Annoying 9

[Kinma]

@Nodas

for mental math, I am skipping imagining any algebraic identities containing a, b. (usually). I also skip the 0’s (thousands, millions, billions etc.)

Personally, I do too. However; I like to explain why things work the way they work. For that reason the algebra comes out.

@Nodas

For 4-digit numbers

1. I just split them into two 2-digit parts
2. take their respective 2-digits quares (requires knowledge/recall of any of the first 100 squares) ,
3. stick their doubled product, in the middle position, and then merge the two-digit numbers
   e.g. 2048 is 20 and 48
   (2048^2=20^2| 22048 | 48^2 = 400 | 1920 | 2304 )
   You imagine that as 4 00 | 19 20 | 23 04
   and then merge any 2-digit near the vertical split line
   thus 4 00 | 19 20 | 23 04 = 4 19 43 04 = 4 million 194 thousands 304

Well said.

When thinking about squaring 4096 it occurred to me that there is an even faster way. Here is another way of doing the (4096^2):

(4096 = 4100 - 4)

I imagine this as 41 | -4

Then I do the square the same way you do:

(41^2 = 1681)
(241-4 = -328)
((-4)^2 = 16)

So the result before the merge is:
1681 | -328 | 16 =

Now merge:
1677 | 72 | 16 =

16,777,216