Number System Conversion

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Number System Conversion

Converting between number systems involves using systematic methods. Below are the detailed steps for converting between Decimal, Binary, Octal, and Hexadecimal systems.

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1. Decimal to Other Systems

Decimal to Binary:

1. Divide the decimal number by 2.
2. Record the remainder (0 or 1).
3. Repeat the division until the quotient is 0.
4. Write the remainders in reverse order.

Example: Convert $25_{10}$ to Binary.

• $25 \div 2 = 12$ R $1$
• $12 \div 2 = 6$ R $0$
• $6 \div 2 = 3$ R $0$
• $3 \div 2 = 1$ R $1$
• $1 \div 2 = 0$ R $1$
  Result: $11001_2$.

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Decimal to Octal:

1. Divide the decimal number by 8.
2. Record the remainder.
3. Repeat until the quotient is 0.
4. Write the remainders in reverse order.

Example: Convert $45_{10}$ to Octal.

• $45 \div 8 = 5$ R $5$
• $5 \div 8 = 0$ R $5$
  Result: $55_8$.

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Decimal to Hexadecimal:

1. Divide the decimal number by 16.
2. Record the remainder (0-9 or A-F).
3. Repeat until the quotient is 0.
4. Write the remainders in reverse order.

Example: Convert $175_{10}$ to Hexadecimal.

• $175 \div 16 = 10$ R $15 (F)$
• $10 \div 16 = 0$ R $10 (A)$
  Result: $AF_{16}$.

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2. Binary to Other Systems

Binary to Decimal:

1. Multiply each binary digit by $2^{\text{position}}$ (from right to left, starting at 0).
2. Sum all the values.

Example: Convert $11001_2$ to Decimal.

• $(1 \times 2^4) + (1 \times 2^3) + (0 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) = 25_{10}$.

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Binary to Octal:

1. Group binary digits in sets of 3, starting from the right.
2. Convert each group to its octal equivalent.

Example: Convert $11001_2$ to Octal.

• Group: $011 \, 001$.
• Convert: $011_2 = 3_8$, $001_2 = 1_8$.
  Result: $31_8$.

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Binary to Hexadecimal:

1. Group binary digits in sets of 4, starting from the right.
2. Convert each group to its hexadecimal equivalent.

Example: Convert $11001_2$ to Hexadecimal.

• Group: $0001 \, 1001$.
• Convert: $0001_2 = 1_{16}$, $1001_2 = 9_{16}$.
  Result: $19_{16}$.

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3. Octal to Other Systems

Octal to Decimal:

1. Multiply each digit by $8^{\text{position}}$ (from right to left, starting at 0).
2. Sum all the values.

Example: Convert $57_8$ to Decimal.

• $(5 \times 8^1) + (7 \times 8^0) = 40 + 7 = 47_{10}$.

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Octal to Binary:

1. Convert each octal digit to its 3-bit binary equivalent.

Example: Convert $57_8$ to Binary.

• $5_8 = 101_2$, $7_8 = 111_2$.
  Result: $101111_2$.

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Octal to Hexadecimal (via Binary):

1. Convert the octal number to binary.
2. Group the binary digits in sets of 4.
3. Convert each group to hexadecimal.

Example: Convert $57_8$ to Hexadecimal.

• $57_8 = 101111_2$.
• Group: $0001 \, 0111$.
• Convert: $0001_2 = 1_{16}$, $0111_2 = 7_{16}$.
  Result: $17_{16}$.

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4. Hexadecimal to Other Systems

Hexadecimal to Decimal:

1. Multiply each digit by $16^{\text{position}}$ (from right to left, starting at 0).
2. Sum all the values.

Example: Convert $1A_{16}$ to Decimal.

• $(1 \times 16^1) + (10 \times 16^0) = 16 + 10 = 26_{10}$.

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Hexadecimal to Binary:

1. Convert each hexadecimal digit to its 4-bit binary equivalent.

Example: Convert $1A_{16}$ to Binary.

• $1_{16} = 0001_2$, $A_{16} = 1010_2$.
  Result: $00011010_2$.

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Hexadecimal to Octal (via Binary):

1. Convert the hexadecimal number to binary.
2. Group the binary digits in sets of 3.
3. Convert each group to octal.

Example: Convert $1A_{16}$ to Octal.

• $1A_{16} = 00011010_2$.
• Group: $000 \, 110 \, 100$.
• Convert: $000_2 = 0_8$, $110_2 = 6_8$, $100_2 = 4_8$.
  Result: $064_8$.

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Would you like to practice any specific conversion or need a detailed explanation of a particular step?

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