Section 26: Miscellaneous #1

Number Systems

Binary: Base 2 \(\rightarrow \{0, 1\}\)
Octal: Base 8 \(\rightarrow \{0, 1, 2, 3, 4, 5, 6, 7\}\)
Decimal: Base 10 \(\rightarrow \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\)
Hexadecimal: Base 16 \(\rightarrow \{0, 1, \dots, 9, A, B, C, D, E, F\}\)

Comparison Table

Decimal	Binary	Octal	Hexadecimal
0	0	0	0
1	1	1	1
2	10	2	2
3	11	3	3
4	100	4	4
5	101	5	5
6	110	6	6
7	111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F

Conversion of Number Systems

1. Decimal to Binary

Method: Divide by 2 and take the remainder from end (bottom) to start (top).

Example: \(50_{10} \rightarrow (?)_{2}\)

\[ \begin{array}{r|l} 2 & 50 \\ \hline 2 & 25 \rightarrow 0 \\ \hline 2 & 12 \rightarrow 1 \\ \hline 2 & 6 \rightarrow 0 \\ \hline 2 & 3 \rightarrow 0 \\ \hline 2 & 1 \rightarrow 1 \\ \hline & 0 \rightarrow 1 \end{array} \]

Result: \(50_{10} = 110010_{2}\)

2. Decimal to Octal

Method: Divide by 8 and take the remainder from end to start.

Example: \(50_{10} \rightarrow (?)_{8}\)

\[ \begin{array}{r|l} 8 & 50 \\ \hline 8 & 6 \rightarrow 2 \\ \hline & 0 \rightarrow 6 \end{array} \]

Result: \(50_{10} = 62_{8}\)

3. Decimal to Hexadecimal

Method: Divide by 16 and take the remainder from end to start.

Example: \(50_{10} \rightarrow (?)_{16}\)

\[ \begin{array}{r|l} 16 & 50 \\ \hline 16 & 3 \rightarrow 2 \\ \hline & 0 \rightarrow 3 \end{array} \]

Result: \(50_{10} = 32_{16}\)

4. Binary to Decimal

Method: Sum of (Digit \(\times\) Base\(^{\text{position}}\)).

Example: \(11110_{2} \rightarrow (?)_{10}\)

\[ \begin{aligned} & 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\ = & 16 + 8 + 4 + 2 + 0 \\ = & 30 \end{aligned} \]

Result: \(11110_{2} = 30_{10}\)

5. Octal to Decimal

Example: \(34_{8} \rightarrow (?)_{10}\)

\[ \begin{aligned} & 3 \times 8^1 + 4 \times 8^0 \\ = & 24 + 4 \\ = & 28 \end{aligned} \]

Result: \(34_{8} = 28_{10}\)

6. Hexadecimal to Decimal

Example: \(1E_{16} \rightarrow (?)_{10}\)

\[ \begin{aligned} & 1 \times 16^1 + E(14) \times 16^0 \\ = & 16 + 14 \\ = & 30 \end{aligned} \]

Result: \(1E_{16} = 30_{10}\)

7. Octal to Binary

Method: Convert each octal digit to a 3-bit binary number.

Example: \(72_{8} \rightarrow (?)_{2}\)

\(7 \rightarrow 111\)
\(2 \rightarrow 010\)

Result: \(72_{8} = 111010_{2}\)

8. Binary to Octal

Method: Group bits in 3s from right to left (LSB to MSB).

Example: \(10110110_{2} \rightarrow (?)_{8}\)

\(110 \rightarrow 6\)
\(110 \rightarrow 6\)
\(010 \rightarrow 2\) (added padding zero)

Result: \(10110110_{2} = 266_{8}\)

9. Hexadecimal to Binary

Method: Convert each hex digit to a 4-bit binary number.

Example: \(3C4_{16} \rightarrow (?)_{2}\)

\(3 \rightarrow 0011\)
\(C \rightarrow 1100\)
\(4 \rightarrow 0100\)

Result: \(3C4_{16} = 1111000100_{2}\) (Leading zeros can be omitted)

10. Binary to Hexadecimal

Method: Group bits in 4s from right to left.

Example: \(10011001010_{2} \rightarrow (?)_{16}\)

\(1010 \rightarrow A\)
\(1100 \rightarrow C\)
\(0100 \rightarrow 4\) (added padding zeros)

Result: \(10011001010_{2} = 4CA_{16}\)

11. Octal to Hexadecimal

Note: There is no direct method. Strategy: Octal \(\rightarrow\) Binary \(\rightarrow\) Hexadecimal.

Example: \(276_{8} \rightarrow (?)_{16}\)

Octal to Binary:
- \(2 \rightarrow 010\)
- \(7 \rightarrow 111\)
- \(6 \rightarrow 110\)
- Binary: \(010111110_{2}\)
Binary to Hexadecimal:
- Group by 4s: 0000 1011 1110
- \(1110 \rightarrow E\)
- \(1011 \rightarrow B\)
- \(0 \rightarrow 0\)

Result: \(276_{8} = BE_{16}\)

12. Hexadecimal to Octal

Note: No direct method. Strategy: Hexadecimal \(\rightarrow\) Binary \(\rightarrow\) Octal.

Example: \(276_{16} \rightarrow (?)_{8}\)

Hexadecimal to Binary:
- \(2 \rightarrow 0010\)
- \(7 \rightarrow 0111\)
- \(6 \rightarrow 0110\)
- Binary: \(001001110110_{2}\)
Binary to Octal:
- Group by 3s: 001 001 110 110
- \(110 \rightarrow 6\)
- \(110 \rightarrow 6\)
- \(001 \rightarrow 1\)
- \(001 \rightarrow 1\)

Result: \(276_{16} = 1166_{8}\)