Convert the following numbers as directed. (a) (41.513)_{10} into binary, octal and hexadecimal (b) (41.513)_{10} into base 2, 8 and 10 numbers (c) (26.86)_{10} to base 2 and 10 numbers

Conversion of Decimal Numbers to Different Bases

Part (a) Convert (41.513)_{10} into binary, octal, and hexadecimal

Step 1: Convert the integer part (41) to binary, octal, and hexadecimal

Binary Conversion:

To convert the integer part to binary, we divide by 2 and keep track of the remainders.

41 ÷ 2 = 20 remainder 1
20 ÷ 2 = 10 remainder 0
10 ÷ 2 = 5  remainder 0
5 ÷ 2  = 2  remainder 1
2 ÷ 2  = 1  remainder 0
1 ÷ 2  = 0  remainder 1

Reading the remainders in reverse order, we get the binary representation: (101001)_2

Octal Conversion:

For octal, we divide by 8.

41 ÷ 8 = 5 remainder 1
5 ÷ 8  = 0 remainder 5

The octal representation is: (51)_8

Hexadecimal Conversion:

For hexadecimal, we divide by 16.

41 ÷ 16 = 2 remainder 9
2 ÷ 16  = 0 remainder 2

The hexadecimal representation is: (29)_16

Step 2: Convert the fractional part (0.513) to binary, octal, and hexadecimal

Binary Conversion:

To convert the fractional part to binary, we multiply by 2 and keep track of the integer parts.

0.513 × 2 = 1.026 (integer part 1)
0.026 × 2 = 0.052 (integer part 0)
0.052 × 2 = 0.104 (integer part 0)
0.104 × 2 = 0.208 (integer part 0)
0.208 × 2 = 0.416 (integer part 0)
0.416 × 2 = 0.832 (integer part 0)
0.832 × 2 = 1.664 (integer part 1)
...

This process can continue until the desired precision is reached. For this example, let's stop here and the binary representation of the fractional part is approximately: (100001)_2

Octal Conversion:

For octal, we multiply by 8.

0.513 × 8 = 4.104 (integer part 4)
0.104 × 8 = 0.832 (integer part 0)
0.832 × 8 = 6.656 (integer part 6)
...

Again, we can continue until the desired precision is reached. The octal representation of the fractional part is approximately: (40)_8

Hexadecimal Conversion:

For hexadecimal, we multiply by 16.

0.513 × 16 = 8.208 (integer part 8)
0.208 × 16 = 3.328 (integer part 3)
...

The hexadecimal representation of the fractional part is approximately: (83)_16

Final Results:

Base	Integer Part	Fractional Part	Combined
Binary	(101001)_2	(100001)_2	(101001.100001)_2
Octal	(51)_8	(40)_8	(51.40)_8
Hexadecimal	(29)_16	(83)_16	(29.83)_16

Part (b) Convert (41.513)_{10} into base 2, 8, and 10 numbers

The conversion for base 2, 8, and 10 numbers is the same as the conversion to binary, octal, and hexadecimal, respectively. Therefore, the results are the same as in part (a).

Part (c) Convert (26.86)_{10} to base 2 and 10 numbers

Convert the integer part (26) to binary

26 ÷ 2 = 13 remainder 0
13 ÷ 2 = 6  remainder 1
6 ÷ 2  = 3  remainder 0
3 ÷ 2  = 1  remainder 1
1 ÷ 2  = 0  remainder 1

The binary representation is: (11010)_2

Convert the fractional part (0.86) to binary

0.86 × 2 = 1.72 (integer part 1)
0.72 × 2 = 1.44 (integer part 1)
0.44 × 2 = 0.88 (integer part 0)
0.88 × 2 = 1.76 (integer part 1)
...

The binary representation of the fractional part is approximately: (1101)_2

Final Results for (26.86)_{10}:

Base	Integer Part	Fractional Part	Combined
Binary	(11010)_2	(1101)_2	(11010.1101)_2
Decimal	26	0.86	26.86

Note: For base 10, the number remains the same as the original decimal number, which is (26.86)_{10}.