Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at markdotmath@gmail.com. I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

Ted’s Question: Can I Graph a Decimal Slope?

Posted by mark schwartz on November 16, 2016

Introduction

We were working on graphing lines using the slope-intercept method.

The equation to graph was y = 4/3x + 2. Traditionally, plot the point (0, 2) first – the y-intercept and from this point, move up 4 units (positive 4 on the y-axis) while moving 3 units to the right (positive 3 on the x-axis). This finds the second point at (3, 6). This process gives an accurate line between these two points.

Ted asked “If I use my calculator to find the value for the slope, I get 1.33 … can I use 1.33 as the slope to graph the line”? Having never heard this question before, I said I wasn’t sure but let’s look at it.

The Story

As it turns out Ted is correct … 1.33 can be used but it’s important to understand how to use it.

It goes back to a basic fraction relationship. In order to preserve the relationship between the numerator and denominator, it is allowable to multiply or divide both the numerator and denominator by the same value. This is what is done when searching to either find an equivalent fraction when reducing a fraction to lowest terms or finding an equivalent fraction for adding or subtracting fractions.

Given this, it’s not that the fraction is converted to a decimal by dividing 4 by 3. Rather the mathematical operation is to divide both the numerator and denominator by 3, giving the fraction 1.33/1. When we do this conversion, we typically don’t note the denominator of 1; it simply is ignored as if it weren’t there.

So, back to plotting the equation. Again starting at (0, 2), we would move up 1.33 (move positive 1.33 on the y-axis) while moving right 1 (move positive 1 on the x-axis). This is valid and falls on the line plotted when using slope = 4/3.

Well, not exactly. Using 1.33 isn’t quite as accurate as using 4/3, simply because, in this case, it is a repeating decimal. But, even without a repeating decimal, there still is the possibility of a loss of accuracy. Of course, for classroom purposes this might be acceptable After all, we’re not designing a spacecraft that needs quite accurate calculations for design and flight.

Using this decimal idea with y = 3/5x + 2, we would have y = .6x + 2. The plot again begins at (0, 2). The issue now is the scale on the x and y axes. If these axes are laid out in .1 increments, then .6 can readily be used with the same accuracy as 3/5, but if the scale is in whole units, the .6 is an ‘eyeball’ estimate and may not be as accurate. As a reminder, in this case, when moving up .6 on the y-axis, move a corresponding 1 on the x-axis. When using a decimal, the denominator (change on the x-axis) is always 1.

However, the question was wonderful and exploring it was interesting and … well, educational.

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