Cisco CCNA: Numbering Systems

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Unit 1. Decimal

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1. Decimal
       1.1 Understanding Base 10
       1.2 Decimal Conversions
              1.2.1 Whole Number
              1.2.2 Fractional
       1.3 Unit 1 Summary

In this course, we will explore three different numbering systems, called decimal, binary, and hexadecimal. Understanding these numbering systems is essential to many computer operations and programming.
In this unit, you will learn about the decimal, or base 10, numbering system.

After completing this unit, you should be able to:

• Describe the decimal numbering system
  
  
• Convert whole numbers to decimals
  
  
• Convert fractions to decimals

This unit does not address any specific Cisco objectives. However, it does provide background information that is essential for the CCNA exam.
In the course index, questions about background information are indicated with the abbreviation BCK and a short description of the question subject matter.

Topic 1.1: Understanding Base 10

*Base 10
Numbering systems represent digits with a method that is based on a particular number. The decimal system represents digits using a base of 10.
Base 10 requires ten different digits, 0–9, to represent numbers. In a decimal number, each successive digit (starting from the decimal point and moving to the left) indicates a successively increasing power of 10.

Topic 1.2: Decimal Conversions

*Whole Numbers as Decimals
Look at the whole numbers in the first column of this table. Because they are whole numbers, the decimal point is usually omitted.
The decimal point is added in the second column to illustrate that the value of the numbers is determined by their position in relation to the decimal point.
                       Expressed with        Total
   Whole Number        Decimal Point         Value
   5                   5.                    5 ones
   50                  50.                   5 tens
   500                 500.                  5 hundreds
   5000                5000.                 5 thousands

*Number Placement and Values
The number immediately left of the decimal point represents 10 to the 0 power, which is 1. So this is called the ones place. The total value of the number 5 is 5 ones or 5. In the second row, you see the 5 is two places to the left of the decimal point, so its value is 10 to the 1 power, or 5 tens + 0 ones, which is 50.
                       Expressed with        Total
   Whole Number        Decimal Point         Value
   5                   5.                    5 ones
   50                  50.                   5 tens
   500                 500.                  5 hundreds
   5000                5000.                 5 thousands

Topic 1.2.1: Whole Number

*Continuing the Concept
This concept follows as you move to the left of the decimal point. A third digit would be multiplied by 10 to the 2 power (100), the next multiplied by 1000, the next by 10,000 and so on.
For example, you can break down the decimal number 382 as seen here:

*Direction Indicates Value
This is a fairly simple concept to grasp since we use the decimal system in everyday applications.
The further left of the decimal point, the larger the value of the number. On the contrary, as you move to the right of the decimal point, the value of the number decreases.

Question 1

Question 2

Question 3

Topic 1.2.2: Fractional

*Decimals as Fractions
The numbers to the right of the decimal point are fractions of a whole number.
The number .1 could be shown as 1/10. The number .01 could be shown as 1/100. Notice that the zero holds a place value in the decimal.

*Moving to the Right
The number immediately right of the decimal point represents tenths of a whole.
   .5     =     5 tenths           or     5/10    
   .05    =     5 hundredths       or     5/100   
   .005   =     5 thousandths      or     5/1000  
   .0005  =     5 ten thousandths  or     5/10,000

*An Everyday Example
The second number to the right of the decimal point represents hundredths. Think about $.05. There are 100 pennies in a dollar. So what we're looking at here is 5 hundredths of a dollar or 5 cents.
   .5     =     5 tenths           or     5/10    
   .05    =     5 hundredths       or     5/100    
   .005   =     5 thousandths      or     5/1000  
   .0005  =     5 ten thousandths  or     5/10,000

*Logical Progression
This pattern continues as you move to the right of the decimal point. The third number to the right of the decimal point represents thousandths of a whole, the fourth number to the right of the decimal point represents ten thousandths of a whole, and so on.

Question 4

Question 5


* Exercise 1
Try some decimal conversions. When you're finished, you can check your answers.
Answers

Examine the following table

Step	Action
1	Convert the following whole and fractional numbers to their decimal equivalent: 1/10 6/10 45 57/100 7/1000 1/10,000
2	Convert the following decimals to their fractional equivalent: .33 .80 .456 .9832 .875 .1265

1. .1
    .6
    45.00
    .57
    .007
    .0001
2. 33/100
    80/100 or 8/10
    456/1000
    9832/10,000
    875/1000
    1265/10,000

Topic 1.3: Unit 1 Summary

In this unit, you learned how the base 10, or decimal numbering system works.
You also got some practice converting whole and fractional numbers to decimals.

Unit 2. Binary

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2. Binary
       2.1 Understanding Base 2
       2.2 The Binary Role in Computing
       2.3 Binary Conversions
              2.3.1 Binary to Decimal
              2.3.2 Decimal to Binary
       2.4 Unit 2 Summary

In this unit, you will learn about the binary, or base 2, numbering system.
You will learn the roles of bits, nibbles, and bytes in computer processing, and you will perform some binary number conversions.

