Floating point representation of a real number is not unique

Question

I am reading about floating point representation of real numbers.
In any real number system with arithmetic base β, a real number x can be represented as
x=(0.d ₁ d ₂ --- d _t d _t+1 ---) _β X β ^e (1)
where 0≤d _i ≤β-1
Now consider (0.5) ₁₀ in binary system.
In binary system, (0.5) _{10} =(0.1) ₂ (2)
But we can see that 1/2=(1/2 ² )+(1/2 ³ )+----
(0.5) ₁₀ =(0.01111---) ₂ (3)

From (2) and (3) we can see that the same number has two floating point representation.

I have read that such ambiguities can be removed by considering the "normalized floating point representation" ie,
d ₁ ≠ 0

But with the normalized floating point representation, we still have the ambiguity.
(0.5) ₁₀ =(0.100000) ₂ (2(a))
(0.5) ₁₀ =((0.1111111----) X 2 ^-1 ) (3(a))

So from (2(a)) and (3(a)) we can see that the ambiguity is not resolved still even with normalized floating point representation. Can somebody please tell whether I am right. And how to resolve this ambiguity?

Answer 1

... with the normalized floating point representation, we still have the ambiguity.

No ambiguity.

---

(2(a)) (0.5) ₁₀ = (0.100000) ₂
(3(a)) (0.5) ₁₀ = ((0.1111111----) ₂ X 2 ^-1

(2(a)) is true.
(3(a)) is false as the number of terms in a floating point value is finite. This is no more true than 1.0 equals 0.999999999999999.

With common binary floating point, all finite values have only 1 encoding, except +0.0 and -0.0.

Answer 2

(3(a)) (0.5)10 = ((0.1111111----)2 X 2-1 is true.  **BUT** real world floats are finite, so 0.1111111---- does not exist.