What is a Number System? : Definition, Types, Conversion Systems, Examples

What is a Number System in Maths?

A number system is a way to represent and work with numbers using a set of symbols and rules. It’s like a language for numbers, helping us perform calculations and solve problems. Different systems use different symbols and bases, which determine how numbers are expressed and understood. From everyday math to complex computing, number systems play a crucial role. In this article, we’ll explore the basics of number systems, look at different types, and learn how to convert numbers between these systems with clear examples. 

Definition of Number System

A number system is a structured way to represent and work with numbers. It defines a set of symbols or digits and rules for combining these symbols to express numbers. Each number system has its own base or radix, which determines how the symbols are used to represent different values.

Components of a Number System

Symbols or Digits:

These are the basic elements of a number system. For example, in the decimal system, the symbols are 0 through 9. In other systems, different symbols might be used, such as 0 and 1 in binary, or letters A through F in hexadecimal.

Base or Radix:

The base is the number of unique symbols used in the system, including zero. It determines how the value of each digit is calculated. For instance, the base of the decimal system is 10, meaning it uses 10 symbols. The base of the binary system is 2, so it uses only 2 symbols.

Place Value:

This is a key concept in understanding how numbers are represented. In a number system, the position of each digit affects its value. For example, in the decimal system, the position of each digit corresponds to a power of 10. This means that each digit has a different weight depending on where it is placed in the number.

How Number Systems Work?

In a number system, each digit’s place determines its value based on the base of the system. For example, in the decimal system, each place represents a power of 10:

• Units Place: 10^0
• Tens Place: 10^1
• Hundreds Place: 10^2
• Thousands Place: 10^3

So, in the number 345:

• The digit 5 is in the units place, representing 5 × 10^0 = 5
• The digit 4 is in the tens place, representing 4 × 10^1 = 40
• The digit 3 is in the hundreds place, representing 3 × 10^2 = 300

Adding these together, we get 345.

Importance of Number Systems

Understanding number systems is fundamental in many areas:

1. Mathematics: They help in solving problems, performing calculations, and understanding numerical relationships. Each number system has its own rules for arithmetic operations.
2. Computer Science: Computers primarily use binary numbers to process and store data. Knowing how to convert between binary and other number systems is essential for programming and digital design.
3. Engineering: Different number systems are used in designing and analyzing circuits, systems, and algorithms. For example, hexadecimal numbers are often used in programming and debugging.

Types of Numbers in Maths

In math, there are different types of numbers, each with its own special features. Here’s a simple guide to the main types:

1. Natural Numbers

• What They Are: These are the numbers we use for counting and ordering things.
• Examples: 1, 2, 3, 4, 5, …
• Characteristics: They start from 1 and go on forever. Zero is not included.

2. Whole Numbers

• What They Are: Whole numbers are like natural numbers but also include zero.
• Examples: 0, 1, 2, 3, 4, …
• Characteristics: They include all natural numbers plus zero.

3. Integers

• What They Are: Integers include all whole numbers and their negative counterparts.
• Examples: -3, -2, -1, 0, 1, 2, 3, …
• Characteristics: They include positive numbers, negative numbers, and zero.

4. Rational Numbers

• What They Are: Rational numbers are numbers that can be expressed as a fraction where both the top (numerator) and bottom (denominator) are integers.
• Examples: 1/2, 4/5, -3/7
• Characteristics: They can be whole numbers, fractions, or decimals that end or repeat.

5. Irrational Numbers

• What They Are: Irrational numbers cannot be written as a simple fraction. Their decimal form goes on forever without repeating.
• Examples: √2 (square root of 2), π (pi)
• Characteristics: They have non-repeating, non-terminating decimals.

6. Real Numbers

• What They Are: Real numbers include all the numbers on the number line. This means they include both rational and irrational numbers.
• Examples: -5, 0.75, √3, π
• Characteristics: They cover all types of numbers you can think of, including fractions, decimals, and irrational numbers.

7. Complex Numbers

• What They Are: Complex numbers include a real part and an imaginary part. The imaginary part involves the square root of -1, written as “i”.
• Examples: 3 + 4i, -2 – 5i
• Characteristics: They are used in advanced math and engineering to solve problems that can’t be solved with just real numbers.

Types of Number System in Maths

A number system is a way of representing and working with numbers. Different number systems use different bases or radix values, which determine how numbers are written and calculated. Here’s a detailed look at the main types of number systems:

1. Decimal Number System (Base-10)

• What It Is: The decimal system is the most common number system we use in everyday life.
• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• How It Works:
  • Each digit’s position in a number represents a power of 10.
  • For example, in the number 345:
    • 5 is in the units place (10^0), so it represents 5.
    • 4 is in the tens place (10^1), so it represents 40.
    • 3 is in the hundreds place (10^2), so it represents 300.
  • Adding these up, 345 = 300 + 40 + 5.

2. Binary Number System (Base-2)

• What It Is: The binary system is used mainly in computers and digital devices.
• Digits Used: 0, 1.
• How It Works:
  • Each digit’s position represents a power of 2.
  • For example, in the binary number 1011:
    • 1 is in the 2^3 place, so it represents 8.
    • 0 is in the 2^2 place, so it represents 0.
    • 1 is in the 2^1 place, so it represents 2.
    • 1 is in the 2^0 place, so it represents 1.
  • Adding these up, 1011 in binary equals 11 in decimal.

3. Octal Number System (Base-8)

• What It Is: The octal system is less common but used in some computing contexts.
• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7.
• How It Works:
  • Each digit’s position represents a power of 8.
  • For example, in the octal number 345:
    • 5 is in the 8^0 place, so it represents 5.
    • 4 is in the 8^1 place, so it represents 32.
    • 3 is in the 8^2 place, so it represents 192.
  • Adding these up, 345 in octal equals 229 in decimal.

