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Decimal to Hexadecimal | Base 10 to base 16

Number System

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.

 

 

Decimal to Hexadecimal 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 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

 

PRACTICE PROBLEMS BASED ON DECIMAL TO HEXADECIMAL CONVERSION-

 

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

 

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.

 

  • 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

 

To gain better understanding about Decimal to Hexadecimal Conversion,

Watch this Video Lecture

 

Next Article- Converting Any Base To Any Base

 

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