After completing this unit, you should be able to:

• Explain the binary numbering system
  
  
• Recognize the differences between bits, bytes, and nibbles
  
  
• Convert decimals to binary numbers and binary numbers to decimals

This unit does not address any specific Cisco objectives. However, it does provide background information that is essential for the CCNA exam.
In the course index, questions about background information are indicated with the abbreviation BCK and a short description of the question subject matter.

Topic 2.1: Understanding Base 2

*What Is Binary?
Binary (meaning "made of two"), is the basic building block of today's electronic technology. In an electronic circuit there can be only two possible states, either ON or OFF.
This is the basis of binary. On a basic level, this ON/OFF logic is how computers and circuits make decisions.

*Base 2
The binary numbering system uses base 2 the same way the decimal numbering system uses base 10.
Base 2 uses two different digits, 0 and 1, to represent numbers. In a binary number, each position as you move from right to left, represents a successively increasing power of 2. Fractional numbers are normally not used in binary.

*Binary Representation
As you can imagine, binary notation requires many more digits than decimal to represent a given value.
It is useful information, however, because the way a binary number represents values is very similar to the way a computer stores integer values in memory.

Question 6

Question 7

Topic 2.2: The Binary Role in Computing

*The Bit
At the most simplified level, all information on your hard drive is stored as 1s and 0s. Every 1 or 0 comprises one bit of information. Binary numbers are typically written as a sequence of bits (BIT is short for Binary digIT).

*The Nibble
A nibble is a collection of 4 bits. Hexadecimal numbers (which will be discussed later) can be represented with a single nibble.

*The Byte
The most important data structure used by the computer is a byte. Eight bits, or two nibbles, make up one byte. It takes one byte to store a single character, for example, the letter A. This table illustrates the relationship between bit, nibble, and byte.

*Multiple Bits and Bytes
The amount of information your hard drive can store, or its size, is measured in bytes. Thousands of bytes are called kilobytes, millions of bytes are called megabytes, and billions of bytes are known as gigabytes. Here is a list of prefixes and abbreviations that you will see in this Cisco CCNA series:

Examine the following table

Prefix or Abbreviation	Meaning
kilo	thousand
mega	million
giga	billion
Kb	kilobit
KB	kilobyte
kbps	kilobits per second
KBps	kilobytes per second

Question 8

Question 9

Question 10

Topic 2.3: Binary Conversions

*Positions and Values
In the binary numbering system, the rightmost position represents the "1's" place, the next one to the left represents the "2's" position, the next is the "4's" position, etc.

Topic 2.3.1: Binary to Decimal

*A Binary Breakdown
Let's look at a breakdown of a binary number. Remember that we start with the digit on the right and move left.

Question 11

Question 12

Topic 2.3.2: Decimal to Binary

*Converting to Binary
Converting a decimal number to its binary equivalent can be a bit tedious. You first need to figure out how many binary places you need to cover the decimal amount. Let's keep it simple and choose a small number like 37. To figure out how many binary places you need to be able to reach 37, write out the position values like this:
                             1       1       1       1       1       1       1      1
                          (128)  (64)  (32)  (16)    (8)    (4)     (2)    (1)

*Adding Values
Since the decimal we are trying to convert is less than 64, we only need to use place values up to "32". The next thing we need to do is to place 1s in the correct positions and 0s to get a total of 37 when they are added like this:
                                        1       0       0       1       0       1
                                      (32)  (16)    (8)    (4)     (2)    (1)
Since 32 + 4 + 1 = 37, the binary equivalent of the decimal 37 is 100101.

Question 13

Question 14


* Exercise 1
Try converting some decimals to and from their binary equivalent. When you're finished, you can check your answers.
Answers

Examine the following table

Step	Action
1	Convert these decimals to their binary equivalent: 21 46 92 119 188 252
2	Convert these binary numbers to their decimal equivalent: 101 1100 1010 1110 101010 1111111

1. 10101
    101110
    1011100
    1110111
    10111100
    11111100
2. 5
    12
    10
    14
    42
    127

Topic 2.4: Unit 2 Summary

In this unit, you learned about the binary numbering system and the important role it plays in computing processes.
In addition, you learned how to convert decimals to their binary equivalent and binary numbers to their decimal form.

Unit 3. Hexadecimal

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3. Hexadecimal
       3.1 Understanding Base 16
       3.2 Hexadecimal Conversions
              3.2.1 Decimal to Hexadecimal
              3.2.2 Hexadecimal to Decimal
       3.3 Unit 3 Summary

In this unit, you will learn about the hexadecimal, or base 16, numbering system.
You will also learn how to convert decimals to hexadecimals, and vice versa.

