NUMBERS CONVERSION - BINARY TO DECIMAL, OCTAL, HEXADECIMAL AND SO:

Introduction Of Numbering Systems (Binary, Decimal, Octal And Hexadecimal):

◙ - ► Binary (Base 2):  0, 1

◙ - ► Octal (Base 8):  0, 1, 2, 3, 4, 5, 6, 7

◙ - ► Decimal (Base 10):  0, 1, 2, 3, 4, 5, 6, 7, 8, 9

◙ - ► Hexadecimal (Base 16):  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

For More About - > Notes For Number Systems (Binary, Decimal, Octal, Hexadecimal):

Conversion Table - Binary, Decimal, Hexadecimal, Octal:

The Binary Concept - Power Of Two:

HEXADECIMAL	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
DECIMAL	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

Power Of Two :

   2^0=1
   2^1=2
   2^2=4
   2^3=8
   2^4=16
   2^5=32
   2^6=64
   2^7=128

The Relationship Between These Number Systems Can Be Seen In The Following Table, Where The Zero Position Refers To The Rightmost Digit Of A Number, And The ^ Symbol Represents Exponentiation:

Position :	Value in Base :
	10	2	8	16
0	10^0 = 1	2^0 = 1	8^0 = 1	16^0 = 1
1	10^1 = 10	2^1 = 2	8^1 = 8	16^1 = 16
2	10^2 = 100	2^2 = 4	8^2 = 64	16^2 = 256
3	10^3 = 1000	2^3 = 8	8^3 = 512	16^3 = 4,096
4	10^4 = 10,000	2^4 = 16	8^4 = 4,096	16^4 = 65,536
5	10^5 = 100,000	2^5 = 32	8^5 = 32,768	16^5 = 1,048,576
6	10^6 = 1,000,000	2^6 = 64	8^6 = 262,144	16^6 = 16,777,216
7	10^7 = 10,000,000	2^7 = 128	8^7 = 2,097152	16^7 = 268,435,456

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BINARY TO DECIMAL CONVERSION

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1. Start The Decimal Result At 0.
2. Remove The Most Significant Binary Digit (Leftmost) And Add It To The Result.
3. If All Binary Digits Have Been Removed, You’re Done. Stop.
4. Otherwise, Multiply The Result By 2.
5. Go To Step 2.

An Example Of Converting 11100000000 Binary To Decimal :

Binary Digits	Operation	Decimal Result	Operation	Decimal Result
11100000000	+1	1	× 2	2
1100000000	+1	3	× 2	6
100000000	+1	7	× 2	14
00000000	+0	14	× 2	28
0000000	+0	28	× 2	56
000000	+0	56	× 2	112
00000	+0	112	× 2	224
0000	+0	224	× 2	448
000	+0	448	× 2	896
00	+0	896	× 2	1792
0	+0	1792

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DECIMAL TO BINARY CONVERSION

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The Decimal (Base Ten) Numeral System Has Ten Possible Values (0,1,2,3,4,5,6,7,8, Or 9) For Each Place-Value. In Contrast, The Binary (Base Two) Numeral System Has Two Possible Values, Often Represented As 0 Or 1, For Each Place-Value.

To Avoid Confusion While Using Different Numeral Systems, The Base Of Each Individual Number May Be Specified By Writing It As A Subscript Of The Number. For Example, The Decimal Number 156 May Be Written As 15610 And Read As "One Hundred Fifty-Six, Base Ten". The Binary Number 10011100 May Be Specified As "Base Two" By Writing It As 10011100 ₂ .

Decimal To Binary Conversion Examples :

• (51)₁₀ = (110011)₂
• (217)₁₀ = (11011001)₂
• (8023)₁₀ = (1111101010111)₂

Decimal Binary Conversion Table:

Decimal	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Binary	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111

An Example Of Using Repeated Division To Convert 1792 Decimal To Binary :

1. Divide The Decimal Number By The Desired Target Radix (2, 8, Or 16).
2. Append The Remainder As The Next Most Significant Digit.
3. Repeat Until The Decimal Number Has Reached Zero.

Decimal Number	Operation	Quotient	Remainder	Binary Result
1792	÷ 2 =	896	0	0
896	÷ 2 =	448	0	00
448	÷ 2 =	224	0	000
224	÷ 2 =	112	0	0000
112	÷ 2 =	56	0	00000
56	÷ 2 =	28	0	000000
28	÷ 2 =	14	0	0000000
14	÷ 2 =	7	0	00000000
7	÷ 2 =	3	1	100000000
3	÷ 2 =	1	1	1100000000
1	÷ 2 =	0	1	11100000000
0

Now We Are Going To See How To Convert Decimal To Binary :

The Decimal or "Denary" Counting System Uses The Base of 10 Numbering System Where Each Digit In A Number Takes On One Of Ten Possible Values From 0 to 9, eg 213₁₀ (Two Hundred And Thirteen). In A Decimal System Each Digit Has A Value Ten Times Greater Than Its Previous Number And This Decimal Numbering System Uses A Set Of Symbols, b, Together With A Base, q, To Determine The Weight Of Each Digit Within A Number. For Example, The Six In Sixty Has A Lower Weighting Than The Six In Six Hundred And In A Binary Numbering System We Need Some Way Of Converting Decimal To Binary.

