Unit 4: Number Systems - Subjective Questions

Number system: A number system is a method of representing numerical values using a set of symbols or digits according to specific rules.

Base or radix: The base of a number system is the total number of distinct digits available in that system. For a base- number system, the permitted digits range from to .

Positional value: The value of a digit depends on both the digit itself and its position. Positions to the left of the radix point have powers , while positions to the right have powers .

For example, in decimal:

Thus, the digit represents , the digit represents , and the digit represents .

The binary number system is a base- number system used by digital computers. It contains only two digits:

Each binary digit is called a bit. The positional weights, from right to left, are powers of : .

For example:

Computers use binary because electronic circuits can conveniently represent two stable states, such as on/off, high/low, or true/false.

In hexadecimal, the symbols have the following decimal values:

Binary, octal, and hexadecimal are closely related because and .

To convert a decimal integer to binary, repeatedly divide the number by and record each remainder. The remainders are then read from bottom to top.

Reading the remainders upward gives:

A positional check is:

Each binary digit is multiplied by the corresponding power of , starting with at the rightmost position:

Evaluating the powers gives:

Therefore:

A decimal integer is converted to octal by repeatedly dividing it by and recording the remainders.

Reading the remainders from bottom to top gives:

Verification:

In the octal system, each position has a weight that is a power of .

Substituting the positional weights:

Therefore:

The digits are valid because every octal digit must lie between and .

To convert a decimal integer to hexadecimal, repeatedly divide it by . Remainders from to are represented by to .

Reading the remainders from bottom to top gives:

Verification:

In hexadecimal, and . Expanding the number according to powers of gives:

Therefore:

Because , each octal digit corresponds to exactly three binary bits.

Binary to octal

Group the bits into sets of three from the radix point toward the left:

Using , , and :

Octal to binary

Replace every octal digit with its three-bit binary equivalent:

Therefore:

This direct grouping method avoids an intermediate conversion to decimal.

Since , one hexadecimal digit corresponds to exactly four binary bits.

Binary to hexadecimal

Divide the binary number into groups of four bits:

Thus:

Hexadecimal to binary

Replace each hexadecimal digit with four bits:

Therefore:

The initial zero may be omitted, giving:

Step 1: Octal to binary

Replace each octal digit with its three-bit binary equivalent:

Therefore:

Step 2: Binary to hexadecimal

Starting from the right, group the binary digits into sets of four. Add leading zeros where necessary:

Thus:

A decimal verification gives , and .

The integer and fractional parts are converted separately.

Integer part

Repeated division of by gives:

Fractional part

Repeatedly multiply the fraction by . At each step, record the integer part of the product:

Reading the integer bits from top to bottom gives .

Combining both parts:

Positions to the left of the binary point use non-negative powers of , while positions to the right use negative powers of .

Evaluating each term:

Therefore:

A number is valid only when all of its digits belong to the permitted digit set of its base.

• Binary: Permitted digits are and .
  • Valid:
  • Invalid: , because it contains .
• Octal: Permitted digits are through .
  • Valid:
  • Invalid: , because it contains .
• Decimal: Permitted digits are through .
  • Valid:
  • Invalid: , because is not a decimal digit.
• Hexadecimal: Permitted symbols are through and through .
  • Valid:
  • Invalid: , because is not a hexadecimal digit.

The largest single digit in a base- number system has the value .

Long binary numbers are difficult for people to read and write. Octal and hexadecimal provide shorter representations while allowing easy direct conversion to and from binary.

• Since , one octal digit represents three binary bits.
• Since , one hexadecimal digit represents four binary bits.

For example:

For octal, add a leading zero and group into threes:

For hexadecimal, group into fours:

Thus, the same value can be written as:

Hexadecimal is commonly used for memory addresses, machine-level values, debugging output, and color codes because it is compact and closely aligned with groups of four bits.

For a positional number in base written as:

its decimal value is obtained from:

where each digit satisfies .

For :

Therefore:

This formula can be used for any positional number system, including binary, octal, decimal, and hexadecimal.

Each number can be converted independently to decimal.

Binary value

Octal value

Hexadecimal value

In hexadecimal, :

Therefore:

This also illustrates how one value may have different representations in different bases.

Leading zeros placed before the most significant nonzero digit do not change the value of a number.

For example:

Leading zeros are often added to complete fixed-size groups during binary-to-octal or binary-to-hexadecimal conversion.

Trailing zeros after a radix point also do not change the value:

However, a zero added to the right of an integer with no radix point changes its positional value:

but

Similarly, inserting or removing a zero between significant digits can change the value. Therefore, zeros leave the value unchanged only when they are insignificant leading zeros or trailing fractional zeros.

Decimal to binary

Repeated division by gives:

Binary to octal

Add a leading zero and group the binary digits into sets of three:

These groups correspond to , , , and :

Binary to hexadecimal

Add a leading zero and group into sets of four:

The groups correspond to , , and :

Verification

Hence: