Nikhilaṃ Navataścaramaṃ Daśataḥ — All from 9 and the last from 10

निखिलं नवतश्चरमं दशतः — Nikhilam Navatascaramam Dasatah

All from 9 and the last from 10. Find how far each number is from the nearest power of 10. Those distances do the multiplication for you.

Why it works: For numbers near base B: let a = B−x, b = B−y (deficiencies). Then a×b = (B−x)(B−y) = B²−B(x+y)+xy = B(a−y)+xy = B(b−x)+xy. The left part is one number minus the other's deficiency; the right part is the product of the deficiencies.

Speed advantage: 3 steps vs 12+ steps for multiplying numbers near a base.

Best for

• Multiplying 2–4 digit numbers near powers of 10

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Originally published at https://www.sutraflow.app/sutras/nikhilam-navatascaramam-dasatah