why 1x1=1 11x11=121 111x111=12321.........

Let the number contains n 1's.
Then the number can be rewritten as 1 + 10 + 10^2 + ... + 10^n
Then, 11....11 x 11....11
= (1 + 10 + 10^2 + ... + 10^(n-1))x(1 + 10 + 10^2 + ... + 10^(n-1))
= (1 + 10 + 10^2 + ... + 10^(n-1)) +
10x(1 + 10 + 10^2 + ... + 10^(n-1)) +
10^2x(1 + 10 + 10^2 + ... + 10^(n-1)) + ... +
10^(n-1)x(1 + 10 + 10^2 + ... + 10^(n-1))
= (1 + 10 + 10^2 + ... + 10^(n-1)) +
(10 + 10^2 + 10^3 + ............. + 10^n +
(10^2 + 10^3 + 10^4 + ........................+ 10^(n+1)) + ... +
(10^(n-1) + 10^n + 10^(n+1) + ... + 10^(2n-2))
= 1 + 2x10 + 3 x 10^2 + 4 x 10^3 + ...+ n x 10^(n-1) +
(n-1) x 10^n + (n-2) x 10^(n+1) + 2 x 10^(2n-1) + 10^(2n-2)
(since the above formula has n rows)

Then, for n = 1, we have 1 x 1 = 1 since there is only one term
for n = 2, we have 11 x 11 = 1 + 2x10 +10^2 = 121
for n = 3, we have 111 x 111 = 1 + 2x10 + 3x10^2 + 2x10^3 + 10^4 = 12321
and so on.

1x1=1------------------------------------------------everyone know this,right?
11x11=121-----------------------------------------11x10+11=110+11=121
111x111=12321----------------------------------111x100+111x10+111=12321
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understand?????????????????

數學定理
呢一個類型好似一個樓梯咁樣
好似咁



1
111
11111
1111111

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其實我都做過
但係好難講比你聽
因為我都唔識講比你知啊

這是數學定律!

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