Numbersensse Question: change 7/16 to a base 4 decimal_________The answer is .13

lfryerda | High School Teacher | (Level 2) Educator

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A base 4 decimal is a decimal number that looks like `0.a_1a_2a_3a_4\ldots` but each `a_k` is a coefficient of the kth negative power of 4.  This works in a similar way to our regular decimal system, but with a base of 4 instead of 10.

For example, in regular decimals, `0.75=7/{10}+5/{10^2}`. With base 4, the fraction needs to be converted into a number that looks like `a_1/4+a_2/{4^2}+a_3/{4^3}+a_4/{a^4}+\cdots`, where we need to determine each of the `a_k`

Since the denominator of `7/{16}` is a power of 4, the conversion will be a finite decimal, which means the decimal will eventually stop (in this case after two digits).

Starting with the fraction `7/{16}`, we see that `7/{16}=1/4+3/{16}`, which means that in this case `a_1=1` and `a_2=3`, with the rest of the `a_k=0`.  This means that the fraction `7/16` in base 4 is 0.13.

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