Continues Numbers & Alike Numbers
1... COUPLES OF CONTINUES NUMBERS
For every two couples of continues numbers, holds that: if the SECOND number from the first couple of numbers is added to the FIRST number
from the second couple of numbers, if we then subtract from the result of the addition the FIRST number from the first couple of numbers
then the result will be the SECOND number of the second couple of numbers, e.g.
| First Couple of Numbers |
Second Couple of Numbers |
| 1 |
101 |
| 2 |
102 |
We add the SECOND number from the first couple of numbers to the FIRST number from the second couple of numbers (2 + 101 = 103).
From the result of the addition we subtract the FIRST number from the first couple of numbers (103 - 1 = 102). The result is
the SECOND number from the second couple of numbers (102).
Proof:
If we assign variables to the numbers we take:
1 = α, 2 = β, 101 = γ, 102 = δ
We now use these variables in order to construct a general equation which will be capable of describing the general case of the example given above.
We take:
(β + γ) - α = δ
Observing that: α = β - 1 and δ = γ + 1
We introduce into the equation the corresponding equivalents of the variables α and δ.
We take:
(β + γ) - (β - 1) = γ + 1
We solve the equation in respect to the first part in order to obtain the second:
(β + γ) - (β -1) = β + γ - β + 1 = β + γ - β + 1 = γ + 1
2... SERIES ADDITIONS BETWEEN ALIKE NUMBERS
6 + 6 = 12
66 + 66 = 132
666 + 666 = 1332
6666 + 6666 = 13332
66666 + 66666 = 133332
[6 * n] + [6 * n] = 1[3 * n-1]2
7 + 7 = 14
77 + 77 = 154
777 + 777 = 1554
7777 + 7777 = 15554
77777 + 77777 = 155554
[7 * n] + [7 * n] = 1[5 * n-1]4
8 + 8 = 16
88 + 88 = 176
888 + 888 = 1776
8888 + 8888 = 17776
88888 + 88888 = 177776
[8 * n] + [8 * n] = 1[7 * n-1]6
9 + 9 = 18
99 + 99 = 198
999 + 999 = 1998
9999 + 9999 = 19998
99999 + 99999 = 199998
[9 * n] + [9 * n] = 1[9 * n-1]8
9 + Two Digit Number
9 + 10 = 19
9 + 11 (1 + 1 = 2) = 20 (2 + 0 = 2)
9 + 12 (1 + 2 = 3) = 21 (2 + 1 = 3)
9 + 13 (1 + 3 = 4) = 22 (2 + 2 = 4)
9 + 14 (1 + 4 = 5) = 23 (2 + 3 = 5)
9 + 94 (9 + 4 = 13 (1 + 3 = 4)) = 103 (1 + 0 + 3 = 4)
9 + Three Digit Number
9 + 111 (1 + 1 + 1 = 3) = 120 (1 + 2 + 0 = 3)
9 + 458 (4 + 5 + 8 = 17) = 467 (4 + 6 + 7 = 17)
9 + Four Digit Number
9 + 5478 (5 + 4 + 7 + 8 = 24) = 5487 (5 + 4 + 8 + 7 = 24)