CCNA - Converting Between Binary and Decimal Numbering Systems

Summary

This video explains the process of converting numbers between the decimal (base 10) and binary (base 2) systems. It begins by discussing place values in the decimal system, where numbers are based on powers of ten, and then compares it to the binary system, which only uses two digits (0 and 1). The video walks through examples of converting decimal numbers like 168 to binary and vice versa, including how to interpret binary numbers in IP addresses. The focus is on understanding place values and simplifying the conversion process for easy comprehension.

Takeaways

• 🔢 The video explains binary to decimal conversion, starting with a review of positional notation or place values.
• 💡 In the decimal system (base 10), each digit represents powers of 10, such as the ones, tens, hundreds, and thousands places.
• ✏️ The number 2168 can be broken down into place values: 2 in the thousands place, 1 in the hundreds place, 6 in the tens place, and 8 in the ones place.
• 🔍 The decimal system uses ten numerals (0-9) in each place value, making it a base-10 system.
• ⚙️ Binary is a base-2 system with only two numerals: 0 and 1. Each place value represents powers of 2.
• 📊 The place values in binary correspond to powers of 2, such as 2^0 = 1, 2^1 = 2, 2^2 = 4, up to 2^7 = 128 for an 8-bit byte.
• 💻 To convert decimal to binary, find the place values where 1 is needed. For example, 168 in binary is 10101000.
• 🔄 The process for converting binary to decimal involves adding up the place values for the positions where a 1 appears. For example, binary 01101101 equals decimal 109.
• 🌐 The video demonstrates how to convert a binary IP address to decimal. Each octet is calculated separately, such as 11000000 for 192.
• 📊 In the example, the binary IP address 11000000.10101000.00000001.01100101 converts to the decimal IP address 192.168.1.101.

Q & A

What is the significance of place values in the decimal number system?

In the decimal system, place values determine the value of a digit based on its position. Each place represents powers of ten, such as ones, tens, hundreds, and thousands.

How are the powers of ten related to the place values in the decimal system?

In the decimal system, each place value is a power of ten. For example, the ones place is 10^0, the tens place is 10^1, the hundreds place is 10^2, and so on.

What does the number 2168 represent in terms of its place values?

The number 2168 represents 2 in the thousands place (2000), 1 in the hundreds place (100), 6 in the tens place (60), and 8 in the ones place (8). Added together, this equals 2168.

How is binary different from the decimal system?

Binary is a base-2 system that uses only two digits, 0 and 1, whereas the decimal system is base-10 and uses digits from 0 to 9. Each place value in binary represents a power of 2.

What are the place values in the binary system?

In binary, the place values are based on powers of 2, such as 2^0 (1), 2^1 (2), 2^2 (4), 2^3 (8), 2^4 (16), 2^5 (32), 2^6 (64), and 2^7 (128).

How do you convert the decimal number 168 into binary?

To convert 168 to binary, you identify which powers of 2 sum to 168. This gives 128 (2^7), 32 (2^5), and 8 (2^3), so 168 in binary is 10101000.

What is the process of converting a binary number to decimal?

To convert binary to decimal, multiply each binary digit by its corresponding power of 2 and sum the values. For example, the binary number 01101101 converts to decimal as 109.

How would you convert the binary number 01101101 to decimal?

The binary number 01101101 equals 64 + 32 + 8 + 4 + 1, which totals 109 in decimal.

What is an IP address, and how is it represented in binary?

An IP address is a 32-bit number divided into four 8-bit segments, or octets. Each octet can be represented in binary, and then converted to decimal for easier reading.

How do you convert a binary IP address to its decimal form?

To convert a binary IP address to decimal, convert each 8-bit binary octet individually to its decimal equivalent, and then combine the four decimal numbers. For example, the binary IP 11000000.10101000.00000001.01100101 converts to 192.168.1.101.

Outlines

00:00

📊 Understanding Decimal Place Values

This paragraph introduces the concept of positional notation in the decimal (base-10) number system using the example of the number 2168. It breaks down the place values as powers of ten, such as the ones, tens, hundreds, and thousands place, showing how each contributes to the overall value of the number. The author explains how each digit represents a multiplication of the digit with the corresponding power of ten. The discussion includes an example of how numbers are added together in their expanded form (e.g., 2000 + 100 + 60 + 8). This provides a foundation for understanding decimal notation as we use it in everyday counting.

