 Page 8

9 16. Convert the following binary string to hex: First, we'll convert the binary string to decimal: Decimal To finish answering the question, convert the decimal 51 to hex. Are there any units of 256 in the decimal 51? No. Are there any units of 16 in the decimal 51? Yes, three, for a total of 48 and a remainder of three. Three units of one give us the hex number "33" Convert the following binary string to hex: First, we'll convert the binary string to decimal: Decimal Now convert the decimal 207 to hex. Are there any units of 256 in the decimal 207? No. Are there any units of 16 in the decimal 207? Yes, twelve of them, for a total of 192 and a remainder of 15. Twelve is represented in hex with the letter "c". Fifteen units of one are expressed with the letter "f", giving us a hex number of "cf". 0 c f 18. Convert the following binary string to hex: First, convert the binary string to decimal: Decimal Now convert the decimal 93 to hex. There are no units of 256, obviously. How many units of 16 are there? Five, for a total of 80 and a remainder of 13. We express the number 13 in hex with the letter "d". The final result is the hex number "5d". 0 5 d 19. Convert the following binary string to hex: Chris Bryant, CCIE #12933 Page 9

10 As always, convert the binary string to decimal first: Decimal Now convert the decimal 157 to hex. There are no units of 256. How many units of 16 are there in the decimal 157? Nine, for a total of 144 and a remainder of 13. You know to express the number 13 in hex with the letter "d", resulting in a hex number of 9d". 0 9 d 20. Convert the following binary string to hex: First, convert the binary string to decimal: Decimal Now convert the decimal 213 to hex. No units of 256, but how many of 16? Thirteen of them, with a total of 208 and a remainder of 5. Again, the number 13 in hex is represented with the letter "d", and the five units of one give us the hex number "d5". 0 d 5 Chris Bryant, CCIE #12933 Page 10