While drawing graphs by hand, how correct is it to take unequal units on the x & y axes?

Question

In exams and schoolwork, teachers allow us to take unequal units on the $x$ & $y$ axes if their values are really far apart for ease of drawing and scalability. For example, if the $x$ values are $1,2,3...$ and the $y$ values are $100,200,300...$ then it is encouraged to take like $1$ small square as a unit in the $x$-axis, but like $10$ small squares as a unit in the $y$-axis.

My question is, how mathematically rigorous is this method? Aren't we distorting the graph? The length of $1$ unit on the $x$-axis & the length of $1$ unit on the $y$-axis are unequal. The length of a unit on the $y$-axis is $10$ times the length of $1$ unit on the $x$-axis in the above example. So, aren't we drawing wrong graphs?

Answer 1

There is no problem with drawing graphs with different scales. The key is that you can't always use the same interpretations of what the graph MEANS.

Following your example, if I draw a line through the origin at a 45 degree angle up from the positive x-axis, I get the line $y=10x$. This will look like the line $y=x$ in standard cartesian coordinates. So the intuition of "how steep is the slope" by visualization needs to shift.

The thing you have to grapple with is what do you mean by "wrong" or "right". In math, we look for our tools to be internally logically consistent (Different mathematical setups lead to different outcomes, just like your different scales means the angles for slopes of lines change, but within their own system they are fine) and then at least one of the following: Useful, beautiful, or interesting. Nowhere is there a "universally objectively correct way" to define math.

To give a simple example, there are lots of ways we define numbers that can lead to $1+1=0$! These aren't the usual real numbers you are used to, but they follow all the arithmetic laws (AKA they form a field).

Answer 2

Not at all, you might be thinking that your units are represented by the grid in your notebook, but remember that whenever you physically draw a graph you will never have a mathematically "rigorous" representation. However even with an ideal graph, you can have one of the axis be contracted and it will still accurately represent whatever you graph.

Think for instance two points $(x_1 ,y_1)$ and $(x_2 ,y_2)$, if your graph shows increments by 1 in the $x$-axis, and by 100 in the $y$-axis, counting how many steps in each direction get you to each coordinate and taking into account that your steps in the $y$ direction are longer, then you will still arrive at the same lengths, so everything related to length will still give the same results. This also applies to ratios and even to calculus, you are just changing your physical representation not the mathematical object itself.

Answer 3

In general not and it really depends on what you want to convey, if indeed you want to draw something to scale to easily estimate or measure slopes with a geo triangle (without additional unit conversions) than same units are desired. But in general there is nothing wrong with using different scales, moreover; when drawing graphs that represent physical quantities, like time vs position, your choice of units(seconds, minutes for time and inches or meters for position) determine the scale and this is in a sense somewhat arbitrary, meaning, you can choose anything you like and in theory you can even make up your won units, as long as you relate them to known units. But again, it depends on what you want to show/convey.

TL;DR: There is nothing wrong with choosing different scales for x and y axes, when representing physical quantities(eg. time vs. position) this will often be required to show what you want to show.