07-29-2016, 06:25 PM
(This post was last modified: 09-01-2022, 01:05 AM by Fearless Community.
Edit Reason: Fixed Encoding
)
Right, here's a more visual representation of what he has done:
![[Image: 50d84171bf814c7e92bfc2a4f453a91e.png]](https://wsrv.nl/?url=i.gyazo.com/50d84171bf814c7e92bfc2a4f453a91e.png&w=1920&h=1080&fit=inside&n=-1&we)
The last line shows 3 different ways of expressing the exact same number: 0.999..., 9/9 and 1.
Another way of expressing 0.999... or 0.9 recurring is by the following infinite (and converging sum), which is equal to 1:
![[Image: 7f0a4c345494213ca72bdec4c945ccdd.png]](https://wsrv.nl/?url=i.gyazo.com/7f0a4c345494213ca72bdec4c945ccdd.png&w=1920&h=1080&fit=inside&n=-1&we)
The above notation effectively means: 0.9+0.09+0.009+... going on forever.
I'll show you that the above sum is true as it is a geometric series, so the sum to infinity is:
![[Image: 7941ffdea884347ebe8decdb2252b8f6.png]](https://wsrv.nl/?url=i.gyazo.com/7941ffdea884347ebe8decdb2252b8f6.png&w=1920&h=1080&fit=inside&n=-1&we)
where a is the first term (in this case 0.9) and r is the common ratio (what we multiply to get the next term) which is 0.1 in this case.
When we apply the formula:
![[Image: b2e59179e8eb3381a64044a27f081b66.png]](https://wsrv.nl/?url=i.gyazo.com/b2e59179e8eb3381a64044a27f081b66.png&w=1920&h=1080&fit=inside&n=-1&we)
Hopefully this clears it up a bit more.
![[Image: 50d84171bf814c7e92bfc2a4f453a91e.png]](https://wsrv.nl/?url=i.gyazo.com/50d84171bf814c7e92bfc2a4f453a91e.png&w=1920&h=1080&fit=inside&n=-1&we)
The last line shows 3 different ways of expressing the exact same number: 0.999..., 9/9 and 1.
Another way of expressing 0.999... or 0.9 recurring is by the following infinite (and converging sum), which is equal to 1:
![[Image: 7f0a4c345494213ca72bdec4c945ccdd.png]](https://wsrv.nl/?url=i.gyazo.com/7f0a4c345494213ca72bdec4c945ccdd.png&w=1920&h=1080&fit=inside&n=-1&we)
The above notation effectively means: 0.9+0.09+0.009+... going on forever.
I'll show you that the above sum is true as it is a geometric series, so the sum to infinity is:
![[Image: 7941ffdea884347ebe8decdb2252b8f6.png]](https://wsrv.nl/?url=i.gyazo.com/7941ffdea884347ebe8decdb2252b8f6.png&w=1920&h=1080&fit=inside&n=-1&we)
where a is the first term (in this case 0.9) and r is the common ratio (what we multiply to get the next term) which is 0.1 in this case.
When we apply the formula:
![[Image: b2e59179e8eb3381a64044a27f081b66.png]](https://wsrv.nl/?url=i.gyazo.com/b2e59179e8eb3381a64044a27f081b66.png&w=1920&h=1080&fit=inside&n=-1&we)
Hopefully this clears it up a bit more.
Regards,
aviator