fractional part

[logistic_guy]

Find the fractional part of each of these numbers:

\(\displaystyle \bold{(a)} \ \frac{8}{5}\)

\(\displaystyle \bold{(b)} \ \frac{1}{7}\)

\(\displaystyle \bold{(c)} \ -\frac{11}{4}\)

\(\displaystyle \bold{(d)} \ 7\)

 

[khansaheb]

Find the fractional part of each of these numbers:

\(\displaystyle \bold{(a)} \ \frac{8}{5}\)

\(\displaystyle \bold{(b)} \ \frac{1}{7}\)

\(\displaystyle \bold{(c)} \ -\frac{11}{4}\)

\(\displaystyle \bold{(d)} \ 7\)

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[logistic_guy]

\(\displaystyle \bold{(a)}\)

\(\displaystyle \frac{8}{5} = 1 + \frac{3}{5}\)

Then, the fractional part is:

\(\displaystyle \frac{3}{5}\)

 

[logistic_guy]

\(\displaystyle \bold{(b)}\)

\(\displaystyle \frac{1}{7} = 0 + \frac{1}{7}\)

Then, the fractional part is:

\(\displaystyle \frac{1}{7}\)

 

[logistic_guy]

\(\displaystyle \bold{(c)}\)

With positive numbers, the above method works, but with negative numbers, we have to use the definition.

The fractional part of \(\displaystyle -\frac{11}{4}\) is defined as:

\(\displaystyle -\frac{11}{4} - \left\lfloor-\frac{11}{4}\right\rfloor = -\frac{11}{4} - \left\lfloor-2-\frac{3}{4}\right\rfloor\)


\(\displaystyle = -\frac{11}{4} - (-3) = -\frac{11}{4} + 3 = -\frac{11}{4} + \frac{12}{4} = \frac{1}{4}\)

 

[logistic_guy]

\(\displaystyle \bold{(d)}\)

\(\displaystyle 7 = 7.0\)

Then, the fractional part is:

\(\displaystyle 0\)