4. Hexadecimal Number System (Base-16)

• What It Is: The hexadecimal system is often used in programming and digital electronics.
• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. (A = 10, B = 11, C = 12, D = 13, E = 14, F = 15)
• How It Works:
  • Each digit’s position represents a power of 16.
  • For example, in the hexadecimal number 1A3:
    • 3 is in the 16^0 place, so it represents 3.
    • A (10 in decimal) is in the 16^1 place, so it represents 160.
    • 1 is in the 16^2 place, so it represents 256.
  • Adding these up, 1A3 in hexadecimal equals 419 in decimal.

Conversion of Number Systems to Other Systems

Converting numbers between different systems is important for various applications, from computer programming to mathematical problems. Here’s a simple guide on how to convert between the most common number systems: decimal, binary, octal, and hexadecimal.

Conversion of Decimal Number System to Other Systems

Decimal to Binary Conversion

To convert a decimal number to binary:

1. Divide the decimal number by 2.
2. Record the remainder (0 or 1).
3. Update the decimal number to the quotient (result of the division).
4. Repeat the process until the quotient is 0.
5. Read the remainders backward (from last to first) to get the binary number.

Example: Convert 13 to binary

• 13 ÷ 2 = 6, remainder 1
• 6 ÷ 2 = 3, remainder 0
• 3 ÷ 2 = 1, remainder 1
• 1 ÷ 2 = 0, remainder 1

Reading the remainder backward, 13 in decimal is 1101 in binary.

Decimal to Octal Conversion

To convert a decimal number to octal:

1. Divide the decimal number by 8.
2. Record the remainder (0 through 7).
3. Update the decimal number to the quotient.
4. Repeat until the quotient is 0.
5. Read the remainders backward to get the octal number.

Example: Convert 156 to octal

• 156 ÷ 8 = 19, remainder 4
• 19 ÷ 8 = 2, remainder 3
• 2 ÷ 8 = 0, remainder 2

Reading the remainders backward, 156 in decimal is 234 in octal.

Decimal to Hexadecimal Conversion

To convert a decimal number to hexadecimal:

1. Divide the decimal number by 16.
2. Record the remainder (0-9 and A-F).
3. Update the decimal number to the quotient.
4. Repeat until the quotient is 0.
5. Read the remainders backward to get the hexadecimal number.

Example: Convert 254 to hexadecimal

• 254 ÷ 16 = 15, remainder 14 (E)
• 15 ÷ 16 = 0, remainder 15 (F)

Reading the remainders backward, 254 in decimal is FE in hexadecimal.

Conversion of Binary Number System to Other System

Binary to Decimal Conversion

To convert a binary number to decimal:

1. Write down the binary number.
2. Multiply each digit by 2 raised to its position power, starting from 0.
3. Add up the results.

Example: Convert 1011 to decimal

• 1 × 2^3 = 8
• 0 × 2^2 = 0
• 1 × 2^1 = 2
• 1 × 2^0 = 1

Adding these, 1011 in binary is 11 in decimal.

Binary to Octal Conversion

To convert binary to octal:

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

Example: Convert 110101 to octal

• Group: 110 101
• 110 in binary = 6 in octal
• 101 in binary = 5 in octal

So, 110101 in binary is 65 in octal.

Binary to Hexadecimal Conversion

To convert binary to hexadecimal:

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

Example: Convert 11010111 to hexadecimal

• Group: 1101 0111
• 1101 in binary = D in hexadecimal
• 0111 in binary = 7 in hexadecimal

So, 11010111 in binary is D7 in hexadecimal.

Conversion of Octal Number System to Other System

Octal to Decimal Conversion

To convert octal to decimal:

1. Write down the octal number.
2. Multiply each digit by 8 raised to its position power, starting from 0.
3. Add up the results.

Example: Convert 345 to decimal

• 5 × 8^0 = 5
• 4 × 8^1 = 32
• 3 × 8^2 = 192

Adding these, 345 in octal is 229 in decimal.

Octal to Binary Conversion

To convert octal to binary:

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

Example: Convert 745 to binary

• 7 = 111
• 4 = 100
• 5 = 101

So, 745 in octal is 111100101 in binary.

Octal to Hexadecimal Conversion

To convert octal to hexadecimal:

1. Convert octal to binary first.
2. Group binary digits into sets of four.
3. Convert each group to its hexadecimal equivalent.

Example: Convert 345 to hexadecimal

• Octal 345 → Binary 110100101
• Group: 0011 0100 1010
• 0011 = 3
• 0100 = 4
• 1010 = A

So, 345 in octal is 34A in hexadecimal.

Conversion of Hexadecimal Number System to Other System

Hexadecimal to Binary Conversion

To convert hexadecimal to binary:

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

Example: Convert 1A3 to binary

• 1 = 0001
• A (10 in decimal) = 1010
• 3 = 0011

So, 1A3 in hexadecimal is 000110100011 in binary.

Hexadecimal to Octal Conversion

To convert hexadecimal to octal:

1. Convert hexadecimal to binary first.
2. Group binary digits into sets of three.
3. Convert each group to its octal equivalent.

Example: Convert 1A3 to octal

• Hexadecimal 1A3 → Binary 0001 1010 0011
• Group: 001 101 000 011
• 001 = 1
• 101 = 5
• 000 = 0
• 011 = 3

So, 1A3 in hexadecimal is 1503 in octal.

These conversion methods help in understanding and working with different number systems used in various fields, including computing and mathematics.

What is a Number System? FAQs