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Number Systems

Crack the Code of Number Systems: A Journey from Everyday Numbers to Digital Mastery

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Introduction

Number systems are the foundation of mathematics, enabling us to count, measure, and perform calculations. They form the basis of arithmetic and various mathematical operations. In this blog, we’ll explore different types of number systems, the classification of numbers, conversions between them, and how to solve problems related to them.

Number System

What is a Number System?

A number system is a way to represent numbers in a consistent manner. It uses a specific set of symbols (called digits) and a base to express quantities. The base is the number of unique digits, including zero, used in a system.

For example, the decimal system (our common system) uses 10 digits: 0 through 9. Number systems are critical in fields like computer science, digital electronics, and everyday arithmetic operations.

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Section 1

Types of Number Systems

There are four primary types of number systems:

1) Binary Number System (Base 2)

The binary number system is a base 2 system, consisting only of two digits: 0 and 1. This system is essential for computers and digital circuits.

  • Example: 1011 (Binary representation of the decimal number 11)
  • Positional Value: In binary, the position of each digit represents a power of 2. For example, in the binary number 1101:
  • 1101=1×23+1×22+0×21+1×20=13

3) Decimal Number System (Base 10)

The decimal number system is the most familiar system, as it’s the one we use in everyday life. It is a base 10 system, meaning it consists of 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

  • Example: 534, 89, 1021
  • Positional Value: In this system, the position of each digit represents a power of 10. For example, in the number 345:
  • 345 = 3×10^2 + 4×10^1 + 5×10^0

2) Octal Number System (Base 8)

The octal number system uses base 8, meaning it consists of 8 digits: 0, 1, 2, 3, 4, 5, 6, and 7. This system is primarily used in computer applications.

  • Example: 17 (Octal representation of the decimal number 15)
  • Positional Value: In the octal system, each position is a power of 8. For example, the octal number 135:
  • 135 = 1×8^2 + 3×8^1 + 5×8^0 = 93

4) Hexadecimal Number System (Bs = 16)

The hexadecimal number system uses base 16, which means it consists of 16 digits. The digits are 0-9 and the letters A, B, C, D, E, and F (where A=10, B=11, C=12, D=13, E=14, F=15).

  • Example: A9 (Hexadecimal representation of the decimal number 169)
  • Positional Value: In the hexadecimal system, each position represents a power of 16. For example, the hexadecimal number 1A3:
  • 1A3= 1×16^2 + 10×16^1 + 3×16^0 = 419
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Section 1

Prime Numbers and Composite Numbers

Discover the distinction between prime and composite numbers, essential building blocks of mathematics.

Prime Numbers

A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. In other words, a prime number is only divisible by 1 and the number itself.

  • Example: 2, 3, 5, 7, 11, 13, 17, 19, etc.

Composite Numbers

A composite number is a number greater than 1 that has more than two divisors. In other words, it is divisible by numbers other than 1 and itself.

  • Example: 4, 6, 8, 9, 12, 15, etc.
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Section 3

Types of Number Systems

Numbers can be classified into different categories:

Natural Numbers

Natural numbers are the counting numbers starting from 1 and going upwards: 1, 2, 3, 4, and so on. They are represented as N.

Whole Numbers

Whole numbers are natural numbers including zero. Hence, whole numbers are 0, 1, 2, 3, 4, and so on.

Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction p/q​, where p and q are integers and q≠0. Rational numbers include both terminating and repeating decimals.

Example: 3/4​, 0.75, 0.333…

Even Numbers

An even number is any integer that is divisible by 2.

Example: 2, 4, 6, 8, 10, etc.

Integers

Integers are whole numbers that can be positive, negative, or zero. This set of numbers is represented as Z.

Example: -3, -2, -1, 0, 1, 2, 3, etc.

Irrational Numbers

Irrational numbers cannot be expressed as a fraction p/q​. Their decimal expansion is non-terminating and non-repeating.

Example: π \ sqrt{2}

Real Numbers

Real numbers include both rational and irrational numbers. They represent all numbers that can be found on the number line.

