To convert decimal numbers to their hexadecimal equivalents, we follow the process of repeatedly dividing the decimal number by 16 and recording the remainders. The hexadecimal system uses base 16, and the digits in hexadecimal are 0-9 and A-F (where 'A' represents 10, 'B' represents 11, and so forth up to 'F' which represents 15).
### (a) [tex]\(256_{10}\)[/tex]
1. Divide 256 by 16. The quotient is 16 and the remainder is 0.
2. Divide the quotient 16 by 16. The new quotient is 1 and the remainder is 0.
3. Finally, divide 1 by 16. The quotient is 0 and the remainder is 1.
Reading the remainders from bottom to top, the hexadecimal representation of [tex]\(256_{10}\)[/tex] is [tex]\(100_{16}\)[/tex].
### (b) [tex]\(478_{10}\)[/tex]
1. Divide 478 by 16. The quotient is 29 and the remainder is 14.
- Note: The remainder 14 in hexadecimal is represented as 'E'.
2. Divide the quotient 29 by 16. The new quotient is 1 and the remainder is 13.
- Note: The remainder 13 in hexadecimal is represented as 'D'.
3. Finally, divide 1 by 16. The quotient is 0 and the remainder is 1.
Reading the remainders from bottom to top, the hexadecimal representation of [tex]\(478_{10}\)[/tex] is [tex]\(1DE_{16}\)[/tex].
### (c) [tex]\(1963_{10}\)[/tex]
1. Divide 1963 by 16. The quotient is 122 and the remainder is 11.
- Note: The remainder 11 in hexadecimal is represented as 'B'.
2. Divide the quotient 122 by 16. The new quotient is 7 and the remainder is 10.
- Note: The remainder 10 in hexadecimal is represented as 'A'.
3. Finally, divide 7 by 16. The quotient is 0 and the remainder is 7.
Reading the remainders from bottom to top, the hexadecimal representation of [tex]\(1963_{10}\)[/tex] is [tex]\(7AB_{16}\)[/tex].
### Summary
Thus, the hexadecimal conversions of the given decimal numbers are:
- [tex]\(256_{10} = 100_{16}\)[/tex]
- [tex]\(478_{10} = 1DE_{16}\)[/tex]
- [tex]\(1963_{10} = 7AB_{16}\)[/tex]
