(a) Why is 0.4999 …=0.5 ? (Try to give more than one reason.) (b) Write 0.234234234 …as a fracti

step 1 Let x be 0 .9, basically 0 followed by infinitely many nines. Let's use this shorthand notation

(a) Why is 0.4999 …=0.5 ? (Try to give more than one...

Key Concepts

Infinite Series and Limits

This concept involves understanding that a repeating decimal can be viewed as the sum of an infinite geometric series. By expressing the repeating part as a series and using the formula for the sum of an infinite geometric series, one can demonstrate that some repeating decimals are exactly equal to their finite counterpart, as in the case of 0.4999… equaling 0.5.

Conversion of Repeating Decimals to Fractions

Converting a repeating decimal into a fraction involves representing the number as an algebraic equation and manipulating it so that the repeating part cancels out. This is often done by multiplying the repeating decimal by an appropriate power of ten, subtracting the original number, and solving for the number in fractional form, thereby showing the equivalence between the repeating decimal and its fractional representation.

Uniqueness of Decimal Representations

This concept addresses the fact that most real numbers have one unique decimal representation, but certain numbers admit two representations due to the behavior of infinite repeating decimals. Specifically, numbers whose decimal expansion terminates can also be represented by an alternative form where the terminal zeros are replaced by an infinite string of 9's, highlighting a subtle nuance in how real numbers are expressed in decimal notation.

Transcript

00:01 Let x be 0 .9, basically 0 followed by infinitely many nines.

00:07 Let's use this shorthand notation to represent that.

00:11 For part a, do you think that x is less than 1 or x equals 1? so i know the answer already, but let's just make a guess here.

00:23 Make your own guess and then record it.

00:25 And then we'll come back to that guess.

00:29 So here we'll end up seeing that it is exactly one, which is quite surprising to most people because it does have a zero on the left of the decimal.

00:41 But we'll see this in part b.

00:42 Let's go ahead and prove it using part b.

00:44 So sum the a geometric series to find the value of x.

00:48 So here all they're saying is let's first write x as a sum and so on.

01:01 Now we can write this as 9 over 10, 9 over 1.

01:03 9 over 100, 9 over 1 ,000, and so on.

01:09 And now it's starting to appear that we have a geometric series.

01:14 And we know the sum of a geometric series, the sum equals the first term over 1 minus the common ratio r.

01:24 This is for a geometric sum.

01:27 So the first term, we see that's 9 over 10.

01:33 And then we divide by 1 minus and what is our value of r? what are we multiplying by each time it's 1 over 10 so we have 9 over 10 divided by 9 over 10 and that's exactly 1 so that verifies that the guess in part a was correct and then see how many decimal representations does the number 1 have well we have one one decimal representation of this form let's just write it that way however we just saw in the part b here that we can also write it is, or if you want, just 1 .0 with a bar.

02:25 So here's, these are two distinct decimal representations of the same real number.

02:34 So two, that'll be our answer for part t.

02:41 And now let's generalize this to other real numbers...

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