Odd Numbers

An odd number is any integer that is not divisible by 2.

Example: 1, 3, 5, 7, 9, etc.

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Section 4

Divisibility Rules

Divisibility rules are shortcuts that help you determine if a number can be evenly divided by another number without actually performing the division. These rules are useful for quickly finding factors of numbers, simplifying fractions, and solving various arithmetic problems.

Let’s go over some of the most common divisibility rules.

Divisibility by 2

Rule: A number is divisible by 2 if its last digit (the unit digit) is even.

Even digits: 0, 2, 4, 6, 8

Examples:

  • 48 is divisible by 2 because the last digit is 8 (even).
  • 123 is not divisible by 2 because the last digit is 3 (odd).

Divisibility by 3

Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.

Steps: Add the digits of the number together, and check if the result is divisible by 3.

Examples:

  • 123: The sum of digits is 1+2+3=6, and since 6 is divisible by 3, 123 is divisible by 3.
  • 529: The sum of digits is 5+2+9=16. Since 16 is not divisible by 3, 529 is not divisible by 3.

Divisibility by 4

Rule: A number is divisible by 4 if the last two digits (tens and units) form a number divisible by 4.

Examples:

  • 324: The last two digits are 24, and since 24 is divisible by 4, 324 is divisible by 4.
  • 679: The last two digits are 79, which is not divisible by 4, so 679 is not divisible by 4.

Divisibility by 5

Rule: A number is divisible by 5 if its last digit is either 0 or 5.

Examples:

  • 145 is divisible by 5 because its last digit is 5.
  • 127 is not divisible by 5 because its last digit is 7.

Divisibility by 6

Rule: A number is divisible by 6 if it is divisible by both 2 and 3.

Examples:

  • 72: The last digit is 2 (even, divisible by 2), and the sum of the digits is 7+2=9 (divisible by 3), so 72 is divisible by 6.
  • 55: The last digit is not even, so it is not divisible by 2, and thus not divisible by 6 either.

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Section 4

Divisibility Rules

Divisibility rules are shortcuts that help you determine if a number can be evenly divided by another number without actually performing the division. These rules are useful for quickly finding factors of numbers, simplifying fractions, and solving various arithmetic problems.

Let’s go over some of the most common divisibility rules.

Divisibility by 2

Rule: A number is divisible by 2 if its last digit (the unit digit) is even.

Even digits: 0, 2, 4, 6, 8

Examples:

  • 48 is divisible by 2 because the last digit is 8 (even).
  • 123 is not divisible by 2 because the last digit is 3 (odd).

Divisibility by 3

Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.

Steps: Add the digits of the number together, and check if the result is divisible by 3.

Examples:

  • 123: The sum of digits is 1+2+3=6, and since 6 is divisible by 3, 123 is divisible by 3.
  • 529: The sum of digits is 5+2+9=16. Since 16 is not divisible by 3, 529 is not divisible by 3.

Divisibility by 4

Rule: A number is divisible by 4 if the last two digits (tens and units) form a number divisible by 4.

Examples:

  • 324: The last two digits are 24, and since 24 is divisible by 4, 324 is divisible by 4.
  • 679: The last two digits are 79, which is not divisible by 4, so 679 is not divisible by 4.

Divisibility by 5

Rule: A number is divisible by 5 if its last digit is either 0 or 5.

Examples:

  • 145 is divisible by 5 because its last digit is 5.
  • 127 is not divisible by 5 because its last digit is 7.

Divisibility by 6

Rule: A number is divisible by 6 if it is divisible by both 2 and 3.

Examples:

  • 72: The last digit is 2 (even, divisible by 2), and the sum of the digits is 7+2=9 (divisible by 3), so 72 is divisible by 6.
  • 55: The last digit is not even, so it is not divisible by 2, and thus not divisible by 6 either.
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Section 5

Conversions Between Number Systems

Converting between number systems is crucial, especially in fields like computer science, where binary, octal, and hexadecimal systems are often used. Let’s go through the most common conversions step by step.

