1/9801 (which is 1/99²) produces a decimal expansion that lists integers in sequence: 0.00010203040506... The trick uses calculus on geometric series, substituting x=1/100 to place each integer in its own decimal block with padding zeros.
The fraction 1 over 99 squared generates as its decimals something that looks a lot like the non negative integers. Why is that? Showing this fraction as this kind of sum may give you a hint, but let's take a look further.
Let's use this geometric series here that.
Holds true when the absolute value of x is less than 1. 1 over 1 minus x is equal to 1 plus x plus x squared plus x cubed and so on. Taking the derivative of both sides, the equality still holds true, resulting in 1 over 1 minus x squared equals 1 plus 2 x plus 3 x squared plus 4 x cubed and so on. Now here's how we get back to the magical fraction that generates something very close to the non negative integers. We plug in 1/100 into X. The result on the left hand side is our fraction 1 over 99 squared.
And on the right hand side, the.
Sum 1 over 100 squared plus 2 over 100 cubed plus 3 over 100 to the power of 4 and so on. And when we take the sum, we get that nice sequence of 0, 1, 2, 3, 4 and so on that we saw before.
Now, if we return to the derivative.
Of the geometric series that we saw before and plug in x equals to 1 over 100, we get the fraction.
That we started the video with.
But we can actually change the number of padding zeros in the decimal sequence that's generated by changing the value of x. For x of 1 over 10, the.
Fraction is 1 over 9 squared or.
1 over 81, and the padding is less. For x of 1 over 1000, the fraction is 1, 1 over 999 squared and the padding is greater.
And you can arbitrarily increase the denominator.
Of x by multiples of 10 to increase the padding on the resulting decimal sequence.
So there you have it.
There's a little bit of math that shows how a strange little fraction can generate a beautiful decimal sequence. You may have noticed that in this case, the number 98 is missing.
The number 98 is not missing in the underlying summation. But since we're doing base 10 arithmetic, eventually the numbers overflow, resulting in a decimal sequence that's missing the number 98 before it starts repeating. So while the underlying summation includes the non negative integers, the resulting representation of the number in decimal form in base 10 notation actually is missing the number 98 and is actually a repeating decimal. I hope you enjoy these little videos. They're easy and fun for me to make.
And allow me to share some basic and advanced ideas in mathematics, computer science, physics, machine learning, and also into the software sciences of psychology, history, philosophy and so on.
I look forward to sharing these with you. And remember, try to learn something new every day. Sa.
