# Is $180^\circ = \pi$?

I want to ask a question that causes confusion. In the trigonometry, we use some units of measure of angle: degree and radian. Which is/are correct?

$$180^{\circ} = \pi$$

or

$$180^{\circ} = \pi \ \ \ \text{radians}$$

In the other words: Is $$180^\circ$$ a real number?

My opinion $$180^\circ$$ is real number like $$3.14$$. Thanks for your comments/answers.

Edit: Some my friends high school mathematics teachers present in the form of a real number of $$180^\circ$$ in the class, my other friends comment: "No, this is not a number. Because they have units". I said that '' We can calculate $$\sin(x)$$ function at the point $$x = \pi/4$$. Hence this argument tells us that $$x$$ is a real number. Because, the sine function is defined in real numbers''. Also, there are a third group. They says ''Don't worry about these! It doesn't matter what this concept is, we can make a calculation and go on our way''.

We could not agree on how to present this concept to students correctly.

• Quantities have magnitudes and units of measurement; pure numbers have no units of measurement. $180^\circ$ is a quantity; $\pi$ is a pure number. However, the radian is a unit of measurement that is equal to $1$. Thus, $\pi$ is a pure number, $\pi$ radians is a quantity, but $\pi=\pi$ radians. – Joel Reyes Noche Dec 28 '18 at 13:42
• Your question is more about mathematics rather than mathematics education, so I am voting to close it. – Joel Reyes Noche Dec 28 '18 at 13:43
• As it stands now, this question is not about the teaching of mathematics. However, it could easily be changed into such a question. If this is about teaching, please give the additional details that makes that clear. If it is not, please deleted this question from this site and perhaps ask it at Mathematics Stack Exchange. – Rory Daulton Dec 28 '18 at 13:47
• This question was debated by a few of my friends mathematics teachers. We could not agree on how to present this concept to students. Thus, I presented you here as a problem with mathematics education. If the question is not suitable for here, I will delete it. Thanks for your efforts. – scarface Dec 28 '18 at 14:10
• '180 degrees = pi radians' is a correct form. Values, one of which has units and another has not, cannot be equal. At best, they can correspond to each other. But no one I know says 'pi radians' colloquially when talking about arc measure of an angle. It is not about real numbers, it about units. Kilogram is not equal to liter, but if you use water you can use then interchangeably with reasonable precision. – Rusty Core Dec 28 '18 at 20:49

It is possible to treat degrees and radians as units, just as we treat inches and centimeters as units. Just as we can convert inches to centimeters (1 in = 2.54 cm), we can convert degrees to radians (1 degree = $$\pi/180$$ radians).

But this point of view overlooks an important fact about radians: The radian measure of an angle is the ratio of two lengths (arc length divided by radius), so as a "unit" it is a pure number (cm/cm, or in/in, or furlong/furlong) independent of any arbitrary choice of units for measuring lengths (or masses or anything else). Because of this, I would prefer to identify "radian" with the pure number $$1$$. As a consequence, I would identify "degree" with $$\pi/180$$. And I would therefore answer the title question affirmatively: $$180^\circ=\pi$$.

• That a unit be dimensionless does not single out the radian. A degree is dimensionless in the same way a radian is. A degree is $1/360$ of a full turn. The numerical value of the ratio of two lengths changes if there are changed the units in which the lengths are measured. The radian can be defined as follows: it is the angle subtended at the center of a circle by an arc equal in length to its radius. It is this relation with circular arc length that distinguishes the radian geometrically. – Dan Fox Dec 29 '18 at 10:53

Both $$180$$ and $$\pi/4$$ are real numbers.

Units are an extramathematical notion related to measurement and indicating how to interpret a real number in some given (for example physical) context.

One can speak of $$\pi/4$$ degrees and $$180$$ radians just as one speaks of $$\pi/4$$ radians and $$180$$ degrees.

One definition of the sine function is that it is the unique solution of the initial value problem, $$\tfrac{d^{2}x}{dt^{2}}= -x$$, $$x(0) = 0$$, $$\tfrac{dx}{dt}(0) = 1$$ (cosine is defined similarly, with different initial conditions). What might be called the natural units of this solution, $$x(t) = \sin{t}$$, are radians and not degrees. The functions $$s(t) = \sin(\tfrac{2\pi}{360}t)$$ and $$c(t) = \cos(\tfrac{2\pi}{360}t)$$ are what what might be called the "sine in degrees" and "cosine in degrees". However, note that their derivatives no longer satisfy the usual identities. For example, $$s^{\prime}(t)$$ is not the cosine in degrees function, rather it is a multiple of it, namely $$s^{\prime}(t) = \tfrac{2\pi}{360}\cos(\tfrac{2\pi}{360}t) = \tfrac{2\pi}{360}c(t)$$. That is $$c^{\prime}(t) \neq -s(t)$$ and $$s^{\prime}(t) \neq c(t)$$. The function $$s(t)$$ solves the initial value problem $$\tfrac{d^{2}x}{dt^{2}}= -\left(\tfrac{2\pi}{360}\right)^{2}x$$, $$x(0) = 0$$, $$\tfrac{dx}{dt}(0) = \tfrac{2\pi}{360}$$.

