Roman Numerals

Points to Remember:

1. Repetition Rule:
   • If any Roman numerals are repeated, their values are added.
     For example, , .
2. Subtraction Rule:
   • When a smaller numeral is placed before a larger numeral, it is subtracted from the larger numeral.
     For example, (5 – 1), (10 – 1).
3. Addition Rule:
   • When a smaller numeral is placed after a larger numeral, it is added to the larger numeral.
     For example, (5 + 1), (10 + 1).
4. Placement Rule:
   • When a smaller numeral is placed between two larger numerals, it is always subtracted from the larger numeral immediately following it.
     For example, (10 + 5 – 1).

Examples:

• Addition:
• Subtraction:
• Complex Examples:

Whole Numbers

Natural Numbers:

• Counting numbers (1, 2, 3, 4, 5, 6, etc.) are called natural numbers.

Whole Numbers:

• All natural numbers together with ‘0’ (zero) are called whole numbers.
  Thus, are whole numbers.
• Every natural number is a whole number, but ‘0’ (zero) is a whole number that is not a natural number.

Successor of a Whole Number:

• If we add 1 to a whole number, the resulting number is called its successor. For example, the successor of 0 is 1, the successor of 1 is 2, the successor of 12 is 13, and so on. Every whole number has its successor.

Predecessor of a Whole Number:

• One less than a given whole number (other than 0) is called its predecessor. For example, the predecessor of 1 is 0, the predecessor of 2 is 1, the predecessor of 10 is 9, and so on.
• The whole number 0 does not have a predecessor.
• Every whole number other than 0 has a predecessor.

Examples:

• Write the successor and predecessor of:
  • 1000
  • 1005399
  • 999999

Solution:

• The successor of 1000 = .
• The predecessor of 1000 = .
• The successor of 1005399 = .
• The predecessor of 1005399 = .
• The successor of 999999 = .
• The predecessor of 999999 = .

Properties of Whole Numbers

1. Closure Property:
   • Whole numbers are closed under addition and multiplication.
     For example, and are whole numbers.
2. Commutative Property:
   • Whole numbers are commutative under addition and multiplication.
     For example, and .
3. Associative Property:
   • Whole numbers are associative under addition and multiplication.
     For example, and .
4. Distributive Property:
   • Multiplication is distributive over addition for whole numbers.
     For example, .

Simplification

Order of Operations:

• In a simplification sum consisting of the operations , , , and , first do the operations in the order of BODMAS:
  • 
  • 
  • 
  • 
  • 
  • 

Examples:

1. Without Brackets:
   • Simplify:
   • Solution:
2. With Brackets:
   • Simplify:
   • Solution: