Binary multiplication

111.10011 x 10.111 = ?

Show steps please...

This works exactly as it does in base ten. The "binary point" behaves like the "decimal point" in that the digit to the left has weight 1, digits to its left have weight 2, 4, 8, etc., and digits to its right have weight 1/2, 1/4, 1/8, etc. Longhand multiplication also works exactly the same as we learned in grade school -- add up shifted versions of the first according to whether the second has a 1 (add it) or 0 (don't add it) in each place:

01111.00110000 (from the 2 in the second number)
00011.11001100 (from the 1/2 in the second number)
00001.11100110 (from the 1/4 in the second number)
00000.11110011 (from the 1/8 in the second number)
============== (add them all up)
10101.11010101 (final sum is the product of original two values)

Note that we can check this by converting to base ten. The two numbers are:

(4 + 2 + 1 + 1/2 + 1/16 + 1/32) = 7 19/32 = 243/32
(2 + 1/2 + 1/4 + 1/8) = 2 7/8 = 23/8

So the product is 5589/256 = 21 213/256

which is the value of the above product