Relationship Between Number Systems

Number systems are different ways of representing numerical values. Although binary, octal, decimal, and hexadecimal use different digits and bases, they can all represent the same value.

Understanding the relationship between these number systems is important in ICT because computers use binary internally, while octal and hexadecimal provide shorter and more readable representations of binary data.

1. Number Systems and Their Bases

A number system is a method of representing numbers using a set of digits and rules. Each number system has a base, also known as its radix, which determines the number of distinct digits it uses.

The four commonly used number systems in ICT are:

Binary Number System

The binary number system has a base of 2 and uses only two digits: 0 and 1.

Computers use binary to represent and process data because digital electronic circuits can operate using two distinct states, commonly represented as 0 and 1.

Octal Number System

The octal number system has a base of 8 and uses eight digits, from 0 to 7.

Octal is closely related to binary because 8 is a power of 2. Each octal digit can represent exactly three binary digits.

Decimal Number System

The decimal number system has a base of 10 and uses ten digits, from 0 to 9.

It is the number system commonly used in everyday calculations. Each position in a decimal number represents a power of 10.

Hexadecimal Number System

The hexadecimal number system has a base of 16 and uses sixteen symbols: 0–9 and A–F.

The letters A, B, C, D, E, and F represent the decimal values 10, 11, 12, 13, 14, and 15, respectively.

Hexadecimal is commonly used in computing to represent binary data in a shorter form, such as memory addresses and colour codes.

2. Why Are Number Systems Related?

All number systems represent numerical values. The main difference between them is the base used to represent those values.

For example, the decimal number 26, the binary number 11010, the octal number 32, and the hexadecimal number 1A all represent the same quantity.

Computers process data using binary. However, long binary numbers can be difficult for humans to read and remember. Octal and hexadecimal provide more compact ways of representing binary values.

The relationship between these number systems is especially useful because the bases of octal and hexadecimal are powers of 2.

• Octal: (8 = 2^3)
• Hexadecimal: (16 = 2^4)

Therefore, binary numbers can be converted into octal and hexadecimal by grouping binary digits.

3. Relationship Between Binary and Octal

The binary and octal number systems are closely related because the base of octal is a power of 2.

Since (8 = 2^3), three binary digits correspond to one octal digit.

Each group of three binary digits can represent a value from 0 to 7, which matches the digits used in the octal system.

Converting Binary to Octal

To convert a binary number into octal:

1. Start grouping the binary digits from the right.
2. Divide the digits into groups of three.
3. Add leading zeros to the leftmost group if necessary.
4. Convert each group into its corresponding octal digit.
5. Write the octal digits in the same order.

Step 1: Group the binary digits into groups of three, starting from the right.

110 010

Step 2: Convert each group into its decimal equivalent.

• (110_2 = 6_8)
• (010_2 = 2_8)

Step 3: Combine the octal digits.

[ 110010_2 = 62_8 ]

Therefore, the binary number 110010 is equivalent to the octal number 62.

Converting Octal to Binary

To convert an octal number into binary, replace each octal digit with its equivalent three-bit binary representation.

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

• (6_8 = 110_2)
• (2_8 = 010_2)

Combine the binary groups.

[ 62_8 = 110010_2 ]

Thus, converting from octal to binary is straightforward because each octal digit corresponds to exactly three binary digits.

4. Relationship Between Binary and Hexadecimal

The binary and hexadecimal number systems are also closely related because hexadecimal has a base of 16.

Since (16 = 2^4), four binary digits correspond to one hexadecimal digit.

A group of four binary digits can represent any value from 0 to 15. These values correspond to the sixteen hexadecimal symbols, 0–9 and A–F.

Converting Binary to Hexadecimal

To convert a binary number into hexadecimal:

1. Start grouping the binary digits from the right.
2. Divide the digits into groups of four.
3. Add leading zeros to the leftmost group if necessary.
4. Convert each group into its corresponding hexadecimal digit.
5. Combine the hexadecimal digits in the same order.

Step 1: Group the binary digits into groups of four.

0001 1010

Step 2: Convert each group into its hexadecimal equivalent.

• (0001_2 = 1_{16})
• (1010_2 = A_{16})

Step 3: Combine the hexadecimal digits.

[ 00011010_2 = 1A_{16} ]

Therefore, the binary number 00011010 is equivalent to the hexadecimal number 1A.

Converting Hexadecimal to Binary

To convert a hexadecimal number into binary, replace each hexadecimal digit with its equivalent four-bit binary representation.

