## Conversion of Bases-

In number system, it is very important to have a good knowledge of how to convert numbers from one base to another base.

In the previous articles, we learnt-

Before you proceed further, make sure that you have gone through these articles. We will use the concepts learnt in these articles here.

In this article, we will learn how to convert a number from any given base to any other base.

Suppose the given number has base x1 and we want to convert the number in base x2.

 (Given Number)base x1   → (?)base x2

To perform such kind of conversions, we will follow the following steps-

### Step-01:

Convert the number from base x1 to base 10 using expansion method.

### Step-02:

Convert the number from base 10 to base x2 using division / multiplication method. ## Problem-01:

Convert (1056)16 to ( ? )8

## Solution-

### (1056)16 → ( ? )10

Using Expansion method, we have-

(1056)16

= 1 x 163 + 0 x 162 + 5 x 161 + 6 x 160

= 4096 + 0 + 80 + 6

= (4182)10

(1056)16 = (4182)10

### (4182)10 → ( ? )8

Using Division method, we have- (4182)10 = (10126)8

Thus,

 (1056)16  = (10126)8

## Problem-02:

Convert (11672)8 to ( ? )16

## Solution-

### (11672)8 → ( ? )10

Using Expansion method, we have-

(11672)8

= 1 x 84 + 1 x 83 + 6 x 82 + 7 x 81 + 2 x 80

= 4096 + 512 + 384 + 56 + 2

= (5050)10

(11672)8 = (5050)10

### (5050)10 → ( ? )16

Using Division method, we have- (5050)10 = (13BA)16

Thus,

 (11672)8  = (13BA)16

## Problem-03:

Convert (2724)8 to ( ? )5

## Solution-

### (2724)8 → ( ? )10

Using Expansion method, we have-

(2724)8

= 2 x 83 + 7 x 82 + 2 x 81 + 4 x 80

= 1024 + 448 + 16 + 4

= (1492)10

(2724)8 = (1492)10

### (1492)10 → ( ? )5

Using Division method, we have- (1492)10 = (21432)5

Thus,

 (2724)8  = (21432)5

## Problem-04:

Convert (3211)4 to ( ? )5

## Solution-

### (3211)4 → ( ? )10

Using Expansion method, we have-

(3211)4

= 3 x 43 + 2 x 42 + 1 x 41 + 1 x 40

= 192 + 32 + 4 + 1

= (229)10

(3211)4 = (229)10

### (229)10 → ( ? )5

Using Division method, we have- (229)10 = (1404)5

Thus,

 (3211)4  = (1404)5

## Problem-05:

Convert (1001001100)2 to ( ? )6

## Solution-

### (1001001100)2 → ( ? )10

Using Expansion method, we have-

(1001001100)2

= 1 x 29 + 0 x 28 + 0 x 27 + 1 x 26 + 0 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 0 x 21 + 0 x 20

= 512 + 64 + 8 + 4

= (588)10

(1001001100)2 = (588)10

### (588)10 → ( ? )6

Using Division method, we have- (588)10 = (2420)6

Thus,

 (1001001100)2  = (2420)6

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## Converting Decimal to Hexa | Base 10 to base 16

In number system, it is very important to have a good knowledge of how to convert numbers from one base to another base.

In the last article, we discussed-

How to convert a Decimal number to Octal number?

We have learnt that any number can be easily converted from base 10 to any other base using division method and multiplication method.

 (Given Number)10  → (?)16

There are two cases possible-

Case-01: For numbers carrying no fractional part

Case-02: For numbers carrying a fractional part

### Case-01: For numbers carrying no fractional part-

To convert the numbers carrying no fractional part from base 10 to base 16, we will use division method.

Division method involves following 2 steps-

### Step-01:

Divide the given number (in base 10) with 16 until the result finally left is less than 16.

### Step-02:

Traverse the result and remainders from bottom to top to get the required number in base 16.

### Example-

Suppose we want to convert a number (2020)10 to base 16.

(2020)10  (?)16

Then using division method, we have- Thus,

 (2020)10  = (7E4)16

### For numbers carrying a fractional part-

Suppose the given number consists of some real part and some fractional part.

