1.........2........2½........2¾.........🤔
What comes after 2¾? Is it 2⅝, or 2⁷⁄₉? 🤔🤔
Fortunately we can calculate this. But for brevity think of a line of numbers - aka the real number line. You are at a some point on that line. numbers behind you are smaller than numbers ahead of you (including the one you are standing on).
Or, if you look at it from another perspective, numbers to the left of you are smaller than numbers to the right.
This is the "real" number line. Some of the points represent fractions. In mathematics they say there are infinitely many real numbers...But...we won't be exploring that.
So let's take an interval on this line - say 0 to 1. We know that half (i.e 0.5 or ½) is a point in this interval. Correctly at the midpoint. There is a formula to get you there, but we usually learn it as part of cartesian geometry.
If you got two numbers on a numbers line, A and B such that A is less than B, you can find a number exactly equidistant from both by computing (A+B)/2.
There are numerous proofs for this. We are not doing that today❗
Now you play the game of iterative substitutions - plug A = 0 and B = 1, and, you get (A+B)/2 = ½ = 0.5
What is between 0.5 and 1? Substitute A = 0.5 and B = 1, and, you get a result 0.75 = ¾
But we humans don't say 2, 2.5, 2.75, ... 🚫
We use the number-with-fraction form. 2 ... 2½ ...2¾... ✅
Do you see the pattern?
What comes after ¾? Use the formula to calculate. The result will be ⅞
The next one will be 15⁄16 , ³¹⁄₃₂, ⁶³⁄₆₄, ...
Do you see the pattern?
Now you can say the numbers and never reach three, just like xeno's paradox. I know I have left you haning on xeno's paradox. that is easily searchable in the modern day.
However, I will pray that your skill of multiplying by 2 is good. 👍😉
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