More Issues With Floating Points

Number Theory Level 5

It is well-known that \(\dfrac{1}{3}\) cannot be represented using a finite a finite decimal representation. \[\dfrac{1}{3} = 0.333 \ldots.\]

Neither can it be represented using a finite binary representation. \[\dfrac{1}{3} = 0.010101 \ldots _2 \]

If we had just 24 bits to store the decimal part of this value, the closest rational number we could store in binary would be

\[\dfrac{5592405}{16777216} \]

What is the closest rational number to \(0.1\) that we could store using binary floating points with only 24 bits (assuming all the 24 bits are being used to store the fractional part of the value)?

If the value comes out to be \(\dfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, submit your as \(a+b\).

Assume that the point occurs before all the 24 bits. So, 111111111111111111111111 will be interpreted as 0.111111111111111111111111


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