To convert such numbers from base 10 to base 16, we will treat the real part and fractional part separately.

### For real part-

To convert the real part of number to base 16, we will use division method as we have used in above example.

### For fractional part-

To convert the fractional part of number to base 16, we will use multiplication method.

Multiplication method involves following steps-

### Step-01:

Multiply the given fractional number with 16 and write the real part and fractional part of result so obtained separately.

### Step-02:

Multiply the fractional part obtained after multiplication in previous step with 16 and write the real part and fractional part of result so obtained separately.

### Step-03:

Recursively apply step-02 until fractional part obtained after multiplication becomes 0.

(In case fractional part does not terminate to 0, we can find the result up to as many places as we want.)

 The series of real part of multiplication results obtained in above steps from top to bottom is the required number in base 16.

Following example illustrates how to apply these steps-

### Example-

Suppose we want to convert a number (2020.65625)10 to base 16.

We will treat the real part and fractional part separately.

### For real part-

The real part (2020)10 will be converted to base 16 in exactly the same manner using division method as we have done above.

So, for real part, we have-

 (2020)10  = (7E4)16

### For fractional part-

The fractional part (0.65625)10 will be converted to base 16 using multiplication method.

Using multiplication method, we have-

 Real part Fractional Part 0.65625 x 16 10 = A 0.5 0.5 x 16 8 0.0

Now, traverse the real part column from top to bottom to obtain the required number in base 16.

Thus,

 (0.65625)10  = (A8)16

### Explanation-

#### Step-01:

Multiply 0.65625 with 16. Result = 10.5. Write 10 (= A in hexadecimal) in real part and 0.5 in fractional part.

#### Step-02:

Multiply 0.5 with 16. Result = 8.0. Write 8 in real part and 0.0 in fractional part.

Since, fractional part becomes 0, so we stop.

Combining the result of real and fractional parts, we have-

 (2020.65625)10  = (7E4.A8)16

## Problem-

Convert the following numbers from base 10 to base 16-

1. (172)10
2. (172.983)10

## Solution-

### 1. (172)10

#### (172)10 → ( ? )16

Using division method, we have- Thus,

 (172)10  = (AC)16

### 2. (172.983)10

#### (172.983)10 → ( ? )16

We will treat the real part and fractional part separately-

### For real part-

• The real part is (172)10
• We will convert the real part from base 10 to base 16 using division method.
• We have already done this in above problem.

Thus,

 (172)10  = (AC)16

### For fractional part-

• The fractional part is (0.983)10
• We will convert the fractional part from base 10 to base 16 using multiplication method.

Using multiplication method, we have-

 Real part Fractional Part 0.983 x 16 15 = F 0.728 0.728 x 16 11 = B 0.648 0.648 x 16 10 = A 0.368 0.368 x 16 5 0.886

Since, the fractional part terminates to 0 after several iterations. So, let us find the value up to 4 decimal places.

Now, traverse the real part column from top to bottom to obtain the required number in base 16.

Thus,

 (0.983)10  = (FBA5)16

Combining the results of real and fractional part, we have-

 (172.983)10  = (AC.FBA5)16

Also read: Converting Decimal to Binary

To gain better understanding of how to convert a Decimal number (Base 10) to a Hexadecimal number (Base 16),

Watch this video

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## Converting Decimal to Octal-

In number system, it is very important to have a good knowledge of how to convert numbers from one base to another base.

In the last article, we discussed-

How to convert a Decimal number to Binary number?

We have learnt that any number can be easily converted from base 10 to any other base using division method and multiplication method.

In this article, we will discuss how to convert any decimal number to Octal number.

 (Given Number)10  → (?)8

There are two cases possible-

Case-01: For numbers carrying no fractional part

Case-02: For numbers carrying a fractional part

### Case-01: For numbers carrying no fractional part-

To convert the numbers carrying no fractional part from base 10 to base 8, we will use division method.

Division method involves following 2 steps-

### Step-01:

Divide the given number (in base 10) with 8 until the result finally left is less than 8.

