More Short Tricks for Fast Calculation

Duration: 1 hr 1 min

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This video is a comprehensive educational lecture on speed mathematics, presented by an instructor from Knowledge Gate. The session begins with an introduction to the concept of a 'tachyon' as a hypothetical particle faster than light, setting a theme of rapid calculation. The core of the lecture focuses on mental math techniques for squaring numbers, starting with a list of prerequisites such as memorizing squares, cubes, and multiplication tables. The instructor demonstrates a systematic method for calculating squares of numbers ending in 5, using the formula (10a + 5)² = 100a(a+1) + 25, which simplifies to writing the product of a and (a+1) followed by 25. This is extended to numbers near 100, where the square is found by subtracting the difference from 100, multiplying it by 2, and adding it to the original number, then appending the square of the difference. The lecture then covers a general shortcut for squaring any number, using the identity (a+b)² = a² + 2ab + b², demonstrated with examples like 103² and 112². The instructor also teaches a pattern for squaring numbers composed entirely of 9s, showing that (99...9)² results in a number with (n-1) 9s, an 8, (n-1) 0s, and a 1. The lesson progresses to cubing numbers, with a similar pattern for (99...9)³, which yields (n-1) 9s, a 7, (n-1) 0s, a 2, and (n-1) 9s. The video concludes with a complex problem involving a number with 2020 nines raised to the power of 3, where the instructor applies the learned pattern to find the total number of digits, the value of the expression, and the sum of its digits. The final segment introduces a method for creating multiplication tables quickly, using a pattern of adding the first digit to the second digit of the product to generate the next row.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card for 'SPEED MATHS' over a chalkboard filled with mathematical equations. It then transitions to a live lecture where the instructor, Yash Jain Sir, introduces the topic. He explains that the course will cover 'tachyon calculation,' a term he defines as a method for rapid calculation, referencing the hypothetical particle 'tachyon' that travels faster than light. The background displays a poster for a 'FREE LIVE COURSE' on 'APTITUDE' and 'TACHYON CALCULATION SHORT TRICKS' scheduled for October 8th at 8:30 PM.

  2. 2:00 – 5:00 02:00-05:00

    The instructor presents a slide titled 'PRE-REQUISITES' listing essential knowledge for the course: squares of 1 to 50, cubes of 1 to 20, tables of 1 to 30, and prime numbers from 1 to 200. He emphasizes the importance of memorizing these foundational facts. He then begins to demonstrate a shortcut for squaring numbers ending in 5, using the example of 25². He writes the formula (10a + 5)² = 100a(a+1) + 25, explaining that for 25, a is 2, so the result is 2*3=6 followed by 25, which is 625.

  3. 5:00 – 10:00 05:00-10:00

    The instructor continues to demonstrate the shortcut for squaring numbers ending in 5. He shows that 35² is 1225 by calculating 3*4=12 and appending 25. He then moves to numbers near 100, explaining that for a number like 98, which is 2 less than 100, the square is found by subtracting 2 from 98 to get 96, and then appending the square of 2, which is 04, resulting in 9604. He applies this to 97², getting 9409. He then demonstrates the method for 103², showing that 103 is 3 more than 100, so the result is (103+3)=106 followed by 09, which is 10609.

  4. 10:00 – 15:00 10:00-15:00

    The instructor demonstrates a general shortcut for squaring any number using the identity (a+b)² = a² + 2ab + b². He uses the example of 103², breaking it down as (100+3)² = 100² + 2*100*3 + 3² = 10000 + 600 + 9 = 10609. He then applies this to 112², showing (110+2)² = 110² + 2*110*2 + 2² = 12100 + 440 + 4 = 12544. He also shows a pattern for squaring numbers like 12, 21, 38, and 83, where the last two digits of the square are the same as the last two digits of the number itself, which is a property of numbers ending in 1, 2, 8, or 9.

  5. 15:00 – 20:00 15:00-20:00

    The instructor explains a pattern for squaring numbers composed entirely of 9s. He shows that 9² = 81, 99² = 9801, 999² = 998001, and 9999² = 99980001. He points out that for a number with n nines, the square has (n-1) 9s, an 8, (n-1) 0s, and a 1. He then applies this to 99999², which is 9999800001. He also demonstrates the pattern for cubing numbers of 9s, showing that 9³ = 729, 99³ = 970299, and 999³ = 997002999, where the result has (n-1) 9s, a 7, (n-1) 0s, a 2, and (n-1) 9s.

  6. 20:00 – 25:00 20:00-25:00

    The instructor presents a complex problem: to find the value of (999...9)³ where there are 2020 nines. He applies the pattern for cubing numbers of 9s, stating that the result will have (2020-1) = 2019 nines, followed by a 7, then (2020-1) = 2019 zeros, a 2, and finally (2020-1) = 2019 nines. He then calculates the total number of digits in the expression, which is 2020, and the value of the expression, which is 999...97000...02999...9. He also calculates the sum of the digits, which is 9*2019 + 7 + 2 + 9*2019 = 36343.

  7. 25:00 – 30:00 25:00-30:00

    The instructor introduces a new topic: creating multiplication tables quickly. He shows a table for 37, where he writes 37 x 1 = 37, 37 x 2 = 74, 37 x 3 = 111, and so on. He points out that the last digit of the product increases by 7 each time, and the first digit increases by 3. He demonstrates this pattern to generate the entire table for 37. He then applies this to 29, showing that 29 x 1 = 29, 29 x 2 = 58, 29 x 3 = 87, and so on, with the last digit increasing by 9 and the first digit increasing by 2.

  8. 30:00 – 35:00 30:00-35:00

    The instructor demonstrates a shortcut for finding the square root of a number. He shows that for a number like 86, the square root is approximately 9.27, since 9² = 81 and 10² = 100. He then shows a method for finding the square root of a number like 37, where he writes 37 x 1 = 37, 37 x 2 = 74, 37 x 3 = 111, and so on, and uses the pattern to find the square root. He also shows a method for finding the square root of a number like 86, where he writes 86 x 1 = 86, 86 x 2 = 172, 86 x 3 = 258, and so on, and uses the pattern to find the square root.

  9. 35:00 – 40:00 35:00-40:00

    The instructor demonstrates a method for creating a multiplication table for a number like 29. He shows that 29 x 1 = 29, 29 x 2 = 58, 29 x 3 = 87, and so on. He points out that the last digit of the product increases by 9 each time, and the first digit increases by 2. He demonstrates this pattern to generate the entire table for 29. He then applies this to 37, showing that 37 x 1 = 37, 37 x 2 = 74, 37 x 3 = 111, and so on, with the last digit increasing by 7 and the first digit increasing by 3.

  10. 40:00 – 45:00 40:00-45:00

    The instructor demonstrates a method for finding the square root of a number like 37. He shows that 37 x 1 = 37, 37 x 2 = 74, 37 x 3 = 111, and so on. He points out that the last digit of the product increases by 7 each time, and the first digit increases by 3. He demonstrates this pattern to generate the entire table for 37. He then applies this to 86, showing that 86 x 1 = 86, 86 x 2 = 172, 86 x 3 = 258, and so on, with the last digit increasing by 6 and the first digit increasing by 8.

  11. 45:00 – 50:00 45:00-50:00

    The instructor demonstrates a method for finding the square root of a number like 86. He shows that 86 x 1 = 86, 86 x 2 = 172, 86 x 3 = 258, and so on. He points out that the last digit of the product increases by 6 each time, and the first digit increases by 8. He demonstrates this pattern to generate the entire table for 86. He then applies this to 37, showing that 37 x 1 = 37, 37 x 2 = 74, 37 x 3 = 111, and so on, with the last digit increasing by 7 and the first digit increasing by 3.

  12. 50:00 – 55:00 50:00-55:00

    The instructor demonstrates a method for finding the square root of a number like 37. He shows that 37 x 1 = 37, 37 x 2 = 74, 37 x 3 = 111, and so on. He points out that the last digit of the product increases by 7 each time, and the first digit increases by 3. He demonstrates this pattern to generate the entire table for 37. He then applies this to 86, showing that 86 x 1 = 86, 86 x 2 = 172, 86 x 3 = 258, and so on, with the last digit increasing by 6 and the first digit increasing by 8.

  13. 55:00 – 60:00 55:00-60:00

    The instructor demonstrates a method for finding the square root of a number like 86. He shows that 86 x 1 = 86, 86 x 2 = 172, 86 x 3 = 258, and so on. He points out that the last digit of the product increases by 6 each time, and the first digit increases by 8. He demonstrates this pattern to generate the entire table for 86. He then applies this to 37, showing that 37 x 1 = 37, 37 x 2 = 74, 37 x 3 = 111, and so on, with the last digit increasing by 7 and the first digit increasing by 3.

  14. 60:00 – 61:15 60:00-61:15

    The video concludes with a final summary of the key concepts covered. The instructor reiterates the importance of the shortcuts for squaring numbers, especially those ending in 5 and those near 100. He emphasizes the pattern for squaring numbers composed of 9s and the method for creating multiplication tables quickly. The final screen displays the Knowledge Gate logo and website address, www.knowledgegate.in.

This video provides a structured and comprehensive lesson on speed mathematics, focusing on mental calculation techniques. The core of the lesson is a series of shortcuts for squaring numbers, which are presented in a logical progression from simple to complex. The instructor begins with foundational prerequisites, then introduces a powerful method for squaring numbers ending in 5, which is a cornerstone of the lesson. This is extended to numbers near 100, demonstrating a clever application of the difference of squares. The lecture then generalizes the technique to any number using the algebraic identity (a+b)², and applies it to numbers like 103 and 112. A significant portion is dedicated to patterns for squaring and cubing numbers composed entirely of 9s, which are shown to follow a predictable structure based on the number of digits. The lesson culminates in a challenging problem that requires applying these patterns to a large number, testing the student's understanding. The video also includes a practical skill for creating multiplication tables quickly, which is a valuable tool for mental arithmetic. The overall teaching style is clear, methodical, and focused on empowering students with efficient calculation methods.