Any Numbering System Can Be Summarised By The Following Relationship :

	N = bi qi
		where:	N Is A Real Positive Number b Is The Symbol q Is The Base Value and integer (i) Can Be Positive, Negative Or Zero
		N = b2 q2 + b1 q1  + b0 q0 + b-1 q-1 ... etc.

In The Decimal Or Denary System, The Columns Have Values Of Units, Tens, Hundreds Etc As We Move From Right To Left And Mathematically These Values Are Written As 10⁰, 10¹, 10², 10³ etc. The Decimal Numbering System Has A Base Of 10 Or Modulo-10 (Sometimes Called MOD-10) With The Position Of Each Digit In The Decimal System Indicating The Magnitude Or Weight Of The Number. For Example, 20 (Twenty) Is The Same As Saying 2 X 10¹ And 400 (Four Hundred) Is The Same As Saying 4 X 10². Likewise, For Fractional Numbers The Weight Of The Number Is Negative, 10^-1, 10^-2, 10^-3 etc.

The Value Of A Decimal Number Is Equal To The Sum Of The Digits Multiplied By Their Respective Weights. For Example:  N = 6163₁₀  (Six Thousand One Hundred And Sixty Three)  In A Decimal Format Is Equal To:

(6×10³) + (1×10²) + (6×10¹) + (3×10⁰) = 6163

Unlike The Decimal Numbering System Which Uses The Base Of 10, Digital Logic Uses Just Two Values Or States, A Logic Level "1" Or A Logic Level "0", So Each "0" And "1" Is Considered To Be A Single Digit In A Base Of 2 or Binary numbering system. In The Binary Numbering System, Each Digit Has A Value Twice That Of The Previous Digit But Can Only Have A Value Of Either "1" Or "0" Therefore, q = "2" And The Position Of Either A "0" Or A "1" Indicates Its Weight.

For Example:

Decimal Digit Value:	256	128	64	32	16	8	4	2	1
Binary Digit Value:	1	0	1	1	0	0	1	0	1

By Adding Together All The Decimal Number Values From Right To Left At The Positions That Are Represented By A "1" Gives Us:  (256) + (64) + (32) + (4) + (1) = 357₁₀ or three hundred and fifty seven in decimal.

Then, The Binary Array Of Digits 101100101₂ Is Equivalent To 357₁₀ In Decimal Or Denary. As The Decimal Number Is A Weighted Number, Converting From Decimal To Binary Will Also Produce A Weighted Binary Number With The Right-Hand Most Bit Being The Least Significant Bit Or LSB, And The Left-Hand Most Bit Being The Most Significant Bit Or MSB. And We Can Represent These As.

MSB	Binary Digit							LSB
28	27	26	25	24	23	22	21	20
256	128	64	32	16	8	4	2	1

Repeated Division-By-2 Method:

Another Method Of Converting Decimal to Binary Number Equivalents Is To Write Down The Decimal Number And To Continually Divide By 2 (Two) To Give A Result And A Remainder Of Either A "1" Or A "0" Until The Final Result Equals Zero.

Example.  Convert The Decimal Number 294₁₀ Into Its Binary Number Equivalent.

Number	294				Dividing Each Number By "2" Gives A Result Plus A Remainder. The Binary Result Is Obtained By Placing The Remainders In Order With The Least Significant Bit (LSB) Being At The Top And The Most Significant Bit (MSB) Being At The Bottom.
Divide By 2
Result	147	Remainder	0  (LSB)
Divide By 2
Result	73	Remainder	1
Divide By 2
Result	36	Remainder	1
Divide by 2
Result	18	Remainder	0
Divide by 2
Result	9	Remainder	0
Divide by 2
Result	4	Remainder	1
Divide by 2
Result	2	Remainder	0
Divide by 2
Result	1	Remainder	0
Divide by 2
Result	0	Remainder	1  (MSB)

Then, The Decimal To Binary Conversion Gives The Decimal Number 294₁₀ equivalent of 100100110₂ In Binary, Reading From Right To Left.

Then The Main Characteristics Of A Binary Numbering System Is That Each "Digit" Or "Bit" Has A Value Of Either "1" or "0" With Each Digit Having A Weight Or Value Double That Of Its Previous Bit Starting From The Lowest Or Least Significant Bit (LSB) And This Is Called The "Sum-Of-Weights" Method. So We Can Convert A Decimal Number To Binary Either By Using The Sum-Of-Weights Method Or By Using The Repeated Division-by-2 method.