Put another way, although $$(\sin^{\prime}(t), \sin(t))$$ parameterizes the unit circle, it is not true that $$(s^{\prime}(t), s(t))$$ parameterizes the unit circle. If we work in degrees rather than radian identities such as the Pythagorean identity $$\sin^{2}(t) + \cos^{2}(t) =1$$ fail in the sense that it is not true that $$s^{2}(t) + (s^{\prime}(t))^{2}$$ equals $$1$$.

Alternatively, the functions $$(\cos(t), \sin(t))$$ in radians yield a unit speed parameterization of the unit circle, whereas the functions $$(c(t), s(t))$$ in degrees only yield a constant speed parameterization of the unit circle. The choice of units corresponds to the choice of normalization of the speed of the parameterization. The non-unit speed parameterization is less aesthetically pleasing because it breaks the symmetry of certain standard identities, and practically a bit unwieldy because of the factors of $$\tfrac{2\pi}{360}$$ it generates.

• Also in yours solution, we can see that radians useful in the trigonometric functions. Thanks. (Your description is gladsome. The site only allows us to choose one of the best answers). – scarface Dec 28 '18 at 22:01
• Sine is neither in degrees nor in radians, it is just a number. Degrees or radians or rads is a measure of angle, which is an argument of sine. Also, if you meant to compare specific angle measures, did not you mean pi/2, not pi/4 ? – Rusty Core Dec 29 '18 at 5:23
• @RustyCore: As follows from the statement about the initial value problem, the graphs of the sine/cosine functions have a canonical parameterization. Whether one wants to call this parameter a "natural unit" or "radians" is a question of context (either could be said equally well to be "just a number"), but "radian" is just the name given to this canonical parameter when speaking in a (extramathematical) context where measurement occurs. Note that there is another possibility for "natural unit": the number of turns. – Dan Fox Dec 29 '18 at 10:46
• It seems very odd to suggest $c(t)$ be the derivative of $s(t)$. I would think $c(t) = \cos \left(\frac{2\pi}{360}t \right)$ in which case $\frac{ds}{dt} = \frac{2\pi}{360}c(t)$ and symmetrically $\frac{dc}{dt} = -\frac{2\pi}{360}s(t)$ and of course naturally $c^2(t)+s^2(t)=1$ with my definition for $c(t)$. Your post was disturbing until I saw what you did... – James S. Cook Jul 15 '19 at 1:09
• @James S. Cook I could not understand the end of Dan's answer until I read your comment. It would look less strange if $c(t)$ had been used to denote $\cos((2\pi/360)t)$, so then $s(t)^2 + c(t)^2 = 1$ for all $t$ but $s'(t) \not= c(t)$ and $c'(t) \not= -s(t)$ in general. – KCd Jul 16 '19 at 12:59

Radians are "pure numbers", inasmuch as they are ratios of the length along the arc subtending the angle to the radius of the circle. As such, they don't require units, and you could (and should) write $$180^{\circ} = \pi$$.

That being said, there's a good reason to specifically write radians. As Dan Fox points out that there is another natural unit for angle measure: the turn. In fact, the turn is arguably the most natural measure of an angle, since everyone understands what it is (though our language confuses the issue: "Turn around" and "Turn halfway around" are used interchangeably).

This is also a ratio (it's the fraction of a full rotation) and as such has no units. So you could just as legitimately write $$180^{\circ} = 1$$.

Unfortunately, this leads to writing $$1 = \pi$$.

So while you could write $$180^{\circ} = \pi$$ and not raise an eyebrow, it's probably better to write $$180^{\circ} = \pi$$ radians.

• – J W Jan 4 '19 at 19:10
• I find this sentence very essential: ''they are ratios of the length along the arc subtending the angle to the radius of the circle''. – scarface Jan 4 '19 at 20:25
• I'd think that this viewpoint is asking for trouble with students, even if it is mathematically defensible on various grounds. To suggest that one side of an equation, with units, is equal to something dimension-less is not a great idea in several ways. – paul garrett Jan 4 '19 at 22:43
• Surely a full rotation is $360^{\circ}$ (not $180^{\circ}$), e.g. as noted in @JW's link. N.B. Some people now use the symbol $\tau$ for this value (as noted under en.wikipedia.org/wiki/Turn_%28geometry%29#Tau_proposals). – Daniel R. Collins Jul 16 '19 at 0:15
• @scarface "Radians are ratios. As such, they don't require units, and you could (and should) write $180^{\circ}=\pi.$" 1.But angles in general--including when measured in degrees--are ratios of lengths. 2.The fact that angles are ratios means that they are dimensionless, but not necessarily that they are unitless. ($180^{\circ}=\pi\text{ rad}$ here.) – Ryan G Feb 6 at 17:08

$$180^{\circ} = \pi$$?

Certainly not : the surface of the unity circle is $$\pi$$. If anyone tells me the surface of anything can be expressed in degrees, I'll tell this person (s)he's wrong.

$$180^{\circ} = \pi$$ radians?

That I can agree to.

• The argument here seems to me to be self-defeating: you argue against $180^\circ = \pi$ on the grounds that $\pi$ doesn't have the units of area. The error is introduced when you consider the unit circle to have a dimensionless radius, not in the comparison of two angles. – Peter Taylor Jan 11 '19 at 8:41
• If the radius of the unit circle is measured in units of $l$, e.g., $l$ is in meters, then the area of the unit circle is $180 \deg\cdot l^2$. – user52817 Jan 11 '19 at 15:17