Is 0.9 recurring equal to 1?Yes — 0.9 recurring (0.999…) is exactly equal to 1.
Here are a few simple ways to see why:
1️⃣ Algebra proof
Let
x=0.999…
Multiply both sides by 10:
10x=9.999…
Now subtract the original equation:
10x-x=9.999…-0.999…
9x=9
x=1
But we said x=0.999…, so:
0.999…=1
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2️⃣ Fraction proof
We know:
1/3=0.333…
Multiply both sides by 3:
1=0.999…
We also know that 1/9=0.1 recurring and 2/9=0.2 recurring etc and in that spirit 9/9 is equal to o.9 recurring. But we also know 9/9 is equal to one so 0.9 recurring must be too.
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3️⃣ Intuition explanation
There is no number between 0.999… and 1.
If they were different, there would have to be some tiny gap — but there isn’t any real number you can put between them. So they must be the same number.
4. Geometric progression
To show that 0.9 ̇=1using a geometric progression, write it as an infinite series.
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Step 1: Express 0.9 ̇as a sum
0.9 ̇=0.9+0.09+0.009+0.0009+⋯
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Step 2: Identify the geometric series
This is a geometric progression with:
First term: a=0.9
Common ratio: r=0.1
So we have:
0.9+0.9(0.1)+0.9(0.1)^2+0.9(0.1)^3+⋯
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Step 3: Use the infinite geometric series formula
For ∣r∣<1, the sum to infinity is:
S=a/(1-r)
Substitute a=0.9and r=0.1:
S=0.9/(1-0.1)
S=0.9/0.9
S=1
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Conclusion
0.9 ̇=1
So using a geometric progression, we’ve shown rigorously that 0.9 recurring is one.
In short, as we get closer to the sum to infinity we get closer to one.
These are powerful tools to turn recurring decimals back to fractions. The Algebra method is the GCSE method. It is nice not only being able to turn fractions to recurring decimals.
