# How to define “axes with the same scale” in Secondary/High School?

It's easy to recognize visually when an orthogonal coordinate system has its axes in the same scale. See, for instance, the following image. But I'm trying to write down a precise definition of it.

After searching the usual channels (Google Scholar, Google Books), my impression is that it seems this knowledge is implicit in High School teaching, that is, the explanation given is mainly visual: the teacher shows an example of an orthogonal coordinate system with the same scale on both axes and another example with different scales. Done. Some people define "the axes are on the same scale if they have the same unit" but, then, what does it mean "to have the same unit"? I'm looking for a precise definition accessible to Secondary School students.

So, my three questions are:

(1) Do you know a precise definition for "axes on the same scale" accessible to Secondary School students?

(2) Do you know some school textbook or a scientific article/book where such definition is presented or discussed?

(3) What do you think about this definition: "We say that an orthogonal coordinate system of the plane has the x and y axes in the same scale if the segment joining (0, 0) to (1, 0) has the same length as the segment joining (0, 0) and (0, 1) when measured with the same ruler.".

While in Analytic Geometry it is always assumed the axes are on the same scale, when studying functions or Statistics, axes on different scales is a necessity.

• While I don't have an answer for you on this, I would love to see someone post a complete enough answer that it can also define semilog and log-log plots too. – kcrisman Apr 6 '20 at 13:08
• Same scale means that the unit intervals are of the same length (can be measured directly with a pair of compasses to demonstrate). Different scale means that the unit intervals have different lengths. – Rusty Core Apr 7 '20 at 20:19
• Others have tried to address this, but I'd like to emphasize: if the two quantities to be plotted are of different physical dimension (like weight and time), then the question of same scale is meaningless, since quantities of different dimension cannot be compared (they cannot have the same unit). But it's not clear if this is what you are asking about. – Michael Bächtold Apr 13 '20 at 16:49

We say that an orthogonal coordinate system of the plane has the x and y axes in the same scale if the segment joining (0, 0) to (1, 0) has the same length as the segment joining (0, 0) and (0, 1) when measured with the same ruler.

I think there is a problem with this definition, which has to do with the difference between how mathematicians think about numbers and how scientists and engineers do.

Suppose that your graph shows the growth of a plant. We graph its height $$h$$ versus the time $$t$$. The height is in mm and the time is in days. Mathematicians think of the units as being part of the definition of the variables, while scientists and engineer think of them as being part of the value of the variables. On a graph labeled "$$h$$ (mm)," a mathematicians says that the "(mm)" defines $$h$$, while a scientist says that the "(mm)" is a unit to be applied to all the numbers on the scale, so that we don't need to write 1 mm, 2 mm, 3 mm, etc.

I think like a physicist, so if I try to apply your definition, I say, "The point (1,0) doesn't exist on this graph. The unitless 1 isn't a possible value for the variable $$t$$." An example of a point that does exist is (1 day,0).

So if you want to be compatible with the way scientists and engineers think, I would do something like this:

We say that an orthogonal coordinate system of the plane has the x and y axes in the same scale if the axes have compatible units and the segment joining (0, 0) to $$(a, 0)$$ has the same length as the segment joining (0, 0) and $$(0, a)$$ when measured with the same ruler.

By this definition, we could have square graph paper with 1 cm boxes on the x axis and 10 mm boxes on the y axis, and that would be OK.

• It's a good thing that your definition includes the "unit-less" units. – Peter Saveliev Apr 7 '20 at 16:06
• I think the point is that axes with the same scale maintain shapes. I think it only applies when the units used on the axes are the same, or when no units are used. – Sue VanHattum Apr 7 '20 at 16:07
• Why is the part "measured with the same ruler" necessary? A certain constant length (say 2 cm) is independent of the ruler we use to measure it, isn't it? – Michael Bächtold Apr 7 '20 at 17:21
• @MichaelBächtold: I think it's necessary because otherwise there are various ways that a student could try to construe length. They could use (1) a ruler, (2) counting squares on graph paper, or (3) the real-number values being graphed. I think the OP's explicit mention of a ruler is also helpful because it clarifies that we care about the geometrical structure of the paper, and the structure we care about is more than just the affine structure. He's postulating that we can rotate our ruler, which is not true in bare affine geometry. – Ben Crowell Apr 7 '20 at 17:31
• Interesting perspective. I don't quite understand what "the '(mm)' defines $h$" means, but I take it that the bridge between the two views is that the mathematician plots instead what the physicist would refer to as the dimensionless quantity $h' = h/(1 \text{ mm})$? – Vandermonde Apr 12 '20 at 18:58

Great question!

And great diagrams: it suddenly occurs to me that a mathematical circle may appear pictorially as an ellipse if the scales are different. This doesn't mean we're talking about a non-circular ellipse; it just means we're drawing the circle under question in a particular way. Honestly, this had never occurred to me before!

To answer your question, how about: The axes of a graph are said to have the same scale if and only if we can rotate said graph in such a way that (a) the $$1$$ on the horizontal axis moves to the position of the $$1$$ on the vertical axis and (b) the new origin coincides with the old origin.

Obviously, you'll have to break down the definition into parts and talk through some examples before students will understand what's being described. Those diagrams will definitely help!

