Math Is Fun Forum

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

It might help to consider what we actually mean by e^x.

What is the definition of e^x?

Hint:

You'd need to:

-Differentiate w(x) with respect to x, to obtain an expression for w'(x)
-Find the values of x for which w'(x) = 0
-Substitute these values of x into your equation for w''(x) to determine which values of x correspond to a minimum and which correspond to a maximum