Notes: Multiplication of Ordinary Decimal Numbers,
Digit-by-digit
2-digit numbers:

a = a1 a0 = a1∙10 + a0
b = b1 b0 = b1∙10 + b0
c = c2 c1 c0 = c2∙100 + c1∙10 + c0
c = a * b = (a1∙10 + a0) * (b1∙10 + b0)
c2 = a1*b1
c0 = a0*b0
c1 = a1*b0 + a0*b1
c1 = (a1 + a0) * (b1 + b0) - (c2 + c0)
8-digit numbers:

c   =    a     *     b    =       c
      23456789 * 98765432 = 2316719898917848

Break up a and b into 4-digit pieces:

a   =   a1 10e4 +   a0    =    a
       23450000 +  6789   = 23456789

b   =   b1 10e4 +   b0    =    b
       98760000 +  5432   = 98765432

c = a * b = (a1 10e4 + a0) * (b1 10e4 + b0) =
   c2 10e8 + c1 10e4 + c0

c2 = a1  *  b1  =    c2
    2345 * 9876 = 23159220

c0 = a0  *  b0  =    c0
    6789 * 5432 = 36877848

c1 = (a1 + a0) * (b1 + b0) - (c2 + c0) =    c1
        9134   *   15308   -  60037068 = 79786204

c =  c2 10e8     +    c1 10e4   +    c0    =       c
2315922000000000 + 797862040000 + 36877848 = 2316719898917848

Recurrence normal:  T(n) = 4 T(n/2) + Θ(n)
   Master theorem: b = 2, a = 4, logb(a) = log2(4) = 2 > 1, so
   Case 1:  T(n) = Θ(nlogb(a)) = Θ(n2)

Recurrence special:  T(n) = 3 T(n/2) + Θ(n)
   Master theorem: b = 2, a = 3, logb(a) = log2(3) = 1.5849625 > 1, so
   Case 1:  T(n) = Θ(nlog2(3)) = Θ(n1.585)