Question

Show that if x is a real number, then \(\left\lceil x \right\rceil - \left\lfloor x \right\rfloor = 1\) if x is not an integer and \(\left\lceil x \right\rceil - \left\lfloor x \right\rfloor = 0\) if x is an integer.

Short answer

\(\left\lceil x \right\rceil - \left\lfloor x \right\rfloor = 1\) if x is not an integer and \(\left\lceil x \right\rceil - \left\lfloor x \right\rfloor = 0\) if x is an integer.

Step-by-step solution

Step: 1

If x is not an integer.

Let\(\left\lfloor x \right\rfloor = n\), then\(n < x < n + 1\)or\( - n - 1 < - x < - n\)

Let\(\left\lceil x \right\rceil = m\), then\(m - 1 < x < m\)we have

\(\begin{array}{l}(m - 1) + ( - n - 1) < x - x < m - n\\m - n - 2 < 0 < m - n\end{array}\)

Step: 2

\(\begin{aligned}{l}\left\lceil x \right\rceil - \left\lfloor x \right\rfloor - 2 < 0 < \left\lceil x \right\rceil - \left\lfloor x \right\rfloor \\0 < \left\lceil x \right\rceil - \left\lfloor x \right\rfloor < 2\end{aligned}\)

Since \(\left\lceil x \right\rceil - \left\lfloor x \right\rfloor \) is integer, it follows that \(\left\lceil x \right\rceil - \left\lfloor x \right\rfloor = 1\)

Step: 3

If x is an integer,

\(\left\lceil x \right\rceil - \left\lfloor x \right\rfloor = x - x = 0\)