05:02

💡 Introduction to Binary Place Values

This section contrasts the decimal system with the binary (base-2) system. In binary, only two digits (0 and 1) are used. It explains how binary place values correspond to powers of two, starting from 2^0 (1) up to higher powers like 2^7 (128). The paragraph includes a detailed example of converting the decimal number 168 into binary by examining which powers of two fit into the number and assigning 1s or 0s to the corresponding place values. This results in the binary representation 10101000 for the decimal number 168. The idea of 8 bits forming a byte in computer processing is also introduced.

Keywords

💡Positional Notation

Positional notation refers to the method of representing numbers in which the position of each digit determines its value. In the video, this concept is introduced through the example of the number 2168, where the value of each digit depends on its place (e.g., thousands, hundreds, tens, ones). This is fundamental to both the decimal (base-10) and binary (base-2) number systems.

💡Base-10 (Decimal System)

The base-10 system, also known as the decimal system, is the most common numerical system used in everyday life. It is based on powers of ten, with ten possible digits (0-9) in each place value. The video explains this using the number 2168, where each digit represents a power of 10, such as 10^0 for the ones place or 10^3 for the thousands place.

💡Base-2 (Binary System)

The binary system, or base-2, is a number system used in computing that only uses two digits: 0 and 1. Each digit in a binary number represents a power of two. The video shows how to convert decimal numbers like 168 into binary (10101000), illustrating the importance of binary in digital systems, such as computer processing and networking.

💡Powers of Ten

Powers of ten refer to the exponents used to express place values in the decimal system. In the video, place values like 10^0, 10^1, and 10^2 represent ones, tens, and hundreds, respectively. Understanding this concept is essential for grasping how numbers in base-10 are structured and how they differ from other systems, such as binary.

💡Powers of Two

Powers of two are critical in the binary system, where each place value represents increasing powers of 2 (e.g., 2^0, 2^1, 2^2). The video illustrates this when converting decimal to binary and vice versa. For instance, the binary number 10101000 represents 128, 32, and 8, which are powers of 2 added together to make 168.

💡Bit

A bit is the most basic unit of data in computing and represents a binary value, either 0 or 1. In the video, bits are used to form binary numbers, with 8 bits grouped together to form a byte. The video explains how 8 bits can represent a range of numbers, and how binary numbers are processed by computers.

💡Byte

A byte consists of 8 bits and is a common data size used in computing. The video discusses how a byte is used to represent numbers in binary and how important this is for computer systems. For example, when converting the number 168 to binary, it is represented by 8 bits, forming a complete byte.

💡Binary to Decimal Conversion

Binary to decimal conversion is the process of translating a binary number into its decimal equivalent. In the video, this is demonstrated with the binary number 01101101, which is converted into the decimal number 109 by adding up the corresponding place values (64 + 32 + 8 + 4 + 1).

💡Decimal to Binary Conversion

Decimal to binary conversion is the reverse process of turning a decimal number into its binary form. The video shows how to convert the decimal number 168 to binary by identifying which powers of two (128, 32, and 8) add up to 168, resulting in the binary number 10101000.

💡IP Address

An IP address is a numerical label assigned to devices on a network, often represented in decimal format. In the video, a 32-bit IP address is shown as a binary number, and each octet (group of 8 bits) is converted to its decimal form to explain how computers use binary numbers to manage network addresses, like 192.168.1.101.

Highlights

Introduction to binary to decimal conversion and overview of positional notation in base 10.

Explanation of place values in the decimal system: ones, tens, hundreds, thousands, etc.

Demonstration of breaking down the number 2168 by place values, showing how it's composed of 2000, 100, 60, and 8.

Clarification that the decimal system is a base 10 system using powers of 10, with digits from 0 to 9.

Comparison between decimal (base 10) and binary (base 2) number systems.

Binary system explained: only two possible values (0 and 1) and place values as powers of 2 (1, 2, 4, 8, 16, 32, etc.).

Illustration of converting the decimal number 168 into binary by assigning 1 or 0 to binary place values.

Step-by-step process of converting 168 to binary: placing 1s and 0s in the appropriate positions (128, 32, and 8).

Explanation of how 168 in decimal equals 10101000 in binary.

Introduction to converting binary numbers back to decimal using place values.

Demonstration of converting the binary number 01101101 to decimal by adding up place values (64, 32, 8, 4, 1).

Conversion result: the binary number 01101101 equals 109 in decimal.

Introduction to IP addresses in binary and how to convert them to decimal.

Breakdown of a 32-bit IP address into four octets for conversion.

Example conversion of binary IP address 11000000.10101000.00000001.01100101 to decimal 192.168.1.101.

