Signum function

Algebra Level 1

∑ x = − 50 50 s g n ( x ) = ? \large \displaystyle\sum_{x=-50}^{50} \mathrm {sgn(x)} = \ ?

Clarification: s g n ( x ) = { 1 , x > 0 − 1 , x < 0 0 , x = 0 \mathrm {sgn(x)} = \begin{cases} {1 \quad , \quad x > 0} \\ {-1 \quad , \quad x < 0} \\ {0 \quad , \quad x =0} \end{cases} .


The answer is 0.

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Parth Lohomi by Parth Lohomi , 20, India

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1 solution

Parth Lohomi
Jun 19, 2015

s g n ( x ) = { − 1 if x < 00 1 if x > 00 0 if x = 00 \mathrm{sgn(x)} =\begin{cases} - 1&\mbox{if}\ x< 00\\ 1&\mbox{if}\ x>00\\0&\mbox{if}\ x=00\end{cases}

So our sum become

50 ( − 1 ) + 0 + . . . . . . . . . . . . + 50 ( 1 ) = 0 50(-1)+0+............+50(1)=\boxed{0}

6 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Another way to do this. is to rearrange the terms as sgn ( 0 ) + ∑ x = 1 50 ( sgn ( x ) + sgn ( − x ) ) = 0 + 50 ( 0 ) = 0 \displaystyle \text{sgn}(0) + \sum_{x=1}^{50} \left( \text{sgn}(x) + \text{sgn}(-x) \right) = 0 + 50(0) = 0 .

Bonus question : Evaluate ∑ x = − 50 50 sgn ( x 2 + 5 x + 6 ) \displaystyle \sum_{x=-50}^{50} \text{sgn}(x^2+5x+6) .

@Parth Lohomi How is this a Calculus question ? I think it should be in Algebra.

Venkata Karthik Bandaru - 5 years, 12 months ago

Pl congrats in advance for your 1000 followers.

Shubhendra Singh - 5 years, 11 months ago

Answering to the Challenge Master's Challenge of ∑ x = − 50 50 s g n ( x 2 + 5 x + 6 ) \sum_{x=-50}^{50}sgn(x^{2}+5x+6) , it will be 3 s g n ( 0 ) + [ ∑ x = 1 50 [ 2 s g n ( x 2 ) + s g n ( − x ) + s g n ( x ) + s g n ( 6 ) ] 3sgn(0)+[ \sum_{x=1}^{50}[2sgn(x^{2})+sgn(-x)+sgn(x)+sgn(6)] or simply 3 ( 0 ) + 2 ( 50 ) + 50 ( 0 ) + 1 ( 50 ) = 150 3(0)+2(50)+50(0)+1(50) = 150

Kay Xspre - 5 years, 9 months ago

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