Multiplication by 9, 99, 999, etc.

One way to multiply a number by 9 is to multiply by 10 and then subtract the number from the product. There is another way to multiply fast by 9 and as the first one it has an analogue for multiplication by 99, 999 and all such numbers. Let's start with the multiplication by 9.

To multiply a one digit number a by 9, first subtract 1 and form b = a - 1. Next, subtract b from 9: c = 9 - b. Then just write b and c next to each other:

9a = bc.

For example, find 6×9 (so that a = 6.) First subtract: 5 = 6 - 1. Subract the second time: 4 = 9 - 5. Lastly, form the product 6×9 = 54.

Next, find 37×99. First, subtract 1: 36 = 37 - 1. Then subtract 63 = 99 - 36. Lastly, form the product: 37×99 = 3663.

Why does this work? For the multiplication by 9, bc = 10b + c:

 bc = 10b + c = 10(a - 1) + (9 - (a - 1)) = 10a - 10 + 10 - a = 9a,

as required. Similarly, for a 2-digit a:

 bc = 100b + c = 100(a - 1) + (99 - (a - 1)) = 100a - 100 + 100 - a = 99a.

Do try the same derivation for a three digit number. As an example,

 543×999 = 1000×542 + (999 - 542) = 999×542 + 999 = 999×543

just by using the distributive law twice.

• Multiplication by 9, 99, 999, (Multiply + Subtract) etc.
• Squaring 2-Digit Numbers
• Division by 5
• Multiplication by 2
• Multiplication by 5
• Multiplication by 9, 99, 999, etc. (Something Special)
• Product of 10a + b and 10a + c where b + c = 10
• Product of numbers close to 100
• Product of two one-digit numbers greater than 5
• Product of 2-digit numbers
• Squaring Numbers in Range 26-50
• Squaring Numbers in Range 51-100
• Squares of Numbers That End in 5
• Squares Can Be Computed Squentially
• How to Compute Fast Any Square
• Adding a Long List of Numbers
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