BINARY TO HEXADECIMAL CONVERSION

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To Convert A Binary Number Into Its Hexadecimal Equivalent, Divide It Into Groups Of Four Bits. If The Number Of Bits Isn't A Multiple Of Four, Simply Insert Extra 0 Bits At The Left Are Called Padding.

For Example:

1010010₂ = 0101 0010 Grouped With Padding = 52₁₆

11011101₂ = 1101 1101 Grouped = DD₁₆

Another Easy Conversion To Do Is Binary To Hexadecimal. The Hexadecimal Number System Uses The Digits 0 To 9 And A, B, C, D, E, F. And Since The Hexadecimal System Is A Power Of 2 (2⁴), We Can Take A Binary Number In Groups Of 4 And Use The Appropriate Hexadecimal Digit In Its Place.

The Steps To Doing So Are Simple. Begin At The Rightmost 4 Bits. If There Are Not 4 Bits, Pad 0s To The Left Until You Hit 4. Repeat The Steps Until All Groups Have Been Converted.

An Example Is Below :

Take The Binary Number (1000101). Using The Above Steps, Here Is The Conversion:

0100 | 0101 	Note That We Needed To Pad A 0 To The Left.
4 | 5

Answer: (45)16

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HEXADECIMAL TO BINARY CONVERSION

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This Conversion Is Also Simplistic. Given A Hexadecimal Number, Simply Convert Each Digit To It’s Binary Equivalent. Then, Combine Each 4 Bit Binary Number And That Is The Resulting Answer.

Hexadecimal To Binary Conversion Examples :

• (1e3)₁₆ = (111100011)₂
• (0a2b)₁₆ = (101000101011)₂
• (7e0c)₁₆ = (111111000001100)₂

Hexadecimal To Binary Conversion Chart Table:

Hex	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
Binary	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111

Take The Hexadecimal Number (A2F)₁₆ And Convert It To Its Binary Form:

A | 2 | F
1010 | 0010 | 1111

Answer: (1010 0010 1111)2

Take the hexadecimal number (55)₁₆ and convert it to it’s binary form:

5 | 5
0101 | 0101

Answer: (1010101)2

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BINARY TO OCTAL CONVERSION

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An Octal System Is Again A Power Of Two (23), We Can Take Group The Bits Into Groups Of 3 And Represent Each Group As An Octal Digit. The Steps Are The Same For The Binary To Hexadecimal Conversions Except We Are Dealing With The Octal Base Now.

Given A Binary Number You Start By Grouping The Binary "1" And "0" Digits In Groups Of 3, Starting At The Right. Then Convert Each Of These Groups Into One Octal Digit.

Take The Binary Number (10011)2 And Convert It To Octal (Converting From One Base (Base 2) To Base 8) :

010 | 011
2 | 3

Answer: (23)8

Example:

binary number = 11001100010
grouped in 3s = 11 001 100 010
3s into octal = 3 1 4 2

So, The Base 2 number 11001100010 equals 3142 in base 8. (Octal Is A Base 8 System And Binary Is A Base 2 System).

Note:To Convert An Octal Number Into A Binary, Just Reverse The Above Process; Starting All The Way To The Right, Convert Each Digit Into A 3-Bit Binary Number.

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DECIMAL TO OCTAL CONVERSION :

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The Steps As Followed To Convert Decimal To Binary Are :

To Convert A Decimal Number Into An Octal Number, Divide It By 8 Repeatedly And Note The Remainders. The Remainders Are The Digits Of The Octal Number. The “Last” Remainder Is The Most Significant Digit, And The “First” Remainder Is The Least Significant Digit.

For Example,

1701(base 10) = 3245(base 8)		1701 ÷ 8 = 212		remainder		5		LSB
		212 ÷ 8 =  26		remainder		4
		26 ÷ 8 =   3		remainder		2
		3 ÷ 8 =   0		remainder		3		MSB

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OCTAL TO DECIMAL CONVERSION :

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Converting Octal To Decimal Can Be Done With Repeated Division.

1. Start The Decimal Result At 0.
2. Remove The Most Significant Octal Digit (Leftmost) And Add It To The Result.
3. If All Octal Digits Have Been Removed, You’re Done. Stop.
4. Otherwise, Multiply The Result By 8.
5. Go To Step 2.

The Conversion Can Also Be Performed In The Conventional Mathematical Way, By Showing Each Digit Place As An Increasing Power Of 8.