### Step-02:

Traverse the result and remainders from bottom to top to get the required number in base 8.

### Example-

Suppose we want to convert a number (1032)10 to base 8.

(1032)10  (?)8

Then using division method, we have- Thus,

 (1032)10  = (2010)8

### For numbers carrying a fractional part-

Suppose the given number consists of some real part and some fractional part.

To convert such numbers from base 10 to base 8, we will treat the real part and fractional part separately.

### For real part-

To convert the real part of number to base 8, we will use division method as we have used in above example.

### For fractional part-

To convert the fractional part of number to base 8, we will use multiplication method.

Multiplication method involves following steps-

### Step-01:

Multiply the given fractional number with 8 and write the real part and fractional part of result so obtained separately.

### Step-02:

Multiply the fractional part obtained after multiplication in previous step with 8 and write the real part and fractional part of result so obtained separately.

### Step-03:

Recursively apply step-02 until fractional part obtained after multiplication becomes 0.

(In case fractional part does not terminate to 0, we can find the result up to as many places as we want.)

 The series of real part of multiplication results obtained in above steps from top to bottom is the required number in base 8.

Following example illustrates how to apply these steps-

### Example-

Suppose we want to convert a number (1032.6875)10 to base 8.

We will treat the real part and fractional part separately.

### For real part-

The real part (1032)10 will be converted to base 8 in exactly the same manner using division method as we have done above.

So, for real part, we have-

 (1032)10  = (2010)8

### For fractional part-

The fractional part (0.6875)10 will be converted to base 8 using multiplication method.

Using multiplication method, we have-

 Real part Fractional Part 0.6875 x 8 5 0.5 0.5 x 8 4 0.0

Now, traverse the real part column from top to bottom to obtain the required number in base 8.

Thus,

 (0.6875)10  = (54)8

### Explanation-

#### Step-01:

Multiply 0.6875 with 8. Result = 5.5. Write 5 in real part and 0.5 in fractional part.

#### Step-02:

Multiply 0.5 with 8. Result = 4.0. Write 4 in real part and 0.0 in fractional part.

Since, fractional part becomes 0, so we stop.

Combining the result of real and fractional parts, we have-

 (1032.6875)10  = (2010.54)8

## Problem-

Convert the following numbers from base 10 to base 8-

1. (172)10
2. (172.878)10

## Solution-

### 1. (172)10

#### (172)10 → ( ? )8

Using division method, we have- Thus,

 (172)10  = (254)8

### 2. (172.878)10

#### (172.878)10 → ( ? )8

We will treat the real part and fractional part separately-

### For real part-

• The real part is (172)10
• We will convert the real part from base 10 to base 8 using division method.
• We have already done this in above problem.

Thus,

 (172)10  = (254)8

### For fractional part-

• The fractional part is (0.878)10
• We will convert the fractional part from base 10 to base 8 using multiplication method.

Using multiplication method, we have-

 Real part Fractional Part 0.878 x 8 7 0.024 0.024 x 8 0 0.192 0.192 x 8 1 0.536 0.536 x 8 4 0.288

Since, the fractional part terminates to 0 after several iterations. So, let us find the value up to 4 decimal places.

Now, traverse the real part column from top to bottom to obtain the required number in base 8.

Thus,

 (0.878)10  = (7014)8

Combining the results of real and fractional part, we have-

 (172.878)10  = (254.7014)8

Also read: Converting Decimal to Hexa

To gain better understanding of how to convert a Decimal number (Base 10) to a Octal number (Base 8),

Watch this video

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## Number System Conversions-

Before you go through this article, make sure that you have gone through the previous article on Basics of Number System.

In number system,

• It is very important to have a good knowledge of how to convert numbers from one base to another base.
• Here, we will learn how to convert any given number from base 10 to base 2. ## Decimal to Binary Conversion-

A given number can be converted from base 10 to any other base using division method and multiplication method.

Following two cases are possible-

### Case-01: For Numbers Carrying No Fractional Part-

• Division Method is used to convert such numbers from base 10 to another base.
• The division is performed with the required base.

