Can anyone prove to me...?

Question

My professor showed our class how 0.999999... could equal to 1, but i was confused. Can anyone explain to me how 0.999... equal to 1?? I have to prove this on my midterm next week.

Answer 1

It is not equal to 1, but because of the proportion it is reeaaaaaally close to it. The more decimals you got, the closer it goes to 1, unless you want to use some calculus and have to approach your function to infinity.

Another way of looking at it,
1 - 0.999999999999 = 0.000000000001. The last number is so small that it can barely affect your calculations, now imagine if you had more decimals.

Answer 2

Note that 0.9999(repeating) can be written as

9/10 + 9/100 + 9/1000 + ......

a[0] = 9/10
r = 1/10

And we have a formula for the sum of an infinite series.

S = a[0]/(1 - r)

Therefore,

S = (9/10) / (1 - (1/10))

Changing 1 into 10/10, we get

S = (9/10) / (10/10 - 1/10)

Calculating the fraction on the denominator,

S = (9/10) / (9/10)

Multiplying the top and bottom by 10, we eliminate the fractions within fractions, and get

S = 9/9

Therefore,

S = 1

Answer 3

Actually, in algebra, 0.9999.... is undefined. 0.999, for example, is defined by its place holding notation as:

(9/10 + 9/100) + 9/1000

and the parentheses can be dropped because of the associative law. We can get any finite number of terms in the sum, but not an infinite number. Algebra simply does not recognize an infinite sum such as

0.999... = 9/10 + 9/100 + 9/1000 + ....

One way of making sense out of this is to use limits. The partial sums:

0.9, 0.99, 0.999, ...

have the property that any neighborhood of 1 will contain these numbers, and that is true only of 1. So we can say that the limit of this series is 1. It is in this sense that 0.9999.... = 1.

Another way to see this is to let x be 0.9999... Then

x = 0.999.... = 0.9 + 0.0999.... = 9/10 + 1/10 (0.999...) = 9/10 + (1/10) x

or

x = (9+x)/10

Solving this for x yields 1. So 0.9999.... = 1. The key in this is to recognize that 0.999... is a self-similar object, similar to many fractals. It contains a copy of itself. But this makes sense only in terms of limits or something similar.

Answer 4

Think of it in a simple way:

1/3 = 0.33333333333333

Multiply both sides by 3
3*(1/3) = 3*(0.3333333333)
1= 0.9999999999999

Hope this helps.

Answer 5

Multiply out the product tan^2q*(1+cot^2q) = tan^2q + 1 = sec^2q Now tan^2q +1 = sec^2q so tna^2q +1 = sec^q-1 +1 = sec^2q QED

Answer 6

sorry-it is now known to me