After completing this unit, you should be able to:

• Explain the hexadecimal numbering system
  
  
• Convert decimals to hexadecimals
  
  
• Convert hexadecimals to decimals

This unit does not address any specific Cisco objectives. However, it does provide background information that is essential for the CCNA exam.
In the course index, questions about background information are indicated with the abbreviation BCK and a short description of the question subject matter.

Topic 3.1: Understanding Base 16

*What Is a Hexadecimal?
A hexadecimal is another type of number that a computer uses during internal processes. This numbering system uses a base of 16. Knowledge of binary and hexadecimal numbering systems is essential to computer operations and for computer programming. Fractional numbers are normally not used in the hexadecimal numbering system.

*Base 16
If you will recall, the decimal system identifies the rightmost digit (to the left of the decimal point) as the "ones" digit, the next one to the left is the "tens" digit, the next one to the left is the "hundreds" digit, etc. The hexadecimal system works the same, only using increasing powers of 16.

*Positions and Values
For hexadecimal, the rightmost digit is the "1's" digit, the next one to the left is the "16's", the next one to the left is the "256's", the next one is the "4096's", etc. For any given position in a hexadecimal number, you can have sixteen possible numbers, represented as seen here, where a–f correspond to the numbers 10–15.
                              0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f

Question 15

Question 16

Topic 3.2: Hexadecimal Conversions

*Getting the Picture
Here is a graphical representation of all three numbering systems: decimal, binary and hexadecimal. The upper numbers represent the decimal numbering system followed by the [hexadecimal numbering system]. The bottom numbers are the binary equivalent of the upper numbers.

Topic 3.2.1: Decimal to Hexadecimal

*Converting from Decimal to Hexadecimal
Let's convert the decimal 156 to hexadecimal. Similar to the binary conversion you learned, you must first decide how many positions you'll need to extend the hexadecimal out to.
In this case, it will be only two, since the third position in hexadecimal is the "256's", which is greater than what we need.

*Completing the Conversion
Now we must fill in the appropriate amounts in each slot to arrive at 156, working with the "16's" position first. What multiple of 16 is closest to (without exceeding) 156?
16 × 9 = 144, so the digit in our left slot will be 9. Then to find the second digit we take 156 - 144 = 12, which in hexadecimal is "c". So the hexadecimal equivalent of the decimal 156 is 9c.

Topic 3.2.2: Hexadecimal to Decimal

*Converting from Hexadecimal to Decimal
Here is a breakdown of a hexadecimal number, 13f8. We start again with the rightmost digit, 8, which is in the "ones" position, and move to the left multiplying by increasing powers of 16.
And recall that the "f" is the equivalent of 15.  So we get (1 × 8) + (15 × 16) + (3 × 256) + (1 × 4096), for a total of 5112.

Question 17

Question 18


* Exercise 1
Try some hexadecimal conversions. When you're finished, you can check your answers.
Answers

Examine the following table

Step	Action
1	Convert the following decimals to their hexadecimal equivalent: 22 54 89 138 183 220
2	Convert the following hexadecimals to their decimal equivalent: b6 4ae 13f 2cd 53ab 17e6

1. 16
    36
    59
    8a
    b7
    dc
2. 182
    1198
    319
    717
    21,419
    6118

Topic 3.3: Unit 3 Summary

In this unit, you learned about the hexadecimal numbering system that uses a base of 16.
You also learned how to convert decimal numbers to and from their hexadecimal equivalent.

Unit 4. Conversion Exercises

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4. Conversion Exercises
       4.1 Decimal Conversions
       4.2 Binary Conversions
       4.3 Hexadecimal Conversions
       4.4 Unit 4 Summary

In this unit, you will gain even more experience performing numeric conversions. You will be presented with several different types of conversions to test your knowledge of the decimal, binary, and hexadecimal numbering systems.

After completing this unit, you should be able to:

• Perform decimal number conversions
  
  
• Perform binary number conversions
  
  
• Perform hexadecimal number conversions

This unit does not address any specific Cisco objectives. However, it does provide background information that is essential for the CCNA exam.
In the course index, questions about background information are indicated with the abbreviation BCK and a short description of the question subject matter.

Topic 4.1: Decimal Conversions

Question 19

Question 20

Question 21

Question 22

Topic 4.2: Binary Conversions

Question 23

Question 24

Question 25

Question 26

Topic 4.3: Hexadecimal Conversions

Question 27

Question 28

Question 29

Question 30

Topic 4.4: Unit 4 Summary

In this unit, you gained some valuable experience converting numbers. Knowledge of the decimal, binary, and hexadecimal numbering systems will allow you to better understand computer logic and programming.
Since you've seen the conversion of decimal to binary, and decimal to hexadecimal, you could take a binary number, convert it to it's decimal equivalent, and then convert that decimal to a hexadecimal. Essentially, you can take a number, whether it is in decimal, binary, or hexadecimal format, and convert it to the format that best meets your needs.