Let p ≥2 be a fixed integer, and let 0<x<1. If x has a finite-length base p decimal expansion, t

Let p ≥2 be a fixed integer, and let 0<x<1. If x has a...

Key Concepts

Base p Numeral System

This concept involves representing numbers in any integer base p (with p ? 2) where numbers are expressed as sums of multiples of powers of p. The digits range from 0 to p?1 and are arranged in a sequence to represent fractional parts as sums of a_i/p^i. This system generalizes the familiar base 10 (decimal) system and is fundamental in understanding how numbers can be expanded using different bases.

Terminating Expansions and Non-Uniqueness

A terminating expansion is one that has only finitely many nonzero digits before the digits eventually become zero; in other words, it is a finite-length base p expansion. A critical point regarding such expansions is that numbers with a finite expansion possess two representations. This is due to the fact that one can write the termination by having a last nonzero digit, or equivalently, by decreasing that digit by one and following it by an infinite string of the maximal digit (p?1). This phenomenon exemplifies a subtle non-uniqueness inherent in positional numeral systems.

Repeating (Periodic) Expansions

A repeating or periodic expansion is one in which after some point a sequence of digits repeats indefinitely. In a base p numeral system, if a number’s expansion repeats immediately (that is, there is no non-repeating prefix) it is called purely repeating. This concept is crucial because it ties directly into the characterization of certain fractions and how their numerator and denominator interact with the base. Repeating expansions reveal inherent cyclic patterns in the way numbers are represented when the denominator (in lowest terms) is relatively prime to the base p.

Eventually Repeating Expansions and Rational Numbers

An eventually repeating expansion is one that, after a finite non-repeating segment, transitions into a repeating cycle. This property is a hallmark of rational numbers: every rational number between 0 and 1, when written in any base p, has either a terminating or an eventually repeating expansion. Conversely, any number that has an eventually repeating (or purely repeating) expansion is rational. The distinction between terminating and eventually repeating expansions further relates to the prime factorization of the denominator; a terminating expansion occurs if and only if, in its reduced form, the denominator contains only the prime factors present in p.

Transcript

00:02 Hi, so we're going to be looking at series today and specifically we have the series right here and we're going to be using this to show that 0 .55 repeating can be written as a rational number.

00:16 So the first thing we're going to do is we're going to rewrite this as five times some.

00:31 So equals one going to infinity.

00:35 Of 1 over 10 to the k.

00:43 And from here, what we're going to do is solve for our a and r values.

00:50 So how we should do that is by plugging in 4k.

00:54 So we can just do that here, and we can say that, so the 5 on the outside, and then we have 1 is equal to k.

01:09 So in this case, that would just be 110, 1 over 10.

01:13 And that would be plus our next value, which would be 1 over 10 squared.

01:22 You could just put the square sign over the 10 because 1 squared is just equal to 1.

01:29 Same thing.

01:30 For the next one, we're going to just say 1 over 10 cubed for k to the 3rd.

01:37 And then we could just say that this would normally repeat.

01:40 We don't need to do anymore.

01:42 And we have our a value, actually, right here.

01:46 This is our a value.

01:50 A is equal to 1 over 10.

01:53 And now to find our r value, what we have to do is take 1 over 10 squared and divide that by 1 over 10.

02:05 So that would equal 1 over 100 times 10.

02:09 10 over 1, which is equal to 1 over 10.

02:15 So this is our r value.

02:18 And now what we have to do is define the sum, we would compute the equation a over 1 minus r.

02:30 So we're gonna do that now.

02:32 And our a value again is 1 over 10.

02:36 So we've 1 over 10 over 1.

02:41 Minus our r value, which is 1 over 10.

02:48 And now we can rewrite this as 1 over 10 over 10 minus 1 over 10, which is equal to 1 over 10 times, 1 over 10 over, sorry, 9 over 10, which is equal to now what we're going to do is because we have the equation we're just going to say this times five again so we have one over ten and that is all over nine over ten and we're going to simplify this and that will give us 0 .55 in a rational like written in a rational way so all we have to do is we have our five we have our one over ten and we're going to multiply that by 10 over 9, and our tens will cross out.

03:57 And if we distribute the 5 through, we will end up with 5 over 9.

04:07 Now you have another problem that we're going to do, which is the series that has 54 of the sum going to infinity with k equal to 1 of 10 to the 9.

04:23 Negative 2k.

04:24 And we're going to do the same thing we did earlier.

04:27 So we're just going to rewrite this.

04:28 We're going to have 54 times the sum.

04:35 And this time we're going to write it as 1 over 10 squared to the k, which is just equal to 54 times the sum of 1 over 100.

05:00 And now, from here, all we're going to do again is we're going to be plugging in k values.

05:05 So we can keep our 54 over here like we did.

05:09 And we're going to say for k is equal to 1, we would end up with 1 over 100, plus for k is equal to 2, it would be 100 squared, and then plus for 1 over again.

05:25 We have 100, when k is 3, it's 100 to the third.

05:34 And then again, continue, but you don't have to.

05:38 And from here, we're just going to say that our a value is equal to 1 over 100.

05:47 So a is equal to 1 over 100.

05:52 And we can compute our r value, which is equal to 1 over 100.

05:57 Equal to like did earlier 1 over 100 squared divided by 1 over 100 which is going to be equal to 1 over 100...

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