Indian Multiplication by Nines (99, 999)

Multiply by 99, 999 and other strings of nines using the complement trick N times 100 minus N, with each step shown.

Input

An Indian complement method for multiplying by 99, 999 and other strings of nines. Enter N and the number of nines to see the product found with just a shift and a subtraction.

Number N

Multiplier (nines)

N is an integer of 1 or more, and the number of nines can be 2 to 6.

Result

47 times 99 equals

4,653

After shift

4,700

−

Subtract N

Product

4,653

Shift factor

100

Shifted number

4,700

Step by step

Rewrite the nines

99 has 2 nines, so it is just one less than 100.

Switch to the complement

N times 99 becomes N times 100 minus N times 1, that is N times 100 minus 47.

Shift N

Multiplying 47 by 100 appends zeros to give 4,700.

Subtract N

Subtract 47 from 4,700 to get 4,653, the answer.

Breakdown

Item	Value
Multiplier M	99
Shift factor	100
Shifted number	4,700
Number to subtract	47
Product	4,653

Check: computing 47 times 99 the ordinary way also gives 4,653.

Idea: a string of nines M equals ten to the power k minus one, so N times M equals N times ten to the power k minus N, just a shift and a subtraction.

How it works

A string of nines M equals ten to the power k minus one (99 equals 100 minus 1, 999 equals 1000 minus 1). This turns the multiplication into a shift and a subtraction.
N times M equals N times ten to the power k minus N. Shift N left by k digits, then subtract N to get the product.
For example, 47 times 99: shift 47 by two digits to get 4700, then subtract 47 to get 4653. One subtraction replaces the whole long multiplication.
Here k is the number of nines. For 99 use k equals 2, for 999 use k equals 3. The shift factor ten to the power k means appending k zeros.
The ordinary product is shown as a check so you can confirm it matches the step-by-step result.

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