### Steps To Convert From Base 10 to Base 2-

• Divide the given number (in base 10) with 2 until the result finally left is less than 2.
• Traverse the remainders from bottom to top to get the required number in base 2.

### Case-02: For Numbers Carrying A Fractional Part-

To convert such numbers from base 10 to another base, real part and fractional part are treated separately.

### For Real Part-

The steps involved in converting the real part from base 10 to another base are same as above.

### For Fractional Part-

• Multiplication Method is used to convert fractional part from base 10 to another base.
• The multiplication is performed with the required base.

### Steps To Convert From Base 10 To Base 2-

• Multiply the given fraction (in base 10) with 2.
• Write the real part and fractional part of the result so obtained separately.
• Multiply the fractional part with 2.
• Write the real part and fractional part of the result so obtained separately.
• Repeat this procedure until the fractional part remains 0.
• If fractional part does not terminate to 0, find the result up to as many places as required.

Required Number in Base 2

= Series of real part of multiplication results obtained in the above steps from top to bottom

Also Read- Conversion to Base 10

## Problems-

Convert the following numbers from base 10 to base 2-

1. (18)10
2. (18.625)10
3. (172)10
4. (172.878)10

## Solution-

### 1. (18)10

#### (18)10 → ( ? )2

Using division method, we have- From here, (18)10 = (10010)2

### 2. (18.625)10

#### (18.625)10 → ( ? )2

Here, we treat the real part and fractional part separately-

### For Real Part-

• The real part is (18)10
• We convert the real part from base 10 to base 2 using division method same as above.

So, (18)10 = (10010)2

### For Fractional Part-

• The fractional part is (0.625)10
• We convert the fractional part from base 10 to base 2 using multiplication method.

Using multiplication method, we have-

 Real part Fractional Part 0.625 x 2 1 0.25 0.25 x 2 0 0.50 0.50 x 2 1 0

### Step-01:

• Multiply 0.625 with 2. Result = 1.25.
• Write 1 in real part and 0.25 in fractional part.

### Step-02:

• Multiply 0.25 with 2. Result = 0.50.
• Write 0 in real part and 0.50 in fractional part.

### Step-03:

• Multiply 0.50 with 2. Result = 1.0.
• Write 1 in real part and 0.0 in fractional part.

Since fractional part becomes 0, so we stop.

• The fractional part terminates to 0 after 3 iterations.
• Traverse the real part column from top to bottom to obtain the required number in base 2.

From here, (0.625)10 = (0.101)2

Combining the results of real part and fractional part, we have-

(18.625)10 = (10010.101)2

### 3. (172)10

#### (172)10 → ( ? )2

Using division method, we have- From here, (172)10 = (10101100)2

### 4. (172.878)10

#### (172.878)10 → ( ? )2

Here, we treat the real part and fractional part separately-

### For Real Part-

• The real part is (172)10
• We convert the real part from base 10 to base 2 using division method same as above.

So, (172)10 = (10101100)2

### For Fractional Part-

• The fractional part is (0.878)10
• We convert the fractional part from base 10 to base 2 using multiplication method.

Using multiplication method, we have-

 Real part Fractional Part 0.878 x 2 1 0.756 0.756 x 2 1 0.512 0.512 x 2 1 0.024 0.024 x 2 0 0.048

• The fractional part does not terminates to 0 after several iterations.
• So, let us find the value up to 4 decimal places.
• Traverse the real part column from top to bottom to obtain the required number in base 2.

From here, (0.878)10 = (0.1110)2

Combining the results of real part and fractional part, we have-

(172.878)10 = (10101100.1110)2

To gain better understanding about Decimal to Binary Conversion,

Watch this Video Lecture

Next Article- Converting Base 10 to Base 8

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## Number System Conversions-

Before you go through this article, make sure that you have gone through the previous article on Basics of Number System.

In number system,

• It is very important to have a good knowledge of how to convert numbers from one base to another base.
• Here, we will learn how to convert any given number from any base to base 10. ## Converting to Base 10-

A given number can be converted from any base to base 10 using Expansion Method.

