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Alex777 [14]
2 years ago
14

For an integer n ≥ 0, show that (n/2) - (-n/2) = n. Use greatest integer function

Mathematics
1 answer:
mars1129 [50]2 years ago
3 0

The greatest integer function returns the largest integer smaller than the number you provide it. That is, if <em>n</em> ≤ <em>x</em> < <em>n</em> + 1, where <em>n</em> is an integer, then the "greatest integer of <em>x</em>" is [<em>x</em>] = <em>n</em>.

• Let <em>n</em> be even. Then we can write <em>n</em> = 2<em>k</em> for some integer <em>k</em> ≥ 0. Now,

[<em>n</em>/2] = [<em>k</em>] = <em>k</em>

while

[-<em>n</em>/2] = [-<em>k</em>] = -<em>k</em>

so that

[<em>n</em>/2] - [-<em>n</em>/2] = 2<em>k</em> = <em>n</em>

<em />

• Let <em>n</em> be odd. Then <em>n</em> = 2<em>k</em> + 1 for some integer <em>k</em> ≥ 0. Every odd integer occurs between two even integers, so that

<em>n</em> - 1 < <em>n</em> < <em>n</em> + 1

or equivalently,

2<em>k</em> < <em>n</em> < 2<em>k</em> + 2

so that

<em>k</em> < <em>n</em>/2 < <em>k</em> + 1

It follows that [<em>n</em>/2] = <em>k</em>.

Similarly, if we negative the previous inequality, we have

-<em>k</em> > -<em>n</em>/2 > -(<em>k</em> + 1), or -<em>k</em> - 1 < -<em>n</em>/2 < -<em>k</em>

which means [-<em>n</em>/2] = -<em>k</em> - 1.

So we make the same conclusion,

[<em>n</em>/2] - [-<em>n</em>/2] = <em>k</em> - (-<em>k</em> - 1) = 2<em>k</em> + 1 = <em>n</em>

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