Revision Notes: Number System | Mathematics Class 6 ICSE PDF Download

What is a Number?

• A number is a value used to count, measure, or label things. 
  For example, 5 apples or 10 books.
• A numeral is a symbol that represents a number. 
  For example, the numeral "31" represents the number thirty-one.
• Numeration is writing a number in words. 
  For example, the number 31 is written as "thirty-one."

Hindu-Arabic Numeration System

• This system uses digits 0 to 9 to form numbers.
• Each digit's value depends on its position in the number.
• For example, 98, 76, 54, 321 are numbers in the Hindu-Arabic system.

The place values in this system are:

Example: The place values In the number 54,321 are as follows

• 5 is in the ten thousands place: 5 × 10,000 = 50,000
• 4 is in the thousands place: 4 × 1,000 = 4,000
• 3 is in the hundreds place: 3 × 100 = 300
• 2 is in the tens place: 2 × 10 = 20
• 1 is in the ones place: 1 × 1 = 1

International Numeration System

• This system is used in many countries and also uses digits 0 to 9, but the place values are grouped differently.
• For example: 987,654,321 is written as 987 million, 654 thousand, 321 in the International system.

The place values are:

Example: The place values in the number 987,654,321 are as follows

• 9 is in the hundred millions place: 9 × 100,000,000 = 900,000,000
• 8 is in the ten millions place: 8 × 10,000,000 = 80,000,000
• 7 is in the millions place: 7 × 1,000,000 = 7,000,000
• 6 is in the hundred thousands place: 6 × 100,000 = 600,000
• 5 is in the ten thousands place: 5 × 10,000 = 50,000
• 4 is in the thousands place: 4 × 1,000 = 4,000
• 3 is in the hundreds place: 3 × 100 = 300
• 2 is in the tens place: 2 × 10 = 20
• 1 is in the ones place: 1 × 1 = 1

Number System 

• Real Numbers (R): This set includes all negative and positive numbers, zero, and fractions, such as -2, 5, 0, 2/3, -4/5.
• Integer (Z): Examples of integers are -3, 2, -1, 0, 1, 2, 3.
• Whole Numbers (W): This set includes 0, 1, 2, 3, and so on, extending infinitely in the positive direction.
• Natural Numbers (N): These are the counting numbers starting from 1, 2, 3, and so forth, continuing indefinitely.

Tests of Divisibility

• Division by 2: A number is divisible by 2 if its last digit is even. For example, 52 is divisible by 2 because its last digit, 2, is even.
• Division by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. For instance, 192 is divisible by 3 because the sum of its digits (1 + 9 + 2 = 12) is divisible by 3.
• Division by 4: A number is divisible by 4 if its last two digits form a number that is divisible by 4. For example, 172 is divisible by 4 because the last two digits, 72, are divisible by 4.
• Division by 5: A number is divisible by 5 if its last digit is either 0 or 5. For instance, 65 and 90 are divisible by 5 because their last digits are 5 and 0, respectively.
• Division by 10: A number is divisible by 10 if its last digit is 0. For example, 1120 is divisible by 10 because its last digit is 0.
• Even Natural Numbers (E): These are natural numbers that are divisible by 2. The set of even natural numbers is E = {2, 4, 6, 8, 10, 12, 14, ...}
• Odd Natural Numbers (O): These are natural numbers that are not divisible by 2. The set of odd natural numbers is O = {1, 3, 5, 7, 9, ...}.
• Prime Natural Numbers (P): A prime number is a natural number greater than 1 that is divisible only by itself and 1. The set of prime natural numbers is P = {2, 3, 5, 7, 11, ...}.