Mathematics: Lecture 1

Importance of Calculations

• Key to placements and success in mathematics.
• Short-term pain for long-term gain, akin to immunizations in childhood.

Essential Memorization

• Squares: Memorize squares from 1 to 100 for guaranteed placement success.
• Cubes: Memorize cubes from 1 to 40 for optimal results.
• Tables: Learn tables from 1 to 50 for better placement chances.
• Prime Numbers: Know prime numbers from 1 to 500.

Techniques to Memorize Squares

• 21-30 Squares: Use the pattern method by recognizing incremental values and their relationships.
• 41-50 Squares: Utilize multiplication of base values and incremental adjustments.
• 51-60, 91-100 Squares: Similar patterns with stepwise additions and subtractions.

Calculation Tricks

1. Tables: Learn to create tables quickly using positional values and addition techniques.
   • Example: 86's Table – Split into units and tens, then sum the results accordingly.
2. Squares Near 100: Use the difference from 100 to simplify calculations.
   • Example: 103² – Recognize it’s 3 more than 100, so 103 + 3 = 106; 3² = 9. Result: 10609.
3. Adjacent Squares: Use known squares to derive the next one with simple additions.
   • Example: 13² from 12² – 12² = 144, then 13 + 12 = 25. Result: 169.

Advanced Examples

• Calculation for 104², 112² involves squaring the difference or sum base values.
• Multi-digit squaring involves breaking down the problem into smaller, manageable parts.

Special Properties of Numbers

• 999...9 Pattern: Useful in IT questions; simplified using specific patterns.
  • Example: 9’s pattern involves reducing one nine, adding zeros, then adding specific values (8 and 1 for squares, 7 and 2 for cubes).
• Multi-step IT Company Questions: Real-world problems requiring not just calculations but understanding of the pattern and properties of numbers.

Examples and Exercises

1. Calculating Squares and Cubes: Practice to understand and memorize specific patterns and relationships used in competitive exams.
2. Understanding Summations and Digit Counts: Specific to digit manipulation and number properties.

Summary

• Key Points: Accurate, quick calculations and correct use of patterns are essential for success in competitive exams and placements.
• Practice Required: Continuous practice and memorization help in mastering these techniques.

Homework

• Calibrate several exercises related to squares, cubes, and tables to reinforce learning.
• Apply learned techniques to similar problems to ensure understanding and retention.