The mystery of 9

salfriz

I have a question which i wonder about from time to time and would appreciate any responses.

1 Why if you multiply 9 by any whole number and add up the reducing digets until they reduce to one diget will yo always get 9

ie: 9x43=387 3+8+7=18 1+8= 9

I cannot see the connection between the multiplication and the addition of digets. Do any other numbers behave the same?

I dont know much about maths so maybe there's a simple solution that i can't see...thanks anyhow.

europerson

Here's a Wiki link.

MathsManiac

That's a nice link. Here's another way of looking at it, without specifically referring to the more general arithmetic check in that article or "modular arithmetic":

When you multiply something by 9, the answer is obviously a multiple of 9. So, the question now transforms into: Why is it that, if you add up all the digits in a number that is a multiple of 9, the answer is a multiple of 9?

Let's take your example of 387. Writing out what "387" means in terms of place value, you see that it's:
3*(100) + 8*(10) + 7*1. Now that's the same as:
3*(99+1)+8*(9+1)+7. But I can multiply out those brackets and rearrange:
3*99+3*1 + 8*9 +8*1 +7*1 =
[3*99+8*9] + (3*1 + 8*1 + 7*1), which is:
[a thing that's clearly a multiple of 9] +(3+8+7).

So, the original number has been rewritten as [a multiple of 9]+(sum of the digits).

So the original number will be a multiple of 9 if and only if the sum of its digits is. (And we can repeat the process until we get down to 9.)

If you think about, and play around with other examples, you should be able to see that this works no matter how many digits were in the number you started with, and also no matter what those digits were.

Does that help?

If you find this kind of thing interesting, you should look up stuff about "modular arithmetic" on the web. If you had been already familiar with modular arithmetic, all of the above could be shown by simply saying "It clearly follows from the fact that 10 is congruent to 1 (mod 9)". And it has lots of other interesting applications.

And yes, it also works for "3", (and for the same reasons).

salfriz

thanks for the explaination..

And yes, it also works for "3", (and for the same reasons).[/quote]

3x5=15
1+5=6...?

3x98=294
2+9+4=15
1+5=6...?

It doesn,t reduce back to 3 tho in the way that 9 x any number does.

ZorbaTehZ

salfriz wrote: »

It doesn,t reduce back to 3 tho in the way that 9 x any number does.

Does it matter? I mean the practical use of this is that one can check that their calculations are correct, which it does allow even with 3s.

MathsManiac

salfriz wrote: »

It doesn't reduce back to 3 tho in the way that 9 x any number does.

Sorry, I was thinking in terms of the congruence to 3. I see now that I should have said that you always land back at a multiple of 3, (that is 3, 6, or 9). This is sort of the same thing as the case for 9; the difference is that 9 is the only single-digit multiple of 9.

Squirrel

The way I always looked at is:

The first multiple of 9 is 9, proceeded by an infinite number of 0s.
The next number is 9+9,or 9+(10-1).

The number in the next place-holder is increasing by 1, and the original place-holder is decreasing by 1. ie: 1 ten and 8 units.

As this is carried one place holder is increasing by 1 and another is decreasing by 1. Except in cases like 99 where it reverts back to the original case of adding the 2 digits, 9+(10-1).

I'm not sure if place-holder is the correct term, I mean Units, Tens, Hundreds etc.

Doppler

salfriz wrote: »

I have a question which i wonder about from time to time and would appreciate any responses.

1 Why if you multiply 9 by any whole number and add up the reducing digets until they reduce to one diget will yo always get 9

ie: 9x43=387 3+8+7=18 1+8= 9

I cannot see the connection between the multiplication and the addition of digets. Do any other numbers behave the same?

I dont know much about maths so maybe there's a simple solution that i can't see...thanks anyhow.

And on the day after the eighth day, He created the number nine, to cofuse and amaze....;)

pointywalnut

The reason is we are doing arithmetic in base 10 (denary system).

If you did arithmetic in base 8 (octal) the same would apply to the number 7 = (8 - 1). 14 in octal is 16 (8*1+6), adding 1+6 together you get 7.

In base 16, the number 15 has that property.

In general; base - 1 has that property.