Transcripts

this video i'm going to discuss binary

to decimal conversion but before i do

this i want to take a look at positional

notation or place values i have the

number

2168 here

if we look at the place values of the

number 2168

we can see that the place values have

the ones place

the tens place the hundreds place the

thousands place ten thousand hundred

thousand and million these are the place

values of the base 10 decimal number

system

you can see that we have the number two

in the one thousands place so we have

two one thousands we have a one in the

hundreds place for one hundred we have

six in the tens place for sixty and we

have eight in the ones place for eight

so effectively we have two one thousands

one one hundred

six tens for sixty

and eight ones for eight

now when we're talking about the place

values in the decimal number system

we're talking about the powers of ten

you can see that the ones place is the

ten to the zero

the tens place ten to the one the

hundreds place ten to the two or ten

times ten which is a hundred the

thousands place is the ten to the three

or ten times ten times ten

and so on and so forth so you can see

that the place values

are based on powers of ten if we look at

the number 2106

then in long form we can see that

effectively we have two one thousands

one one hundred

six tens

and eight ones and two thousand plus one

hundred plus sixty plus eight

totals two thousand one hundred and

sixty eight

this is the type of counting and

addition that we learn as children

the decimal system

is base ten

it's based on the fact that you have one

powers of ten

but more importantly you have ten

characters or ten numerals in this

counting system from zero all the way up

to nine so that means that in each place

value you can have anywhere from the

number zero up to the number nine in

other words if i had the number

9168 i'd simply replace the two here

with a nine and now i have

nine

one thousands totaling nine thousand

in the one thousands place

so in any one of these place values you

can have the number zero all the way up

to 9. this is the base 10 decimal number

system

if we consider binary and look at it in

the same light as decimal

binary is a base 2 number system there's

only two characters or two numbers zero

and one

so under the place values

we can only have zeros or ones

the place values go from one which is

two to the zero

to two two to the one four two to the

two eight two to the three or two times

two times two is eight

two times two times two times two is

sixteen that's two to the fourth power

place value of sixteen

the thirty-two's place the sixty-fours

place and the one twenty eighths place

notice that i extended the table to

eight place values that's because eight

bits is an important grouping of numbers

eight bits makes a byte in computer

processing so now i have the place

values

for essentially 8 bits

if i want to write the number

168 in binary i just have to find the

corresponding place values and plug in

either a one or a zero

so i'll go to the hundred and twenty

eighths place and ask myself do i need

128 to reach 168 yes i do so i'll put a

one there

now do i need a 64. i already have a 128

if i add 64

i would get 192 because 128 plus 64 is

192. so the answer is no so i put a zero

i still have 128 now now do i need a 32

128 plus 32 is 160 so yes

i could use a 1 here

now i have 160

do i need a 16 no that would make 176

which would go over my target number of

168.

i'll put a zero here

what about an eight if i add an eight

i'll hit the number perfectly 128 plus

32 plus eight is 168. i'll follow this

up with zeros in the fourth place

the two's place

and the ones place

and 168 in binary equals one zero one

zero one zero zero zero

i now have

one one hundred and twenty eight

i have one thirty two

and i have one eight and a hundred and

twenty eight plus thirty two plus eight

equals a hundred 168.

if we go to the next slide

you can see now that i'm now charged

with converting the number 0 1 1 0

1 1 0 1

to decimal

if i want to go the opposite way and

convert this binary number to decimal

all i need to do is plug it into the

place values

i'll put it in here 0

1

1

0.

1 1

0

1

and then add it up

i have a 64

and i have a 32

64 plus 32 is

96. i have an eight that makes a hundred

and four

plus four makes a hundred and eight

plus one makes a hundred and nine

this number converted to decimal is the

number one hundred and nine

now let's look at a full ip address in

binary

i'll go to my next slide

and you can see in this next slide i now

have

a 32-bit ip address

4 octets

or 32 bits total

if i want to convert this binary ip

address to decimal all i need to do is

count up each individual octet

let's start with the first one

right here

we can see that one

one

zero

zero zero zero zero 0

128 plus 64 is

192.

now the next octet has 1 0 1 0 1 let's

do that one

zero

one

zero one and then all zeros

if we count up the numbers

128

plus 32 is 160 plus eight is one sixty

eight

the next octet is all zeros

with a one in the last place in the ones

place

this is easy this is the number

one

all zeros and a one in the ones place

makes the number one

and then finally we have a number here

i'll plug it in here into my table

[Music]

and i have the number zero one one zero

zero

one zero one

we can see that 64 plus 32 we've already

said

is 96

plus four

is a hundred plus one

is one hundred and one

so the conversion of this binary ip

address to decimal is 192.168.1.101