6.99999999999999999?

Question

what is the easiest or a few ways at which to arrive at this number
(i mean it to continue forever)

Answer 1

The easiest way is: 7.

In general any number like x.99999999... is equal to x+1. (And, for instance, 1.30299999... = 1.303, and so on.)

There are a variety of ways to prove it, but I don't know what if anything you've learned about geometric series or limits or the properties of real numbers, so I don't know if you will be able to follow them.

If you've studied geometric series, you can spot that this is 6 + (0.9 + 0.09 + 0.009 + ... )
where the bit in brackets is a geometric series with first term 0.9 and ratio 0.1, so has infinite sum 0.9 / (1 - 0.1) = 1. So we get 6 + 1 = 7.

If you know about limits, you can view 6.999... as the limit of the sequence (6.9, 6.99, 6.999, ...). Here the n'th term is 7 - 10^(-n), which has limit 7 as n -> ∞.

But I, personally, think the most persuasive argument is this: if x = 6.9999... is not equal to 7, then |x-7| = d for some d > 0 (since it's a property of real numbers that for any real numbers x and y, |x-y| >=0 and |x-y| = 0 <=> x = y; formally, this is because the real numbers are a metric space with the metric d(x, y) = |x-y|). But if d > 0 then we can find an integer N such that 0 < 10^-N < d, and if we take more than N decimal places of x, we can see that |x-7| < 10^-N. So d < 10^-N and 10^-N < d, a contradiction; so 6.999.... must be equal to 7.

Answer 2

The easiest way to generate this number longhand is dividing 7 by 1:
. 6.99
1)7.0
. 6
. 1.0
. . . 9
. . . 10
. . . . 9
Obviously, it continues this way forever.

Using a spreadsheet and concatenation of strings,
. . . . A . . . |
1 | "6." . . . |
2 |=A1&"9" |
3 |=A2&"9" |
4 |=A3&"9" |
Use Fill down for as many digits as you desire.
(If you used actual numbers, Excel would eventually round to 7)

Using WORD, type 6. and hold down the 9 key

Mathematically again,

. . . . . . . n
N = 6 + ∑ 9/10^i
. . . . . . i=1
Just don't take the limit, because that's 7.

Answer 3

You can write it as 6.9 with a dot or a bar over the 9, but its value is 7.

You could consider the number to be:
x = 6 + ( 0.9 + 0.09 + 0.009 + .....) to infinity .........(1)
The sum to infinity of a geometric series with first term 0.9 and common ratio 0.1 is:
0.9 / (1 - 0.1) = 0.9 / 0.9 = 1.
Substituting this value in (1):
x = 7.

If I use [ ] to indicate the recurring decimal, you could write:
x = 6.[9] ..............(2)
Multiply by 10:
10x = 69.[9] ........(3)
Subtract (2) from (3):
9x = 63
Divide by 9:
x = 7.

These two techniques are not really different, as the second is the method used to derive the formula for the sum to infinity of a geometric series.

All methods for finding the value of a recurring decimal use the concept of a limit in one way or another.

Answer 4

there's a simple formulation,

(whole number without regarding the point - not circulating part) / (FOR AFTER THE POINT PART 9s as many as the circulating number and 0s as many as the non circulating number)

for this 6.9

(69-6)/9 = 7

another example

0.33333333

(3-0)/9 = 1/3

0.6262626262

(62-0)/99 = 62/99

Answer 5

I've seen the first three answers. There's another way to prove this is equal to 7 (Yes, it is equal to 7, not just "considered as" 7 because it's nearly there!!)

Put x = 6.999999999....
Now multiplying by 10 shifts the decimal point one place to the right, and so
10x = 69.999999999....

Subtracting gives
9x = 63 (exactly!)
and so x = 7.

Now I see the person just above me has already posted this method.

Answer 6

Use the formula for the sum of descending geometric series:

6.99999999999.... = 6+Σ(n=1,inf) 9*10^-n = 6 + 0.9(1 - 0.1) =
= 6 + 0.9/0.9 = 6 + 1 = 7

Answer 7

The number is so close to 7 that it is mathematically considered as 7. There's no fraction that will effectively produce this number unfortunately.

Answer 8

By dividing 6999999999999999999999999999999999999 as many 9 as you want with 1 followed by 0 with as many nines you have..

Answer 9

write down 6.9 (with a little bar over the 9.. it means to keep going forever)

Answer 10

x=6.9999999999...
10*x=69.99999999...
10*x-x=9x=69.99999-6.99999=63
9x=63
x=63/9
x=7