Yes — really! 0.9 with nines that go on forever is exactly equal to 1.
That tiny … means the nines never stop: 0.99999999… going on and on and on, forever.
It sounds impossible. But by the end of this lesson, you'll see three clever ways to prove it. Let's go! 🚀
Every time we add another 9, the number gets closer to 1. The little gap left over keeps shrinking!
Gap left until 1:
Gap = 0.1
Tip: keep tapping. The gap gets tiny — and with infinite nines it vanishes completely.
Here's a rule about numbers: if two numbers are different, you can always find another number squeezed between them.
Try it: between 0.2 and 0.4 sits 0.3. Easy!
So… what number could fit between 0.999… and 1? Let's hunt for one on the next slide. (Spoiler: there's nowhere to put it! 😉)
Tap a number and we'll check: does it fit in between? Remember, to fit it must be bigger than 0.999… but smaller than 1.
Share 1 whole chocolate bar fairly between 3 friends. Each gets one third — and as a decimal that's 0.333… (threes forever).
Tap each piece to give it back. What do 3 pieces add up to?
Let's give our mystery number a nickname: x. Tap to reveal each step!
If 9x = 9, then what is x?
Both proofs landed in the same place — and the gap hunt found nowhere to hide a number. That can only mean one thing:
The thirds way:
⅓ = 0.333… → ⅓ × 3 = 0.999…
but ⅓ × 3 = 1
The x10 way:
9x = 9 → x = 1, and x = 0.999…
It's not almost 1. It's not nearly 1. It is exactly, truly 1. 🌟
Here's the big secret you just unlocked:
You just understood something that surprises grown-ups too. Maths is full of secrets like this — keep exploring! 🚀