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Category: Forgotten Mathematics

Arithmetic often cannot prove some of its strongholds with vague means. In these cases, we need more general algebra methods. For this kind of arithmetic theorems, which justified in an algebraic manner, many rules for shortened calculation operations result. However this project is a sub-project of forgotten math and a part of Secrets fo Algebra.   

 


Speed multiplication:

 


In former time, the time without computers or calculator, great arithmetician used many simple algebraic tricks; in order to make their life easier:

 Let us look on:


 988²=?

Can you solve it in your mind?

It is very easy to do this, let us make a closer look:


988 x 988 = (988 + 12) x (998 -12) + 12² = 1000 x 976 + 144 = 976 144
It is simple  to understand what happens:
(a + b)(a – b)  + b² = a² – b² + b² = a²
O.K. so far so good. Now we try calculate fast, even such combinations like
986 x 997, without calculator:
986 x 997 = (986 - 3) x 1000 + 3 x 14 = 983 042

What happened here? We can write the factors in the following way:
(1000 – 14) x (1000 - 3)  
1000 x 1000 – 1000 x 14 – 1000 x 3 + 14 x 3
Let us play with the factors:
1000(1000 - 14) – 1000 x 3 + 14 x 3 =
1000 x 986 – 1000 x 3 + 14 x 3 =
1000(986 - 3) + 14 x 3
That’s all.  




Let us study another powerful technic of algebra one ca use to calculate some math operations in our mind, based on:

a² = (a + b) x (a - b) + b²


Examples:

27² = (27 + 3) x (27 - 3) + 3²  = 30 x 24 + 9 = 729
63² = 66 * 60 + 3² = 3 969
54² = 58 x 50 + 4² = 2 916

The best fun is when the last number is 5:
35²: 3 x 4 = 12 ; 5² = 25 = 1 225
65²; 6 x 7 = 42 ;  5² = 25 = 4 225