According to expansion method, if abc.de is any given number in base x, then its value in base 10 is given as-

(abc.de)x = (ax2 + bx + c + dx-1 + ex-2)10

### Explanation-

To use expansion method for conversion,

• Assign position number to each digit of the given number.
• Digits to the left of decimal are numbered starting from 0.
• Digits to the right of decimal are numbered starting from -1.
• Write a term for each digit as digit x (base of given number)position number of digit
• Perform the addition of all terms to obtain the number in base 10.
• This formula can be expanded for any number of digits. ## PRACTICE PROBLEMS BASED ON CONVERSION TO BASE 10-

Convert the following numbers to base 10-

1. (10010)2
2. (254)8
3. (AC)16
4. (10010.101)2
5. (254.7014)8
6. (AC.FBA5)16
7. (0.1402)8
8. (0.ABDF)16

## Solutions-

### 1. (10010)2

#### (10010)2 → ( ? )10

Using expansion method, we have-

(10010)2

= ( 1 x 24 + 0 x 2+ 0 x 22 + 1 x 21 + 0 x 20 )10

= ( 16 + 0 + 0 + 2 + 0 )10

= ( 18 )10

### 2. (254)8

#### (254)8 → ( ? )10

Using expansion method, we have-

(254)8

= ( 2 x 82 + 5 x 8+ 4 x 80 )10

= ( 128 + 40 + 4 )10

= ( 172 )10

### 3. (AC)16

#### (AC)16 → ( ? )10

Using expansion method, we have-

(AC)16

= ( A x 161 + C x 160 )10

= ( 10 x 16 + 12 x 1 )10

= ( 160 + 12 )10

= ( 172 )10

### 4. (10010.101)2

#### (10010.101)2 → ( ? )10

Using expansion method, we have-

(10010.101)2

= ( 1 x 24 + 0 x 2+ 0 x 22 + 1 x 21 + 0 x 20 + 1 x 2-1 + 0 x 2-2 + 1 x 2-3 )10

= ( 16 + 0 + 0 + 2 + 0 + 0.5 + 0.125 )10

= ( 18.625 )10

### 5. (254.7014)8

#### (254.7014)8 → ( ? )10

Using expansion method, we have-

(254.7014)8

= ( 2 x 82 + 5 x 8+ 4 x 80 + 7 x 8-1 + 0 x 8-2 + 1 x 8-3 + 4 x 8-4 )10

= ( 128 + 40 + 4 + 0.875 + 0.0019 + 0.0009 )10

= ( 172.8778 )10

### 6. (AC.FBA5)16

#### (AC.FBA5)16 → ( ? )10

Using expansion method, we have-

(AC.FBA5)16

= ( A x 161 + C x 160 + F x 16-1 + B x 16-2 + A x 16-3 + 5 x 16-4 )10

= ( 10 x 16 + 12 x 1 + 15 x 16-1 + 11 x 16-2 + 10 x 16-3 + 5 x 16-4 )10

= ( 160 + 12 + 0.9375 + 0.0429 + 0.0024 + 0.0001 )10

= ( 172.9829 )10

### 7. (0.1402)8

#### (0.1402)8 → ( ? )10

Using expansion method, we have-

(0.1402)8

= ( 0 x 80 + 1 x 8-1 + 4 x 8-2 + 0 x 8-3 + 2 x 8-4 )10

= ( 0 + 0.125 + 0.0625 + 0 + 0.0005 )10

= ( 0.188 )10

### 8. (0.ABDF)16

#### (0.ABDF)16 → ( ? )10

Using expansion method, we have-

(0.ABDF)16

= ( 0 x 160 + A x 16-1 + B x 16-2 + D x 16-3 + F x 16-4 )10

= ( 0 x 1 + 10 x 16-1 + 11 x 16-2 + 13 x 16-3 + 15 x 16-4 )10

= ( 0 + 0.625 + 0.0429 + 0.0032 + 0.0002 )10

= ( 0.6713 )10

To gain better understanding about Conversion to Base 10,

Watch this Video Lecture

Next Article- Decimal to Binary Conversion

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