Number System and Conversions NIMCET - CUET Applied Maths

Number Systems and Conversions

Number systems are foundational in mathematics and computing, involving different bases such as binary, octal, decimal, and hexadecimal. Understanding conversions between these systems is crucial for competitive exams and computer science fundamentals. Each system uses a specific set of digits and base, and conversions require systematic methods.

Types of Number Systems and Their Bases

There are four primary number systems: binary, octal, decimal, and hexadecimal. Each system is defined by its base and the range of digits it uses. Recognizing the base and valid digits is essential for accurate conversions and answering direct exam…

Number Systems and Conversions

Number systems are foundational in mathematics and computing, involving different bases such as binary, octal, decimal, and hexadecimal. Understanding conversions between these systems is crucial for competitive exams and computer science fundamentals. Each system uses a specific set of digits and base, and conversions require systematic methods.

Types of Number Systems and Their Bases

There are four primary number systems: binary, octal, decimal, and hexadecimal. Each system is defined by its base and the range of digits it uses. Recognizing the base and valid digits is essential for accurate conversions and answering direct exam questions.

• Binary (Base 2): Digits 0, 1
• Octal (Base 8): Digits 0 to 7
• Decimal (Base 10): Digits 0 to 9
• Hexadecimal (Base 16): Digits 0 to 9 and A to F (A=10, B=11, C=12, D=13, E=14, F=15)

Binary to Decimal Conversion

To convert a binary number to decimal, multiply each bit by $2$ raised to the power of its position, starting from the rightmost digit (least significant bit). For fractional binary numbers, use negative powers for digits after the point.

• Start from the rightmost digit, multiply by $2^0$, then $2^1$, $2^2$, etc., moving left.
• For fractional parts, multiply by $2^{-1}$, $2^{-2}$, etc., moving right from the point.
• Add all results to get the decimal equivalent.

Example:
Binary $1101.001$

• Integer part: $1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 4 + 0 + 1 = 13$
• Fractional part: $0 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} = 0 + 0 + 0.125$
• Decimal equivalent: $13.125$

Decimal to Binary Conversion

Converting decimal to binary involves dividing the integer part by 2 and recording remainders, and multiplying the fractional part by 2 and recording the integer parts.

• For the integer part:
  • Divide by 2, record the remainder.
  • Repeat with the quotient until it is less than 2.
  • Write remainders from bottom to top.
• For the fractional part:
  • Multiply by 2, record the integer part.
  • Repeat with the new fractional part.
  • Write results from top to bottom.

Example:
Decimal $42.50$

• Integer part: $42 \div 2 \rightarrow$ remainders (from last to first): $1, 0, 1, 0, 1, 0$ → $101010$
• Fractional part: $0.50 \times 2 = 1.00$ (record 1), $0.00 \times 2 = 0$ (record 0) → $.10$
• Binary: $101010.10$

Binary–Octal and Octal–Binary Conversion

Binary and octal conversions use grouping of bits. Three binary digits correspond to one octal digit. For numbers with points, group from the point outward, adding zeros as needed.

• Binary to Octal:

  • Group binary digits in sets of three from right (integer) or left (fractional).
  • Add leading/trailing zeros if necessary.
  • Convert each group to its octal equivalent using a table.

• Octal to Binary:

  • Replace each octal digit with its 3-bit binary equivalent.

Example:
Binary $1010110$

• Groups: $1\,010\,110$ (add leading zero to make three digits: $001\,010\,110$)
• Octal: $1\,2\,6$

Octal $734$

• $7 \rightarrow 111$, $3 \rightarrow 011$, $4 \rightarrow 100$
• Binary: $111011100$

Binary–Hexadecimal and Hexadecimal–Binary Conversion

Four binary digits correspond to one hexadecimal digit. Grouping is done in sets of four, and zeros are added as needed. Hexadecimal digits A–F represent values 10–15.

• Binary to Hexadecimal:

  • Group binary digits in sets of four from right (integer) or left (fractional).
  • Add zeros as needed.
  • Convert each group to its hexadecimal equivalent.

• Hexadecimal to Binary:

  • Replace each hexadecimal digit with its 4-bit binary equivalent.

Example:
Binary $1000010$

• Groups: $0100\,0010$ (add leading zero: $0100\,0010$)
• Hexadecimal: $4\,2$

Hexadecimal $7C$

• $7 \rightarrow 0111$, $C \rightarrow 1100$
• Binary: $01111100$

Octal–Hexadecimal and Hexadecimal–Octal Conversion

There is no direct single-step method for octal–hexadecimal conversion. The process involves two steps: convert to binary, then to the target system.