Replace each hexadecimal digit with its four-bit binary equivalent.

• (1_{16} = 0001_2)
• (A_{16} = 1010_2)

Combine the groups.

[ 1A_{16} = 00011010_2 ]

Therefore, the hexadecimal number 1A is equivalent to the binary number 00011010.

5. Representing the Same Value in Different Number Systems

A numerical value can be represented using different number systems. Although the digits and bases change, the actual value remains the same.

The decimal number 26 can be represented in binary, octal, and hexadecimal.

All four representations have the same numerical value.

Verifying the Binary Representation

The binary number 11010₂ can be expanded using positional weights.

[ \begin{aligned} 11010_2 &= 1 \times 2^4\ &+ 1 \times 2^3\ &+ 0 \times 2^2\ &+ 1 \times 2^1\ &+ 0 \times 2^0\ &=16+8+0+2+0\ &=26 \end{aligned} ]

Verifying the Octal Representation

The octal number 32₈ can be expanded using powers of 8.

[ 32_8 = 3 \times 8^1 + 2 \times 8^0 ]

[ =24+2=26 ]

Verifying the Hexadecimal Representation

The hexadecimal number 1A₁₆ contains the digits 1 and A. The hexadecimal digit A represents decimal 10.

[ 1A_{16}=1\times16^1+10\times16^0 ]

[ =16+10=26 ]

These calculations confirm that all four numbers represent the same value.

6. Relationship Between Binary, Octal, and Hexadecimal

Octal and hexadecimal are useful when working with binary numbers because they provide shorter representations.

The number of binary digits grouped depends on the relationship between the bases.

Binary to Octal Grouping

Every three binary digits form one octal digit.

[ 110\quad010 ]

[ 110_2=6_8 ]

[ 010_2=2_8 ]

Therefore:

[ 110010_2=62_8 ]

Binary to Hexadecimal Grouping

Every four binary digits form one hexadecimal digit.

First, add a leading zero to make the number of digits a multiple of four.

[ 0001\quad1010 ]

Convert each group:

• (0001_2=1_{16})
• (1010_2=A_{16})

Therefore:

[ 0011010_2=1A_{16} ]

The leading zero does not change the numerical value.

7. Converting Between Octal and Hexadecimal

Octal and hexadecimal do not have a direct one-digit-to-one-digit relationship. However, they can be converted through binary.

The conversion process is:

Step 1: Convert octal to binary.

Each octal digit is replaced with three binary digits.

[ 3_8=011_2 ]

[ 2_8=010_2 ]

Therefore:

[ 32_8=011010_2 ]

Step 2: Group the binary digits into groups of four from the right.

[ 0001\quad1010 ]

Step 3: Convert each group into hexadecimal.

• (0001_2=1_{16})
• (1010_2=A_{16})

Therefore:

[ 32_8=1A_{16} ]

This demonstrates that octal and hexadecimal can represent the same value through their relationship with binary.

8. Why Are Octal and Hexadecimal Used in Computing?

Computers internally process binary data, but long binary sequences can be difficult for people to read.

Octal and hexadecimal make binary values more compact.

For example:

[ 11111111_2 ]

The same value can be represented as:

[ 377_8 ]

or

[ FF_{16} ]

Instead of writing eight binary digits, hexadecimal uses only two digits.

Hexadecimal is commonly used in memory addresses, machine-level programming, and RGB colour codes. Octal is also used in some computing contexts, including Unix file permissions.

9. Important Examination Points

Students should remember the following facts when answering questions about the relationship between number systems.

• All number systems represent numerical values, but they use different bases and digits.
• Binary has base 2, octal has base 8, decimal has base 10, and hexadecimal has base 16.
• Octal is related to binary because (8=2^3). Therefore, three binary bits represent one octal digit.
• Hexadecimal is related to binary because (16=2^4). Therefore, four binary bits represent one hexadecimal digit.
• The same numerical value can have different representations in different number systems.
• Binary-to-octal and binary-to-hexadecimal conversions can be performed by grouping binary digits.
• Octal-to-hexadecimal conversion can be performed through binary.

10. Summary

Number systems are different methods of representing the same numerical values using different bases. Binary is the number system used internally by computers, while octal and hexadecimal provide shorter representations of binary data.

The relationship between binary and octal is based on powers of 2, with three binary bits corresponding to one octal digit. Similarly, four binary bits correspond to one hexadecimal digit.

Understanding these relationships makes it easier to convert numbers between systems and interpret how numerical data is represented in computers.