What is decimal expansion: Definition and 17 Discussions
A decimal representation of a non-negative real number r is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
Here . is the decimal separator, k is a nonnegative integer, and
b
0
,
…
,
b
k
,
a
1
,
a
2
,
…
{\displaystyle b_{0},\ldots ,b_{k},a_{1},a_{2},\ldots }
are digits, which are symbols representing integers in the range 0, ..., 9.
Commonly,
b
k
≠
0
{\displaystyle b_{k}\neq 0}
if
k
>
1.
{\displaystyle k>1.}
The sequence of the
a
i
{\displaystyle a_{i}}
—the digits after the dot—is generally infinite. If it is finite, the lacking digits are assumed to be 0. If all
a
i
{\displaystyle a_{i}}
are 0, the separator is also omitted, resulting in a finite sequence of digits, which represents a natural number.
The decimal representation represents the infinite sum:
Every nonnegative real number has at least one such representation; it has two such representations (with
b
k
≠
0
{\displaystyle b_{k}\neq 0}
if
k
>
0
{\displaystyle k>0}
) if and only if one has a trailing infinite sequence of 0, and the other has a trailing infinite sequence of 9. For having a one-to-one correspondence between nonnegative real numbers and decimal representations, decimal representations with a trailing infinite sequence of 9 are sometimes excluded.
Hi All,
The famous proof of the theorem: ## 1 = 0.9999999...## seems to point to a statement more or less like this:
"There is no uniqueness in decimal expansions of real numbers, specially if one wishes to compare numbers (and their decimal expansions) extremely close of one another."
Is this...
Homework Statement
How many decimal expansions terminate in an infinite string of 9's?
How many dont?
The Attempt at a Solution
If we have a number terminate with an infinite amount of 9's then it will be a rational number.
So there would be countably many of these.
And since...
Homework Statement
Prove that every real number x in [0,1] has a decimal expansion.
Homework Equations
Let x\in{[0,1]}, then the decimal expansion for x is an infinite sequence (k_{i})^{\infty}_{i=1} such that for all i, k_i is an integer between 0 and 9 and such that...
Homework Statement
How to compute the base 3 decimal expansion of 1/4?
Homework Equations
The Attempt at a Solution
I tried sums of geometric sequences,but i need a clue for the computation.
I am given the number $.334444\ldots$
So we have $0+33\times 10^{-2} + \sum\limits_{n=3}^{\infty}4\times 10^{-n}$
Is there a way to put this all together in one geometric series?
I read here in Physics Forums that a number can have more than one decimal expansion.
Really? Can someone explain how?
Is it that any number can have more than one decimal expansion or only some numbers?
I have the number .1212212221222212222122222122222221222222221222222222... (notice that the number of 2s increase by one each time)...
Is there a way to come up with an equation that would tell you the number that's in the nth digit?
I am asked to prove that a real number is rational if and only if it has a periodic decimal expansion.
I have shown that any periodic decimal expansion has an integer p such that multiplication returns an integer. For the case of showing that all rational numbers have a periodic decimal...
Homework Statement
Use the fact that every real number has a decimal expansion to produce a 1-1 function that maps S into (0,1). Discuss whether the formulated function is onto.
Homework Equations
S={(0,1):0<x, y<1}
The Attempt at a Solution
I don't even know where to begin. The...
Homework Statement
1/7=?
Homework Equations
The Attempt at a Solution
Consider $$y=\overline{145827}.0=\dots145827145827145827.0$$,
then $$1000000y=\dots 145827145827000000.0$$ and $$-999999y=145827$$ .
Therefore, $$y=-\dfrac{145827}{999999}$$ . Son in fact...
"An infinite decimal expansion has the following meaning:
0.a1 a2 a3 a4... =
∞
∑ ak / 10k =
k=1
sup{∑ ak / 10k : n E N}
where ∑ is the sum from k=1 to k=n."
======================================
Why is the second equality (about the supremum) true? (or why does it make sense?)
Thanks.
Homework Statement
Let n/m be a positive rational number in lowest terms. By examining the long division algorithm, show that the decimal expansion of n/m is eventually periodic, and that the period divides phi(m). For simplicity,you may assume that (m, 10) = 1.
Homework Equations...
My teacher gave us as excercices this:
I'm pretty certain you have to prove it by contradiction, but I don't get how to represent to periodic decimal expension in a proof?
Any hint is welcome, thanks in advance.