Binary to Decimal Conversion

To convert a binary number to a decimal number, each binary digit is multiplied by 2 raised to the power of its position (starting from 0 at the rightmost bit). Then, all these values are summed together to get the decimal equivalent.

Steps to Convert Binary to Decimal:

  • Write down the binary number.
  • Assign powers of 2 starting from the right (from 2^0).
  • Multiply each binary digit by the corresponding power of 2.
  • Sum the results to get the decimal number.

Example: Convert Binary 1010​ to decimal.

= 1×2^3 + 0×2^2 + 1×2^1 + 0×2^0

= 1×8 + 0×4 + 1×2 + 0×1

= 8 + 0 + 2 + 0

= 10​

Decimal to Binary Conversion

To convert a decimal number to a binary number, repeatedly divide the decimal number by 2, keeping track of the remainders. The binary equivalent is obtained by reading the remainders in reverse order.

Steps to Convert Decimal to Binary:

  • Divide the decimal number by 2.
  • Write down the remainder (0 or 1).
  • Continue dividing the quotient by 2 until the quotient is 0.
  • The binary number is the sequence of remainders read from bottom to top.

Example: Convert 10 decimal​ to binary.

  • 10÷2= 5 remainder0
  • 5÷2= 2 remainder 1
  • 2÷2= 1 remainder 0
  • 1÷2= 0 remainder 1

Reading the remainders from bottom to top, we get: 1010 binary.

Octal to Binary Conversion

To convert an octal number to binary, each octal digit is converted to its 3-digit binary equivalent.

Steps to Convert Octal to Binary:

  • Write down the octal number.
  • Convert each octal digit to its 3-bit binary equivalent.
  • Concatenate all the binary groups.

Example: Convert 56 Octal to binary.

Convert each octal digit to binary:

5 Octal ​=101 Binary

​6 Octal​=110 Binary

Binary to Hexadecimal Conversion

To convert binary to hexadecimal, group the binary digits in sets of four, starting from the right. Each group is then converted to its hexadecimal equivalent. Example: Convert 11011011

Binary 1101​ =D Hexa​,

Binary​ 1011=B Hexa​

8 4 2 1 Dec
0 0 0 0 0
0 0 0 1 1
0 0 1 0 2
0 0 1 1 3
0 1 0 0 4
0 1 0 1 5
0 1 1 0 6
0 1 1 1 7
1 0 0 0 8
1 0 0 1 9
1 0 1 0 A (10)
1 0 1 1 B (11)
1 1 0 0 C (12)
1 1 0 1 D (13)
1 1 1 0 E (14)
1 1 1 1 F (15)

Binary to Octal Conversion

Binary numbers can be converted to octal by grouping the binary digits in sets of three, starting from the right. Each group is then converted to its octal equivalent.

Steps to Convert Binary to Octal:

  • Start from the right, group the binary digits in sets of three.
  • Convert each group to its octal equivalent.
  • If the leftmost group has fewer than three digits, add leading zeros.

Example: Convert 101001​ to octal.

  • First, group the digits in sets of three: 101 001 binary
  • Now, convert each group to its octal equivalent:

Binary 101​=5 Octal Conversion​

Binary 001​=1 Octal Conversion

Hexadecimal to Binary Conversion

To convert a hexadecimal number to binary, each hexadecimal digit is converted to its 4-bit binary equivalent.

Steps to Convert Hexadecimal to Binary:

  • Write down the hexadecimal number.
  • Convert each hexadecimal digit to its 4-bit binary equivalent.
  • Concatenate all the binary groups.

Example: Convert 3F​ to binary.

3 Hexa = 0011 Binary

​F Hexa = 1111 Binary

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Section 5

Conversions Between Number Systems

Converting between number systems is crucial, especially in fields like computer science, where binary, octal, and hexadecimal systems are often used. Let’s go through the most common conversions step by step.

Binary to Decimal Conversion

To convert a binary number to a decimal number, each binary digit is multiplied by 2 raised to the power of its position (starting from 0 at the rightmost bit). Then, all these values are summed together to get the decimal equivalent.

Steps to Convert Binary to Decimal:

  • Write down the binary number.
  • Assign powers of 2 starting from the right (from 2^0).
  • Multiply each binary digit by the corresponding power of 2.
  • Sum the results to get the decimal number.

Example: Convert Binary 1010​ to decimal.

= 1×2^3 + 0×2^2 + 1×2^1 + 0×2^0

= 1×8 + 0×4 + 1×2 + 0×1

= 8 + 0 + 2 + 0

= 10​

Decimal to Binary Conversion

To convert a decimal number to a binary number, repeatedly divide the decimal number by 2, keeping track of the remainders. The binary equivalent is obtained by reading the remainders in reverse order.

Steps to Convert Decimal to Binary:

  • Divide the decimal number by 2.
  • Write down the remainder (0 or 1).
  • Continue dividing the quotient by 2 until the quotient is 0.
  • The binary number is the sequence of remainders read from bottom to top.

Example: Convert 10 decimal​ to binary.

  • 10÷2= 5 remainder0
  • 5÷2= 2 remainder 1
  • 2÷2= 1 remainder 0
  • 1÷2= 0 remainder 1

Reading the remainders from bottom to top, we get: 1010 binary.

Octal to Binary Conversion

To convert an octal number to binary, each octal digit is converted to its 3-digit binary equivalent.

Steps to Convert Octal to Binary:

  • Write down the octal number.
  • Convert each octal digit to its 3-bit binary equivalent.
  • Concatenate all the binary groups.

Example: Convert 56 Octal to binary.

Convert each octal digit to binary:

5 Octal ​=101 Binary

​6 Octal​=110 Binary

Binary to Hexadecimal Conversion

To convert binary to hexadecimal, group the binary digits in sets of four, starting from the right. Each group is then converted to its hexadecimal equivalent. Example: Convert 11011011

Binary 1101​ =D Hexa​,

Binary​ 1011=B Hexa​

8 4 2 1 Dec
0 0 0 0 0
0 0 0 1 1
0 0 1 0 2
0 0 1 1 3
0 1 0 0 4
0 1 0 1 5
0 1 1 0 6
0 1 1 1 7
1 0 0 0 8
1 0 0 1 9
1 0 1 0 A
1 0 1 1 B
1 1 0 0 C
1 1 0 1 D
1 1 1 0 E
1 1 1 1 F

Binary to Octal Conversion

Binary numbers can be converted to octal by grouping the binary digits in sets of three, starting from the right. Each group is then converted to its octal equivalent.

Steps to Convert Binary to Octal:

  • Start from the right, group the binary digits in sets of three.
  • Convert each group to its octal equivalent.
  • If the leftmost group has fewer than three digits, add leading zeros.

Example: Convert 101001​ to octal.

  • First, group the digits in sets of three: 101 001 binary
  • Now, convert each group to its octal equivalent:

Binary 101​=5 Octal Conversion​

Binary 001​=1 Octal Conversion

Hexadecimal to Binary Conversion

To convert a hexadecimal number to binary, each hexadecimal digit is converted to its 4-bit binary equivalent.

Steps to Convert Hexadecimal to Binary:

  • Write down the hexadecimal number.
  • Convert each hexadecimal digit to its 4-bit binary equivalent.
  • Concatenate all the binary groups.

Example: Convert 3F​ to binary.

3 Hexa = 0011 Binary

​F Hexa = 1111 Binary

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MCQ Test

Number Systems Quiz

Here are 30 basic-level General Aptitude questions, each with multiple-choice options and detailed answers

min
Beginner Level
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Moderate Level
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Challenge Level
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Expert Level
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Section 6

Conclusion

Number systems play a crucial role in mathematics and computer science. By understanding different types of number systems, how to convert between them, and basic operations like divisibility rules, you’ll have a solid foundation in number theory. Whether you’re preparing for competitive exams or delving into computer science, mastering these concepts is essential.

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