Why is 0.99... equal to 1?
As a math enthusiast, I was very curious in my school days (coz now I am in college!). In the 9th grade, once our algebra teacher taught us how to represent recurring decimals into p/q that is, in rational form. And you know what rational numbers are; any number in p/q form where p and q are any integers and q is not equal to 0.
Here is how to represent them:
For example 0.77...
Let 0.77...=x
∴ 10x = 0.77... × 10
∴ 10x = 7.77...
∴ 10x-x = 7.77... - 0.77...
∴ 9x = 7
∴ x = 7/9
But x is 0.77...
Thus, 0.77... is 7/9 in p/q form.
Similarly, 0.33... is 3/9, 0.55... is 5/9.
If we again divide the p/q form, we get the original recurring number. Like, 0.55... is 5/9 and 5÷9 is again 0.55...
So my curious mind asked me what about 0.99...? We will express 0.99... in p/q form as the above procedure and we will get 9/9. So, 0.99... is 9/9. But when we divide 9/9, we get 1 and not 0.99... As we know 0.99... is approximately equal to 1. But the above procedure is contradicting this statement.
I was confused. Even though 0.99... is approximately equal to 1, I got 0.99... equal to 1. I asked my algebra and geometry teacher too. They appreciated my doubt but weren't able to justify it.
Soon then, the next day, we had a seminar in our school conducted by Sir Vinay Nair. He is the founder of SOVM and has an expertise in mathematics.
Then I asked him my doubt and he gave me a satisfying answer. He said that 0.99... is very close to 1 but not equal. And we follow decimal system of numbers where numbers from 0 to 9 are taken. Thus there is a problem with 0.99... when it comes to p/q form. In binary, the same problem would have been with 1.
Thus 0.99... is not equal to 1 in maths but in decimal system, Yes, it is.
Posts before October 2021 have been marked as "Old Posts". Less likely, but they might have out dated or incorrect information, ugly looking bits of code, no labels, etc. Don't get me wrong, many of these posts are top-notch and interesting too.
I thought it would be better not to delete or revamp these posts, even if they suck. The bitter truth is that old works always suck, but I take that as a positive tool to convey that I am growing. Besides there's no better way to showcase my journey without these old, messy, poorly written posts!
I thought it would be better not to delete or revamp these posts, even if they suck. The bitter truth is that old works always suck, but I take that as a positive tool to convey that I am growing. Besides there's no better way to showcase my journey without these old, messy, poorly written posts!
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