Ratios 10/13/21

Math with Richard Bleil

Ratios and fractions have always fascinated me.  There’s an old joke that five out of four people don’t understand fractions.  It’s not really a surprise because they behave oddly.  Yes, Glenn, they do.

Here’s an example.  Today, the high is expected to be 86oF here in Omaha.  It’s the end of a hot spell as tomorrow’s high is only supposed to be 76.  The difference is 10oF, so you might think it will be (100*10/86=)12% lower, but that’s very misleading.  If we convert to degrees Celsius, then today will be 30oC, and tomorrow will be 24.4.  The difference in Celsius then is 5.6oC, and the temperature will seem to be (100*5.6/30=)19% lower.  But how is this possible?  Is it 12% lower, or 19? 

It’s actually neither.  See, Fahrenheit and Celsius are both relative temperature scales (for example, the Celsius scale is set relative to the boiling and freezing point of water, set to be 100 and 0 respectively).  I’ve been told that Fahrenheit is set relative to the human body temperature, but they didn’t do a very good job (or their equipment wasn’t very good) which is why normal body temperature is close to 100, but I’m not sure I believe this, and I’ve never heard how the lower end of the scale was set.  No, to get the true percentage drop, we have to convert to an absolute temperature scale. 

An absolute temperature scale is one that can never go through zero, because the low end is always set to absolute zero (impossible to achieve except in my ex-wife’s heart).  The absolute scale that most people know of is Kelvin, which corresponds to the Celsius scale (add 273.15).  In Kelvin, the high today should reach 303.15K (note that in absolute scales we don’t use the “degree” symbol), and tomorrow will be 297.55.  The difference here is 6K, and the percent difference is (100*6/303.15=)2%. 

Let’s test this hypothesis, shall we?  We should get the same percentage drop for any absolute temperature scale.  Just as Celsius has Kelvin, the temperature scale few people know is Rankine, the absolute temperature scale corresponding to the Fahrenheit relative scale.  To convert to Rankine, we add 459.67 to Fahrenheit.  So today, the high will be 545.67R, and tomorrow should be 535.67.  The difference between the two is 10R, so the percent drop is (100*10/545.67=)2%.  Because both scales absolute, the percent drop is the same. 

Nothing shows the odd nature of fractions like relative ages.  I’ll confess that I’ve dated women who are quite a bit younger than I.  For example, when I was 39, I was dating a woman who was 21.  This is an 18-year age difference, and I was (100*18/21=)86% older than she was.  She, on the other hand, was only (100*18/39=)46% younger than I, which sounds far less daunting.  But here’s the interesting thing.  Today, I am 58, and she is 40.  The difference in age is still 18 years, but I am only (100*18/40=)45% older than she is.  She, on the other hand, is only (100*18/58=)31% younger than I. 

In age, the absolute difference in age between any two people is always fixed, and yet, as time races inexorably on to our imminent demise (I think I heard that as a quote somewhere; I like it!), the relative ages get ever closer.  If I live to be 100, she will be 82, and only 18% younger than I.  In calculus, we speak of “limits”.  That is to say, what happens if we take the “limit” of this age spectrum as far out as mathematically (not physiologically) possible.  The biggest number in mathematics, as you all know, is infinity.  Our absolute difference, when I’m infinite years old (as I feel I am today), will still and always be 18 years.  But when I’m infinitely old, the percentage of our age differences will be (100*18/infinity=)0%.  In absolute terms, we will be eighteen years apart in age, and yet relatively speaking we will be the exact same age.  The age difference is always the same, and both the numerator and denominator are increasing at the exact same rate, and yet the ratio becomes ever smaller.

A lot of people don’t enjoy mathematics.  I guess I’ll never understand that.  To me, math is a fascinating world, and a powerful one at that.  My doctoral training was all about finding ways to explain natural phenomena mathematically.  With the language of mathematics, so much information can be packed into one little equation.  They say that one photo is worth a thousand words, but one equation must surely be worth at least a thousand photos.

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