## 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 16.

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 16-

• Divide the given number (in base 10) with 16 until the result finally left is less than 16.
• Traverse the remainders from bottom to top to get the required number in base 16.

### 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 16-

• Multiply the given fraction (in base 10) with 16.
• Write the real part and fractional part of the result so obtained separately.
• Multiply the fractional part with 16.
• 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 16

= 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 16-

1. (2020)10
2. (2020.65625)10
3. (172)10
4. (172.983)10

## Solution-

### 1. (2020)10

(2020)10  (?)16

Using division method, we have-

From here, (2020)10 = (7E4)16

### 2. (2020.65625)10

#### (2020.65625)10 → ( ? )8

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

### For Real Part-

• The real part is (2020)10
• We convert the real part from base 10 to base 16 using division method same as above.

So, (2020)10 = (7E4)16

### For Fractional Part-

• The fractional part is (0.65625)10
• We convert the fractional part from base 10 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

### 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.

• The fractional part terminates to 0 after 2 iterations.
• Traverse the real part column from top to bottom to obtain the required number in base 16.

From here, (0.65625)10 = (0.A8)8

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

(2020.65625)10 = (7E4.A8)16

### 3. (172)10

#### (172)10 → ( ? )16

Using division method, we have-

From here, (172)10 = (AC)16

### 4. (172.983)10

#### (172.983)10 → ( ? )16

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 16 using division method same as above.

So, (172)10 = (AC)16

### For Fractional Part-

• The fractional part is (0.983)10
• We 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 0.888

• 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 16.

From here, (0.983)10 = (0.FBA5)8

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

(172.983)10 = (AC.FBA5)16

Also Read- Decimal to Octal Conversion