Mathematical Operations and Commutativity

Study Notes on Arithmetical Operations

• Unit Duration
  • This unit will take approximately one hour to study.
  • It is acknowledged that learning paces vary; do not worry if it takes longer than expected.
  • Focus on achieving the set objectives.

Section A: Arithmetical Operations

• Familiarity with the four basic mathematical operations is essential:

  • Addition
  • Subtraction
  • Multiplication
  • Division

• During the final examination, students will be permitted to use a non-programmable calculator.

  • Despite this allowance, proficiency in performing operations mechanically is crucial.

Example 1: Addition of Large Numbers

• Task: Find the sum of the following numbers: 1024, 768, and 23476.

• Guideline: Arrange large numbers in columns for easier addition.

  • Format the numbers as follows:
  • Ten thousands: 1 (from 23476)
  • Thousands: 0
  • Hundreds: 2
  • Tens: 4
  • Units: 6

  - Columns arranged:

  Ten thousands	Thousands	Hundreds	Tens	Units
  1	0	2	4	6
  0	7	6	8	8
  0	2	3	4	7
  0	0	0	0	0

• Working Steps:

  1. Start from the rightmost column (Units):
  • Units: $6 + 8 + 7 = 21$
  • Write down 1 and carry over 2 to the Tens column.
  1. Tens: $4 + 6 + 2 (carried) = 12$
  • Write down 2 and carry over 1 to the Hundreds.
  1. Hundreds: $2 + 7 + 1 (carried) = 10$
  • Write down 0 and carry over 1 to the Thousands.
  1. Thousands: $0 + 0 + 1 (carried) = 1$
  2. Ten Thousands: write down 1.
  • Final Sum: The final answer is 25868.

• Note: When a number in a column exceeds 9, it is 'carried over' to the column to its left.

• The addition of more than two numbers can be performed in any order because addition is commutative.

Example 2: Simplifying Addition

• Task: Add the following numbers: 218, 365, and 382.
• Thinking Process: It may be easier to first consider adding 218 and 382, as:
  • Simplification: $18 + 82 = 100$
  • Thus, the addition simplifies to:
  • First Step: $(218 + 382) + 365 = 100 + 218 + 365 = 600 + 365 = 965$
  • Final Answer: 965

Glossary

• Commutative: Refers to mathematical operations (specifically addition and multiplication) that can be performed in any order without changing the result.