# Why ADCs Use Integer Math

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### The Problem With Fractions, Decimals, and Heximals

Just as numbers have place value to the left of the decimal point, so they have place value to the right. 3.14159 shows 1×10^{-1} in the first post-decimal location, 4×10^{-2} in the second place, and so on. In binary, one can work with "binary decimals" (to mix number systems), and the place values after the "decimal point" are 2^{-1}, 2^{-2}, and so on. Interestingly, some finite decimals in base 10 are repeating decimals in base 2 and vice versa. 1.2 (base 10) = 1 + 0/2 + 0/4 + 1/8 + 1/16 + 0/32 + 0/64 +1/128, with a remainder of 0.00468755 (base 10), giving a binary representation that start 1.0011001... . Because 2 is a factor of 10, any exact binary decimal can be expressed as an exact base 10 decimal. This means that any attempt to represent a rational decimal quantity in binary risks round-off error. While Hex has place values of 1/16, 1/256, etc., the essential issue doesn't change. 1.2 base 10 = 1.33... in Hex. Exact representation of fractions is a chimera.