345 octal = (3 * 8²) + (4 * 8¹) + (5 * 8⁰) = (3 * 64) + (4 * 8) + (5 * 1) = 229 decimal

Take The Octal Number (764)₈ And Convert It To Decimal:

7 * 82 + 6 * 81 + 4 * 80
448 + 48 + 4 = 500

Answer: (500)10

Take The Octal Number (5771)₈ And Convert It To Decimal:

5 * 83 + 7 * 82 + 7 * 81 + 1 * 80
2560 + 448 + 56 + 1 = 3,065

Answer: (3065)10

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DECIMAL TO HEXADECIMAL CONVERSION

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The Only Addition To The Algorithm When Converting From Decimal To Hexadecimal Is That A Table Must Be Used To Obtain The Hexadecimal Digit If The Remainder Is Greater Than Decimal 9.

Decimal To Hexadecimal Conversion Examples:

• (79)₁₀ = (4F)₁₆
• (120)₁₀ = (78)₁₆
• (1728)₁₀ = (6C0)₁₆

Decimal Hex Conversion Chart Table:

Decimal	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Hex	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F

A2DE Hexadecimal :

= ((A) * 16³) + (2 * 16²) + ((D) * 16¹) + ((E) * 16⁰)
= (10 * 16³) + (2 * 16²) + (13 * 16¹) + (14 * 16⁰)
= (10 * 4096) + (2 * 256) + (13 * 16) + (14 * 1)
= 40960 + 512 + 208 + 14
= 41694 decimal

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OCTAL TO HEXADECIMAL CONVERSION

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Octal To Hexadecimal Is Converting A Number Is An Octal Number System To An Equivalent Number In The Hexadecimal Number System. That Is Converting A Number From Base 8 To Base 16. The Conversion Cannot Be Done Directly.

Example -1:

Convert The Octal Number 536 To Its Equivalent Hexadecimal Number.

Convert 536 (Octal) First To Its Binary Equivalent We Get :

(536)₈ = (101) (011) (110)
           = (101011110)₂

Now forming the groups of 4 binary bits to obtain its hexadecimal equivalent we get,
(101011110)₂= (1) (0101) (1110)
                      = (0001) (0101) (1110)
                      = (15E)₁₆

Example -2:

Convert The Hexadecimal Number 3DE To Its Octal Equivalent :

The Hexadecimal Number 3DE Is First Converted To Its Binary Equivalent.

(3DE)₁₆ = (0011) (1101) (1110)
              = (001111011110)₂

Now The Above Binary Equivalent Is Divided Into Groups Of 3 Bits To Obtain Its Octal Equivalent.
(001111011110)₂ = (001) (111) (011) (110)
                             = (1736)₈

Below Are The Examples On Converting Octal To Hexadecimal -

Solved Examples

Question 1: Convert 1002₈ to hexadecimal
Solution:

 
The given number is 1002₈

1002₈ = (1 * 8³)+ (0 * 8²) + (0 * 8¹) + (2 * 8⁰)

=1 * 512 + 0 * 64 + 0 * 8 + 2 * 1

= 512 + 0 + 0+ 2

= 514(decimal number)

Now we convert the above decimal to hexadecimal

16 | 514
16 | 32   --2
        2 -- 0

The hexadecimal number is 202

1002₈ = 202₁₆
 

Question 2: Convert 100₈ to hexadecimal
Solution:

 
The given number is 100₈

10028 = (1 * 82) + (0 * 81) + (0 * 80)

= 1 * 64 + 0 * 8 + 0 * 1

= 64 + 0 + 0

= 64(decimal number)

Now we convert the above decimal to hexadecimal

16 | 64
16 | 4   --0
16 | 0   -- 4

The hexadecimal number is 40

100₈ = 40₁₆
 

Question 3: Convert 15₈ to hexadecimal
Solution:

 
The given number is 15₈

15₈ =(1 * 8¹) + (5 * 8⁰)

= 1 * 8 + 5 * 1

= 8 + 5

= 13(decimal number)

Now we convert the above decimal to hexadecimal

13 is less than 16 so the equivalent hexadecimal number is D

The hexadecimal number is D

15₈ = D₁₆
 

For More About - > Notes For Number Systems (Binary, Decimal, Octal, Hexadecimal):

Conversion Table - Binary, Decimal, Hexadecimal, Octal:

The Binary Concept - Power Of Two:

†

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CONCLUSION:

The Goal Of This Article Is To Give An Easy Way To Understand The “Numbers Conversion - Binary To Decimal, Octal, Hexadecimal And So:" Hope This Article Will Help Every Beginners Who Are Going To Start Cisco Lab Practice Without Any Doubts.

Some Topics That You Might Want To Pursue On Your Own That We Did Not Cover In This Article Are Listed Here, Thank You And Best Of Luck.

This Article Written Author By: Premakumar Thevathasan. CCNA, CCNP, CCIP, MCSE, MCSA, MCSA - MSG, CIW Security Analyst, CompTIA Certified A+.

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