SEQUENCES “Concave Mirror” CS is now available to order from Falt (France). Cassette housed in full colour poster, artwork by the label. You can preview/purchase it directly via the FALT Bandcamp page (link in bio). #sequences #falt #experimental #ambient #electronic #cassette #audiovisualsatmosphere
DSC_7417 OUTPOST Studio/Cyanotype Process Painting by Russell Moreton Via Flickr: Pattern and Chaos/Liminality/Tectonics Architectural surface for a Library, raw materials, light, silence and solitude.
Here’s another perspective on why .999… repeating is exactly equal to 1.
For any two distinct real numbers, we can always find a rational number strictly between them, i.e. that rational number must be able to be expressed as a terminating decimal or a repeating decimal. To be clear, that rational number is strictly between the two values; it is not allowed to be equal to either.
Suppose k is a rational number strictly between 1 and 0.9999…. If this is possible, then, I can write k exactly as either a decimal with finite digits, or I can write k as a repeating decimal. The problem is, there are no decimals with finite digits between 1 and 0.999… , and there is no way to write a repeating decimal that is greater than 0.999… and still less than 1. Either way, a k strictly between 1 and 0.999… does not exist. The only way this can be true is if those two numbers are not actually distinct. That is to say, 1 = 0.999…..
My favourite math fact is that 0.9999999.. is equal to 1. Exactly. Not approximately. Not as a rounded number. 0.9999 (recurring) is exactly 1.
This is a fun fact with a lot of interesting explanations! While every real number has a unique decimal expansion, sometimes you can express a number in more than one way, provided you bend the rules a bit and use infinitely many digits.
Here’s a little explanation I just came up with, involving a convergent sequence.
No matter how hard you try, you can never write out 0.999… because it has infinitely many digits. However, we can get as close as we please using numbers we can write out.
Start with 0.9. Then add another decimal place, 0.99. Then 0.999. You can imagine this pattern continuing indefinitely, and each term has a finite number of digits. Every step gets closer to 1, but never reaches it. We say that this sequence converges, and the value it converges to is 1. The notation might look something like this:
Infinity is a tricky thing and isn’t exactly real, but you can imagine the only way to get a term that “equals” 1 would be to go infinitely far. @secondaristh gave a great explanation that kind of ties into this sequence: for any number k you choose that is less than 1, you can always find a number in this sequence that is greater than it.
