what's 0.9 recurring as a fraction? please explain?

To turn a recurring decimal into a fraction, take the repeating part and put it over the same number of nines.

For example:
0.272727... = 27/99 = 3/11
0.135135... = 135/999 = 5/37
0.142857... = 142857/999999 = 1/7
Confirm the answers above with long division or a calculator.

When you only have one digit repeating, you use a single nine in the denominator:
0.111111... = 1/9
0.222222... = 2/9
0.333333... = 3/9 = 1/3
0.444444... = 4/9
0.555555... = 5/9
0.666666... = 6/9 = 2/3
0.777777... = 7/9
0.888888... = 8/9

Make sense so far?

Therefore:
0.999999... = 9/9 which is the same as 1

If you have learned about infinite geometric series:

0.99999.....
= 9/10 + 9/100 + 9/1000 + ....

Sum of an infinite geometric series: S = a/(1-r)
a = 9/10 ----- the first term
r = 1/10 ----- the common ratio

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

All numbers recur when divided by 9 therefore 999,999/1,000,000 and to arrive at it the no. is 1

0.99999999... = 1/1 = 1

A simple proof is from the definition of real numbers as a continuum. A property of real numbers is that there is no "next number". If you have two real numbers which are different from each other, there must be a third number in-between the two.

In mathematical terms:
if x and z are real numbers, and x < z, there exists a real number y such that x < y < z.

There is clearly no number between 0.99999999... and 1, therefore 0.99999999.... and 1 are the same number.

Its really simple.

x = 0.999999 Pick a value for the decimal
10x = 9.999999 Multiply everything by 10
10x - x = 9 Minus the original decimal
9x = 9 Tidy up
x = 9/9 Divide through
x = 1 Voila!

recurring means 0.999999 etc which = 1

Proof:
3 x (1/3) = 1
1/3 = 0.33333...
3 x (1/3) = 3 x (0.3333...) = 0.9999... = 1

Do the branch. as an occasion evaluate one million/3. 3 (divide into) one million.00000=.333333 a repeating decimal. yet in a diverse way is with a calculator; enter 3 and press the single million/x button. this is tough to coach a thank you to do some those solutions through loss of the right symbols

There is a rule for converting a repeating decimal number into a fraction. If the repeating pattern starts immediately after the decimal point, with zero before the decimal point, then we can use the following:

Let d = the fraction

d is equal to the infinite sum of A/10^n + A/10^2n + A/10^3n + A/104n.... This is a geometric series and by a well known formula its value is A/[(10^n) - 1]

So, in the case of 0.9, A = 9, n = 1.
Therefore,
d = 9/[10^1 - 1] = 9/(10 - 1) = 9/9 = 1

teddy boy

0.9 recurring = 0.9999... = 0.9 /(1 -- 0.1) = 0.9/0.9 = 1.0

0.9 x 100%
= 0.9 x 100/100
= 90/100
= 9/10