• I like this approach, anf I think it's simpler to just say "can be rotated around the origin" and drop (b). – Jasper Apr 15 '20 at 18:11
• Rotation implies there is an inner product in the background. If you have an inner product, then you already have a way of comparing distances on the two axes. – user52817 Apr 15 '20 at 20:01
• "Rotation implies there is an inner product in the background." What? A Graph is something on paper, and one can rotate paper happily without an inner product. – Jasper Apr 16 '20 at 8:59
• @Jasper the OP asks for a precise definition, not an intuitive and informal one. The mathematical definition of a rotation is a certain type of motion that preserves an inner product. Thus “inner product” precedes “rotation.” And if you have an inner product, then you already have a way of comparing the length of a vector on one axis with that of one on the other axis. – user52817 Apr 22 '20 at 21:22

Since the axes you're talking about are real things on paper or on a screen, I think it's easy and I'd go with something very similar to your proposal:

A coordinate system has axes with the same scale if every number on one axis has the same distance to the origin as the same number on the other axis.

I think we can get around the "with the same ruler" part because using different rulers to measure the length of segments on paper doesn't make sense.

By using the term "number", we make clear that the concept is useless for most "physical" graphs with different units on both axes. For physical graphs with same units on both axes, one could substitute number with "quantity" in thw definition above.

This is a good question, and brings up some subtle issues. A clue is to observe the tacit assumption, made when drawing the two graphs in the original post, that the axes are perpendicular. We draw the axes to appear perpendicular, even though orthogonality might have no intrinsic sense in the underlying mathematics or physics.

What we start with is a pair of one-dimensional vector spaces $$L$$ and $$M$$, with not particular basis designated. Think of $$L$$ as "time" and $$M$$ a "temperature." We form the two-dimensional vector space $$V=L\oplus M$$, which still has no designated basis. Now here is the key point: we select non-zero vectors $$l\in L,\ m\in M$$. This is the point where we choose "units" like seconds and degrees Celsius. Furthermore, we identify $$V$$ with $$({\bf R}^2,\langle\cdot,\cdot\rangle)$$ where $$\langle\cdot,\cdot\rangle$$ is the canonical inner product on $${\bf R}^2$$.

So at this point, we have identified $$V=L\oplus M$$, which was endowed with neither an inner product or a basis, with the canonical inner product space $$({\bf R}^2,\langle\cdot,\cdot\rangle)$$. This is what allows us to graph in the Cartesian plane.

It is interesting to observe that the notion of area under the curve has meaning, and the arena for this is the tensor product $$L\otimes M$$, which is a one-dimensional vector space. Now think of $$L$$ as "distance" and $$M$$ as "force" (scalar). If we measure distance with the units of feet and force in pounds, then area is "work" and has units of $$\hbox{foot}\cdot\hbox{pounds}$$. This mathematical quantity (area) lives in the tensor product $$L\otimes M$$, which inherits a basis from the choice of units in $$L$$ and $$M$$, i.e., foot and pound.

Now--how to bring this to secondary/high school? Evidently, this perspective is too abstract for that. I suggest a brief mention of oblique coordinates, and the move on quickly!

• For future reference, the commands for pointy-braces in TeX / MathJax are \langle (for the left brace) and \rangle (for the right brace). For example, $\langle\cdot,\cdot\rangle$ is rendered as $\langle\cdot,\cdot\rangle$. – Xander Henderson Apr 7 '20 at 17:04
• This doesn't seem to address the question, since you never offer a definition of what would not qualify as axes with the same scale. In your notation, the issue becomes (1) whether or not there is some natural isomorphism between $L$ and $M$, and (2) if so, whether $l$ and $m$ match up under this isomorphism. In your example using time and temperature, I would say that there is no such natural isomorphism, so it fails based on criterion #1 to be a graph with axes having the same scale. Is that what you intended? – Ben Crowell Apr 7 '20 at 17:26
• @BenCrowell my point is that the whole notion of "axes with the same scale" is mostly artificial, especially in applications. You cannot compare a "second" with a "Kelvin degree." Yet we plot temperature versus time. And when we do this, we are usually unaware that we are making $L\oplus M$ an inner product space, even though the metric has no real significance. It's just a construct we use to visualize with a graph. – user52817 Apr 7 '20 at 18:01
• @user52817 +1 I like to make these points when I teach linear algebra. – Steven Gubkin Apr 11 '20 at 21:34
• Temperatures can be stated as degrees Celsius or degrees Fahrenheit, but the Kelvin is not accompanied by "degree". – Jasper Apr 15 '20 at 18:09

I find the first sentence of the Wiki article on Cartesian coordinates sufficient for me.

Although I am timid of my rigor righteousness when someone where asks for "precise", even with added bold to hammer the request harder. ;-)

"A Cartesian coordinate system (UK: /kɑːˈtiːzjən/, US: /kɑːrˈtiʒən/) is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length."

https://en.wikipedia.org/wiki/Cartesian_coordinate_system

P.s. Another answer commented on the preference for perpendicularity. The one area I see this waived a lot is in crystallography (or properties of crystals) where having coordinate systems that are non-perpinducular (or non equal in length) makes sense as it corresponds to the anisoptropic substance. So the norm is absolutely to use a monoclinic coordinate system with monoclinic crystals. Not gyrate around with a cubic coordinate system and do all the conversions.