1=0.99999 (math thingy) ?

[Murakumo]

I doubt we have any mathematicians here but..... here's my dilemma:

"Prove that 1=0.99999"

That's the problem my teacher threw into our faces earlier in class. but we were all utterly clueless. How the hell can 1 be equal to a clearly lower value? Are all my years of basic math wrong? Am I just being too hardheaded here?

 

[Frankincense]

No matter how improbable an equation appears to be, it's wrong to jump to conclusions. That's the purpose of mathematical proof. =)

Regarding your teacher's problem, 1 is greater than 0.99999 indeed, but if your teacher meant that to be a recurring fractions (0.999...) then it's a totally different story.

 

[Murakumo]

I believe he was referring it to as a number on the number line, so if it's considered a number, then it's true?

 

[Frankincense]

A few very simple proofs exist. I'll just put them in spoilers in case you want to solve it yourself.

Spoiler: 1.

1/9 = 0.111...
9*1/9 = 9*0.111...
1 = 0.999...

Spoiler: 2.

x = 0.999...
10x = 9.999...
10x = 9+0.999...
10x = 9+x
9x = 9
x = 1
0.999... = 1

 

[KuroiHana]

You could actually make Proof 1 slightly simpler (it's basically the same but w/e ww):

Spoiler: Proof

3 * 1/3 = 1
1/3 = 0.333333......
3 * 0.3333.... = 0.9999.... = 3 * 1/3
-> 0.999... = 1

The difference between 0.999.... and 1 is infinitely small and as such it's considered to be 0 - which makes those values equal