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-
- (2020)_{10}
- (2020.65625)_{10}
- (172)_{10}
- (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:
Step-02:
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,
Next Article- Converting Any Base To Any Base
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