Review of Number Systems and Number Base Conversions

Definition

A number system is a method of representing numbers using a set of digits and a base (also called radix). The base determines how many unique digits are used and the positional value of each digit.

A base conversion is the process of changing a number from one number system to another without changing its actual value.

For example:

• Decimal uses base 10 and digits 0–9
• Binary uses base 2 and digits 0–1
• Octal uses base 8 and digits 0–7
• Hexadecimal uses base 16 and digits 0–9 and A–F

The general positional value of a number can be written as:

$$(d_n \times b^n) + (d_{n-1} \times b^{n-1}) + \cdots + (d_1 \times b^1) + (d_0 \times b^0)$$

where:

• $d$ = digit
• $b$ = base of the number system
• $n$ = position index

Main Content

1. Number Systems and Their Place Values

Decimal number system

The decimal system is the most familiar number system in daily life. It has base 10 and uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Its positional values are powers of 10. For example, $345_{10}$ means: $$3 \times 10^2 + 4 \times 10^1 + 5 \times 10^0 = 300 + 40 + 5$$ This system is used because humans naturally count using ten fingers, making it intuitive for arithmetic and measurement.

Binary, octal, and hexadecimal systems

Binary is base 2 and uses only 0 and 1. It is the language of digital computers because electronic circuits can easily represent two states such as ON/OFF, HIGH/LOW, or TRUE/FALSE. Octal is base 8 and hexadecimal is base 16. These are compact ways of writing binary numbers. For example, the binary number $1111_2$ is written as $17_8$ in octal and $F_{16}$ in hexadecimal.
A place-value illustration for decimal and binary:

Decimal: $$583 = 5 \times 10^2 + 8 \times 10^1 + 3 \times 10^0$$

Binary: $$1011_2 = 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 11_{10}$$

Importance of positional notation

In all number systems, the value of a digit depends on its position. The same digit may represent very different values depending on where it appears. This is why reading a number requires knowledge of its base. For instance, $101$ in decimal means one hundred and one, but $101_2$ equals five in decimal. Understanding positional notation is the first step in mastering number-base conversion.

2. Binary, Octal, and Hexadecimal Relationships

Binary as the core representation

Binary is the fundamental internal representation in digital systems. Every piece of data in a computer—numbers, text, images, and instructions—is ultimately stored as bits (binary digits). A bit can be 0 or 1, and groups of bits are used to represent larger values. For example, 8 bits form a byte, which can represent values from 0 to 255 in unsigned form.

Octal and hexadecimal as shorthand forms

Since binary numbers can become long and difficult to read, octal and hexadecimal are used as compact notations. Octal groups binary digits in sets of 3 because $2^3 = 8$. Hexadecimal groups binary digits in sets of 4 because $2^4 = 16$. This grouping makes conversion easy and helps reduce errors.
Example: $$110101111010_2$$ Group into 3s from right for octal: $$110\ 101\ 111\ 010_2 = 6572_8$$ Group into 4s from right for hexadecimal: $$1101\ 0111\ 1010_2 = D7A_{16}$$

Hexadecimal digit mapping

Hexadecimal uses the symbols A to F for values 10 to 15:

• A = 10
• B = 11
• C = 12
• D = 13
• E = 14
• F = 15

This is especially useful in computer memory addressing, machine code representation, color coding in web design, and low-level programming. For example, the decimal value 255 is represented as FF in hexadecimal and 11111111 in binary.

3. Number Base Conversion Methods

Conversion from any base to decimal

To convert a number from any base to decimal, expand it using positional weights and add the results. This method works for binary, octal, hexadecimal, or any base.
Example: Convert $101101_2$ to decimal: $$1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$$ $$= 32 + 0 + 8 + 4 + 0 + 1 = 45_{10}$$ Example: Convert $2F_{16}$ to decimal: $$2 \times 16^1 + 15 \times 16^0 = 32 + 15 = 47_{10}$$

Conversion from decimal to another base

The most common method is repeated division for integers and repeated multiplication for fractional parts.
For an integer, divide by the base repeatedly and record the remainders. The remainders read upward give the converted number.
Example: Convert $25_{10}$ to binary:

• 25 ÷ 2 = 12 remainder 1
• 12 ÷ 2 = 6 remainder 0
• 6 ÷ 2 = 3 remainder 0
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Reading the remainders from bottom to top: $$25_{10} = 11001_2$$

Direct conversion between binary, octal, and hexadecimal

Since octal and hexadecimal are powers of 2, conversions among them and binary are done by grouping bits.
Binary to octal: group in 3s
Binary to hexadecimal: group in 4s
Octal to binary: replace each octal digit with 3-bit binary
Hexadecimal to binary: replace each hex digit with 4-bit binary

Example: $$7C_{16}$$ Convert each digit:

• 7 = 0111
• C = 1100

So, $$7C_{16} = 01111100_2$$

Working / Process

1. Identify the base of the given number and the target base

Before starting, determine whether the number is decimal, binary, octal, or hexadecimal. Also identify whether the conversion is to decimal or between non-decimal bases. This step is crucial because the method depends on the base involved.

2. Apply the correct conversion technique

• For base-to-decimal, use positional expansion.
• For decimal-to-base, use repeated division for whole numbers and repeated multiplication for fractional parts.
• For binary-octal or binary-hexadecimal, group bits in 3s or 4s.
• For octal/hexadecimal to binary, convert each digit separately into fixed-bit binary.

3. Verify the result and interpret the value correctly

After conversion, check whether the answer is consistent. You can verify by converting back to the original base or by comparing decimal equivalents. Ensure the correct use of digit symbols, especially in hexadecimal where letters A–F represent values beyond 9.

Advantages / Applications

Easy communication between humans and computers

Humans commonly use decimal, but computers use binary. Base conversions allow us to understand machine-level data and express it in a readable form.

Efficient representation of large binary numbers

Binary numbers are long and difficult to read. Octal and hexadecimal provide a shorter and cleaner way to represent the same data, reducing mistakes in programming and digital design.

Useful in computer architecture and digital electronics

Number systems are used in memory addresses, instruction codes, bitwise operations, color models, register values, and debugging. Hexadecimal is especially common in programming and hardware documentation.

Summary

• Number systems represent values using digits and a base, and their meaning depends on positional place value.
• Binary is the fundamental language of computers, while octal and hexadecimal are convenient compact forms of binary.
• Base conversion can be done using positional expansion, repeated division, repeated multiplication, or bit grouping.
• Number base conversion is essential for understanding digital systems, programming, and computer hardware.
• Common terms to remember include base, radix, digit, positional notation, binary, octal, hexadecimal, bit, nibble, and byte.