Speed Math Tricks

Squaring A Number-

1. Aitken, mathematician took advantage of the following algebraic identity-
                 A² = (A-d)(A + d) + d²
Naturally, this formula works for any value of d, but we should choose d to be the distance to a number close to A that is easy to multiply.

Example- To square the number 23, we let d = 3 to get
                  23²= 20 *26 + 3²
                         = 520 + 9 
                          = 529
2. If  'n' is a number between 0 to 50 
    Then,
            n² = (n-25)*100 + (50-n)² 
     If  'n' is a number between 50 to 100 
    Then,
            n² = (2n-100)*100 + (100-n)² 

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Close Together Method-

For Multiplying any two no.   
           (z + a)(z + b) = z(z + a + b) + ab
where z is typically a number that ends in zero.

Example- For the problem 23*26, z = 20, a = 3, b = 6, leads to

23*26 = 20*29 + 3*6 = 580 + 18 = 598

Note-  On both sides of the equation, the algebra reveals that

the two-digit numbers being multiplied will have the same sum (23 + 26 = 49 = 20 + 29).

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Cubing A Number- 

To cube a two-digit number, we can exploit the algebra

         A³= (A - d) A (A + d) + d²A

Example- Thus to cube the number 23, we can do

23³ = 20* 23* 26+3²*23

      = 20* 598 + 9* 23 = 11,960 + 207 = 12,167

where we used the close-together method to do 23*26.

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To Find Any Root Of A Number-

                   
                    where, n = Index or root no.

                               A = Radicand or any number.
                                x = Perfect root of no. near to required root.
                               Δx = Difference between x and A
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