Math Is Fun Forum

Knewlogik

Member

Registered: 2020-05-11

Posts: 24

Repeating numbers

So what do you decimal number that when multiplied by another number doesn't change but rather the decimal point does for example 2.040 816 326 530 612 244 897 959 183 673 469 387 755 102*2=4.081 632 653 061 224 489 795 918 367 346 938 775 510 24

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,385

Re: Repeating numbers

Hi Knewlogik,

See the links:

Decimal Numbers

Rational Numbers

Irrational Numbers

Transcendental Numbers

Complex Numbers

and

Squares and Square Roots.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Repeating numbers

hi Knewlogik

That number is so long it has blown my mind.  So I'll start small.  Are you talking about this sort of thing?:

2 x 142857 = 285714
3 x 142857 = 428571

etc etc.

And if you are, what is your question?

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Knewlogik

Member

Registered: 2020-05-11

Posts: 24

Re: Repeating numbers

Why does the number repeat the 42 digits repeat with all multplied  numbers except multpiles of 7

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Repeating numbers

Have you got any more examples, with less digits?

How did you stumble upon this number in the first place?

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

Website

Re: Repeating numbers

This looks like the decimal representation of

, which also explains why you don't get the same sort of pattern when you multiply your number by 7, since you'd get

which has a different string of recurring decimal places.

Since

is a rational number, then its decimal expansion will recur (i.e. repeat). The length of the string of digits which recurs is usually called the 'period', and the string of digits which repeats itself is sometimes called the 'repetend'. If you multiply this by any number which is coprime to 49, then you'll end up with the same sequence of decimal places, but 'shifted along' by some number.

Bob's example is a good one because 1/7 is a cyclic number, i.e. if you calculate successive multiples of 1/7 (i.e. 2/7, 3/7, 4/7, etc...) then you're shifting the decimal places by some amount each time (but preserving their 'order'). In other words, you get cyclic permutations. 1/7 has a recurring decimal expansion with period 6.

You can sometimes use properties of periods to determine what the length of the period is (or at least reduce the number of possibilities to a more manageable size!). For example, it's true that if a is coprime to b (that means that a and b share no common factors apart from 1), then the period of a/b is the same as the period of 1/b. This means, for example, that the period of 100/49 is the same as the period of 1/49. But since 49 is the power of a prime (it's 7 x 7 = 49), then since:

then you can say that

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Repeating numbers

Hi zetafunc,

Excellent answer!  Many thanks.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob