finish README

Author: appleJax

src/algorithms/math/karatsuba-multiplication/README.md

@@ -2,6 +2,40 @@
 
 Karatsuba is a fast multiplication algorithm discovered by Anatoly Karatsuba in 1960. Given two n-digit numbers, the "grade-school" method of long multiplication has a time complexity of O(n<sup>2</sup>), whereas the karatsuba algorithm has a time complexity of O(n<sup>1.59</sup>).
 
+## Recursive Formula
+
+```
+x = 1234
+y = 5678
+
+karatsuba(x, y)
+```
+
+1. Split each number into numbers with half as many digits
+```
+a = 12
+b = 34
+
+c = 56
+d = 78
+```
+
+2. Compute 3 subexpressions from the smaller numbers
+  - `ac = a * c`
+  - `bd = b * d`
+  - `abcd = (a + b) * (c + d)`
+
+3. Combine subexpressions to calculate the product
+```
+A = ac * 1000
+B = (abcd - ac - bd) * 100
+C = bd
+
+x * y = A + B + C
+```
+
+_**Note:**_ *The karatsuba algorithm can be applied recursively to calculate each product in the subexpressions.* (`a * c = karatsuba(a, c)`*). When the numbers get smaller than some arbitrary threshold, they are multiplied in the traditional way.*
+
 ## References
 [Stanford Algorithms (YouTube)](https://www.youtube.com/watch?v=JCbZayFr9RE)
-[Wikipedia](https://en.wikipedia.org/wiki/Karatsuba_algorithm)
+[Wikipedia](https://en.wikipedia.org/wiki/Karatsuba_algorithm)