0.999… = 1? Two Simple Proofs That 0.99999 Equals 1

Step by step

1. Does 0.999… (forever) equal 1?
   Does zero point nine nine nine, forever, equal one?
2. Yes? No? Or almost?
   Yes, no, or almost? Decide before we prove it.
3. Many say: "always a tiny bit less" · let's prove it
   Many people say it's always a tiny bit less than one. Let's prove it, two ways.
4. 1 ÷ 3: remainder 1 every time → 1/3 = 0.333…
   Way one: divide one by three. Ten divided by three is three, remainder one. Bring down a zero: three again, remainder one. It never stops. One third is zero point three three three, forever.
5. × 3: 3/3 = 1 = 0.999…
   Now multiply both sides by three. Three thirds is one. Three times zero point three three three is zero point nine nine nine. So they're equal!
6. Way 2: x = 0.999… → 10x = 9.999…
   Way two, the test way. Call the number x. Multiply by ten: ten x is nine point nine nine nine, forever.
7. Subtract: the nines cancel → 9x = 9 → x = 1
   Subtract. The endless nines line up and cancel exactly. Nine x equals nine, so x is one.
8. 0.9 → 0.1 short · 0.99 → 0.01 · 0.999 → 0.001
   But it feels a little less! Zoom in on the number line. Zero point nine is a tenth short of one. Zero point nine nine, a hundredth. Zero point nine nine nine, a thousandth.
9. Gap smaller than all of these → gap = 0
   With nines forever, the gap is smaller than every one of these. The only gap that small is zero. No number fits between them, so they're the same number.
10. Repeating decimal → fraction: 0.777… = 7/9
    Where you'll see it: this exact trick turns any repeating decimal into a fraction. Zero point seven seven seven is seven ninths.
11. On a test: 10x − x = 9 → x = 1
    On a test: let x equal zero point nine nine nine. Then ten x is nine point nine nine nine. Subtract, and nine x equals nine. So x equals one.