• Octal to Hexadecimal:

  • Convert octal to binary (each digit to 3 bits).
  • Group binary digits in sets of four for hexadecimal.
  • Convert each group to hexadecimal.

• Hexadecimal to Octal:

  • Convert hexadecimal to binary (each digit to 4 bits).
  • Group binary digits in sets of three for octal.
  • Convert each group to octal.

Example:
Octal $642$

• $6 \rightarrow 110$, $4 \rightarrow 100$, $2 \rightarrow 010$ → Binary: $110100010$
• Group as $0011\,0100\,010$ (add leading zero): $0011\,0100\,0010$
• Hexadecimal: $3\,4\,2$

Conversion to Decimal from Octal and Hexadecimal

To convert octal or hexadecimal numbers to decimal, multiply each digit by the base raised to the power of its position, starting from the rightmost digit.

• Octal to Decimal:

  • Multiply each digit by $8^n$, where $n$ is the position from right (starting at 0).
  • Add all results.

• Hexadecimal to Decimal:

  • Replace A–F with 10–15.
  • Multiply each digit by $16^n$, where $n$ is the position from right (starting at 0).
  • Add all results.

Example:
Octal $726$

• $6 \times 8^0 = 6$
• $2 \times 8^1 = 16$
• $7 \times 8^2 = 448$
• Total: $470$

Hexadecimal $C8A4$

• $C=12$, $A=10$
• $4 \times 16^0 = 4$
• $A \times 16^1 = 160$
• $8 \times 16^2 = 2048$
• $C \times 16^3 = 49152$
• Total: $51364$

Conversion from Decimal to Octal and Hexadecimal

Decimal to octal and hexadecimal conversions use repeated division by the target base, recording remainders.

• Decimal to Octal:

  • Divide by 8, record remainders.
  • Write remainders from bottom to top.

• Decimal to Hexadecimal:

  • Divide by 16, record remainders.
  • Replace remainders 10–15 with A–F.
  • Write remainders from bottom to top.

Example:
Decimal $93$ to octal:

• $93 \div 8 = 11$, remainder $5$
• $11 \div 8 = 1$, remainder $3$
• $1 \div 8 = 0$, remainder $1$
• Octal: $135$

Decimal $90$ to hexadecimal:

• $90 \div 16 = 5$, remainder $10$ ($A$)
• $5 \div 16 = 0$, remainder $5$
• Hexadecimal: $5A$

Handling Fractional Parts in Conversions

For numbers with fractional parts, the conversion method differs for the part after the point. Multiplication is used for the fractional part, and division for the integer part.

• Decimal Fraction to Binary:

  • Multiply the fractional part by 2.
  • Record the integer part.
  • Repeat with the new fractional part.
  • Write results from top to bottom.

• Decimal Fraction to Octal/Hexadecimal:

  • Multiply the fractional part by 8 (for octal) or 16 (for hexadecimal).
  • Record the integer part.
  • Repeat as needed.

Example:
Decimal $25.625$ to binary:

• Integer: $25 \div 2$ repeatedly → $11001$
• Fraction: $0.625 \times 2 = 1.25$ (record 1), $0.25 \times 2 = 0.5$ (record 0), $0.5 \times 2 = 1.0$ (record 1)
• Binary: $11001.101$

Decimal $0.625$ to hexadecimal:

• $0.625 \times 16 = 10.0$ (record 10, which is $A$)
• Hexadecimal: $.A$

Key Conversion Shortcuts and Rules

Certain shortcuts and rules help in remembering and applying conversions efficiently. The base of the system determines the multiplication or division factor.

• To convert to decimal: multiply each digit by the base raised to the appropriate power.
• To convert from decimal: divide by the target base, record remainders.
• For fractional parts: multiply by the target base, record integer parts.
• For octal–hexadecimal or hexadecimal–octal: always convert via binary as an intermediate step.
• For binary–octal: group in threes; for binary–hexadecimal: group in fours.

Common Pitfalls and Exam Tips

Mistakes often occur in direction of grouping, handling fractional parts, and replacing hexadecimal digits. Careful attention to these details ensures accuracy.

• Always group from the correct side of the point (left for integer, right for fraction).
• Add leading or trailing zeros to complete groups as needed.
• Replace hexadecimal remainders above 9 with A–F.
• For fractional conversions, 3–4 steps are usually sufficient to match options in exams.
• No direct conversion exists between octal and hexadecimal; always use binary as an intermediate.
• Calculators are not allowed; practice manual methods for speed and accuracy.

Reference Table for Quick Conversion

The following table summarizes the mapping between binary, octal, and hexadecimal digits as explicitly described: