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30MSC 2023 #6: Using Bar Numbers.

30MSC 2023 #6: Using Bar Numbers.

Using bar numbers is an ingenious method of writing numbers, avoiding large digits which usually result in simpler computations. 49 is 50 – 1 and can be simply written as 5 then 1 with a bar on top. The bar above the 1 is called a vinculum.

Cubing a number can be done in the several ways:

  1. Direct multiplication – n x n x n =
  2. Binomial Expansion – (a + b) ^3 = a^3 + 3(a^2)(b) +3(a)(b^2) + b^3
  3. Proportionately – This technique of using the ratio of b to a simplifies the computation of binomial expansion. Note that we use the following computations to get the first line of the solution:a^3 x b/a = a^2(b), a^2(b) x  b/a = a(b^2) and a)b^2) x b/a = b^3.

For the second line, we just double the results of the 2nd and 3rd column. Then adding the first and second rows will give the final answer.

1)                            a^3      +  a^2(b)     + a(b^2)    + b^3

2)                        +                 2a^2(b)       2a(b^2)

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(a + b)^3    =       a^3      +  3a^2(b)     + 3a(b^2)    + b^3

Using this technique, we can easily compute the cube of 12, for example: b/a = 2/1 = 2

                                             1        2         4         8

                                       +            4         8

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                                            1      6         12         8 

The result is interpreted as 1000 + 600 + 120 + 8. Thus 12^3 = 1728

But for our featured example, which is a question in the seniors category of the 3rd International Vedic Mathematics  Olympiad (IVMO 2023) which was held last Nov 25, 2023, computing the cube of 79 using any of the three given methods would be time consuming.

With large digits 7 and 9, even the Proportionately technique would involve lengthy calculations and be prone to errors: b/a = 9/7

7^3 = 343

343 x 9/7 = 441

441 x 9/7 = 567

651 x 9/7 = 729

               343        441        567        729

                              882      1134

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               343      1323    1701         729

79^3  = 343,000 + 132,300 + 17,010 + 729 = 493,039

Now, since 79 is 80 -1, we can use the bar notation for faster calculations. (for this post, we will use parentheses to indicate negative or bar digits). Thus 79 is 8(1).

8^3 = 512 and the ratio of the digits, b/a = (1)/8.

               512        (64)        8             (1)

                           (128)      16

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             512       (192)      24           (1)

8(1)^3 = 512,000 – 19,200 + 240 – 1 = 493,039

More applications of bar numbers are discussed in Chapter 6 of our e-book, 30 Master Strategies in Computing, Volume I.  The rewritten version will be available before the end of the year.                              

Follow us on our Facebook page, MATH-Inic Philippines at https://www.facebook.com/MATHInicPhils to see tomorrow’s 30MSC 2023 #7: The Last Digits Adding up to 10.

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