How 9801 generates a nice integer sequence

00:00:00.000 | The fraction 1/99^2 generates as its decimals something that looks a lot like the non-negative

00:00:12.960 | Showing this fraction as this kind of sum may give you a hint.

00:00:17.400 | But let's take a look further.

00:00:20.720 | Let's use this geometric series here that holds true for when the absolute value of

00:00:24.800 | x is less than 1.

00:00:26.920 | 1/(1-x) is equal to 1+x+x^2+x^3 and so on.

00:00:33.880 | Taking the derivative of both sides, the equality still holds true, resulting in 1/(1-x^2)=1+2x+3x^2+4x^3

00:00:44.760 | and so on.

00:00:45.760 | Now here's how we get back to the magical fraction that generates something very close

00:00:51.360 | to the non-negative integers.

00:00:53.700 | We plug in 1/100 into x.

00:00:57.800 | The result on the left hand side is our fraction, 1/99^2.

00:01:03.640 | And on the right hand side, the sum 1/100^2+2/100^3+3/100^4 and so on.

00:01:13.640 | And when we take the sum, we get that nice sequence of 0, 1, 2, 3, 4 and so on that we

00:01:19.280 | saw before.

00:01:20.840 | Now if we return to the derivative of the geometric series that we saw before and plug

00:01:25.160 | in x=1/100, we get the fraction that we started the video with.

00:01:30.000 | But we can actually change the number of padding 0s in the decimal sequence that's generated

00:01:34.760 | by changing the value of x.

00:01:37.280 | For x of 1/10, the fraction is 1/9^2 or 1/81 and the padding is less.

00:01:44.240 | For x of 1/1000, the fraction is 1/999^2 and the padding is greater.

00:01:52.680 | And you can arbitrarily increase the denominator of x by multiples of 10 to increase the padding

00:01:59.040 | on the resulting decimal sequence.

00:02:01.580 | So there you have it.

00:02:02.580 | There's a little bit of math that shows how a strange little fraction can generate a beautiful

00:02:07.760 | decimal sequence.

00:02:09.740 | You may have noticed that in this case, the number 98 is missing.

00:02:14.980 | The number 98 is not missing in the underlying summation.

00:02:18.780 | But since we're doing base 10 arithmetic, eventually the numbers overflow, resulting

00:02:22.680 | in a decimal sequence that's missing the number 98 before it starts repeating.

00:02:27.280 | So while the underlying summation includes the non-negative integers, the resulting representation

00:02:34.120 | of the number in decimal form in base 10 notation actually is missing the number 98 and is actually

00:02:42.480 | a repeating decimal.

00:02:44.080 | I hope you enjoyed these little videos.

00:02:45.560 | They're easy and fun for me to make and allow me to share some basic and advanced ideas

00:02:50.160 | in mathematics, computer science, physics, machine learning, and also into the softer

00:02:55.040 | sciences of psychology, history, philosophy, and so on.

00:03:00.320 | I look forward to sharing these with you.

00:03:02.040 | And remember, try to learn something new every day.

00:03:07.160 | Page 1 of 2

00:03:07.160 | Page 2 of 3

00:03:08.160 | Page 3 of 4

00:03:08.160 | Page 5 of 6

00:03:13.160 | Page 6 of 7