Uncover the Mystery of Numbers: Always Ending up with 9

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In summary, when you take any whole number and add its digits down to a single number, then subtract this number from the original number, and add the resulting digits down to a single number, it will always end up being 9. This is because a number and its sum of its decimal digits are equivalent to each other mod 9, so their difference is divisible by 9. This technique is also known as "casting out nines" and has been used in old-fashioned arithmetic methods. For more fun with numbers, you can check out the links provided by Chuckles in the conversation.
  • #1
Dave9600
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Take any whole number and add the digits down to a single number. Now, subtract this number from the original number. With this answer add the digits down to a single number and it will always end up being 9.

Example: 22 (2+2=4) 22-4=18 (1+8=9)

or
14567 (1+4+5+6+7=23 and 2+3=5) 14567-5=14562 (1+4+5+6+2=18 and 1+8=9)

Why does you always end up with 9 and is there a name for this?
 
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  • #2
A number and its sum of its decimal digits are equivalent to each other mod 9, so their difference is divisible by 9.
 
  • #3
Hi Dave9600! :smile:

Because when you replace eg 50000 by 5, you reduce the number by 5 times 9999, which is obviously divisible by 9.

Add all the 50000s etc, and add all the 5s etc, and then subtract, and the result is divisible by 9.

In numbers: ∑ an10n - ∑ an

= 9(∑ an ∑i<n10i) :wink:
 
  • #5
And for a bit of fun, see

http://mensanator.com/rotanasnem/cherries/cherries.htm"
 
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  • #6
Chuckles
 

FAQ: Uncover the Mystery of Numbers: Always Ending up with 9

How can numbers always end up with 9?

The phenomenon of numbers always ending up with 9 is known as the "9 trick" or "digital root method". It involves adding up all the digits in a number and repeating the process until a single digit is obtained. If the final digit is not 9, the process is repeated until it is. This trick works for all numbers except multiples of 9.

Why does this trick only work for non-multiples of 9?

This trick works because of the number system we use, which is based on 10. Since 9 is one less than 10, when you add 9 to any number, the final digit will always be one less than the original number. This means that when you add up the digits of a number that ends in 9, the final digit will always be 9. However, this does not work for multiples of 9 because they can be divided evenly by 9 and therefore do not have a remainder of 9.

Can this trick be used for numbers with more than two digits?

Yes, this trick can be used for any number with more than two digits. Simply add up all the digits in the number and repeat the process until a single digit is obtained. The final digit will always be 9 if the original number is not a multiple of 9.

Is there a mathematical explanation for this phenomenon?

Yes, there is a mathematical explanation for this phenomenon called the "casting out nines" rule. This rule states that if you add the digits of a number and the resulting sum is divisible by 9, then the original number is also divisible by 9. This rule can be used to prove why the 9 trick works for non-multiples of 9.

Are there any practical applications for this trick?

While this trick may seem like just a fun mathematical curiosity, it actually has practical applications. It can be used to check the accuracy of calculations or to find mistakes in calculations. It is also used in some forms of divination and numerology to determine the meaning of numbers and their significance in a person's life.

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