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I have a question about the measure of angle. we know that $ 0^∘≤m∠XOY≤360^∘$ or $ 0≤m∠XOY≤2π $.

Is $$m∠XOY\in \mathbb{N}$$ or is $$ m∠XOY \in \mathbb{R}$$ ?

ahorn
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elham.mj
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2 Answers2

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How an angle $\angle(XOY)$ in the plane is measured: Draw rays $OX$ and $OY$ and intersect them with a unit circle centered at $O$. The length of the shorter arc between the two points of intersection is the (unsigned) angle defined by the given configuration. This angle can be any real number between $0$ and $\pi$ inclusive.

In daily life we measure angles in degrees instead, whereby $1^\circ:={\pi\over180}$. Fractions of degrees can be expressed in decimals, like $23.7114^\circ$, or using (angle) minutes $\ ':={1^\circ\over60}\ $ and (angle) seconds $'':={1^\circ\over60\cdot60}$. This leads to angle data of the form $61^\circ 34'22.61''$.

But note that, whatever units are used, an angle is a "continuous real variable".

For more sophisticated versions of the notion of angle, see this question at MSE: What is the exact and precise definition of an ANGLE?

Community
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Christian Blatter
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Unless stated otherwise, an angle is a real number (which allows us to be precise). $m$ is not used to denote an angle - rather, just write $\angle XOY$ to denote the angle subtended by the lengths $XO$ and $OY$.

ahorn
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  • Does it exist $\sqrt{2}^∘$ ? and What is؟ – elham.mj Oct 31 '16 at 15:40
  • Yes, an angle can be an irrational number. – ahorn Oct 31 '16 at 15:49
  • ahorn: $\angle XOY$ is a label for an angle being referenced, i.e. it's a name given to an angle. The OP uses $m(\angle XOY),$ which refers to the measure of the angled named $\angle XOY$. In any case it is true that the measure of an angle is a real number, whatever the unit used. – amWhy Nov 03 '16 at 12:29