6 ms·

Why?

margalabargala 2y ago
Humans are notoriously terrible about estimating volumes when things are curved and volume functions are exponential. A great example of this done in 8th grade science classes across the US is to put 100ml of water in a 100ml graduated cylinder, 150ml in a 1L beaker, and ask the class which has more. Humans are awful at estimating how much volume the increased radius adds, and usually will say the 100ml. The problem only gets worse as we graduate from cylinders to spheres. We can all visually see which sphere is bigger, but cannot come close to estimating how much bigger one is than another.
- AnimalMuppet 2y ago
  Volume functions are not exponential. They are polynomial. (Fair point that people are lousy at estimating even polynomial functions, though...)
  - dylan604 2y ago
    > They are polynomial I don't think this word means what you think it does. Or I don't. Exponents are just the number the value is raised. Squaring a value just uses an exponent of 2 where cubing uses an exponent of 3. Polynomials are x^2 + x + 1 type of equations. But admittedly, it has been 30+ years since I've thought about them at that level, so maybe I'm the one with fuzzy groking
    - AnimalMuppet 2y ago
      "Polynomial" meaning x^n. "Exponential" meaning e^x. Exponentials eventually grow much faster than polynomials, no matter what the exponent is. I mean, look, in v = x^3, the "3" is an exponent. But it's not an exponential function because the variable isn't in the exponent.
      - sfink 2y ago
        > Exponentials eventually grow much faster than polynomials, no matter what the exponent is. Since we're being pedantic, that last clause should be: "as long as the exponent is greater than 1."
        margalabargala 2y ago
        We can go a step further with the pedantry, and say that the commenter above is using an unreasonably narrow definition of the work "exponential" and that there are others which allow x^2 to be described as "exponential". https://www.merriam-webster.com/dictionary/exponential https://www.merriam-webster.com/dictionary/exponential
        dylan604 2y ago
        Or one step further https://www.merriam-webster.com/dictionary/exponent https://www.merriam-webster.com/dictionary/exponent which is how I was taught. I only went to CalIII back in the early 90s, so who knows what's being taught now???
        sfink 2y ago
        We could, but I would describe it as "mathematically accurate". Which is not incompatible with "unreasonably narrow", given that the definition of "exponential" has recently gotten polluted enough that it is now often synonymous with "fast growing". But what's the point of arguing over definitions if we're going to start with a baseline of saying that there is no basis upon which to argue definitions other than recent conventional usage? > there are others which allow x^2 to be described as "exponential". Those same definitions allow x*1000 to be described as "exponential". (x*1000000 would be "more exponential"!) If you're describing something as exponential, then either you're just saying "fast growing", or you're trying to describe the type of growth. If you're describing the type of growth, then neither x*1000 nor x^2 is exponential. The fact that x^2 has an exponent in it is no more relevant than saying that x*1000=x*10^3 and x*10^3 has an exponent in it. (Again, I sadly accept that in today's world, "exponentially" is being used to mean "fast growing", or sometimes more specifically "faster than linear". If I'm trying to understand what someone means, then it doesn't matter whether I find that usage to be a good idea or not.)
    - pessimizer 2y ago
      Exponential is c^x=y Polynomial is x^c=y Logarithmic is c^y=x
  - tommiegannert 2y ago
    Already eight years ago, I complained that people were using "exponential" where it doesn't make any sense. (See these two data points? Clearly exponential growth happend there. They're so far apart!) I believe the problem has increased exponentially since then. Now everyone is using exponentially in literally the same way as literally.
  - margalabargala 2y ago
    Thanks for adding the mathematical definition! You might be interested to know that the first definition of "exponential" is "of or relating to an exponent". The second definition is, as you say, "involving a variable in an exponent". https://www.merriam-webster.com/dictionary/exponential https://www.merriam-webster.com/dictionary/exponential As this is an internet forum and not a rigorous mathematical setting, I assert that my use of "exponential" is correct in context and to claim otherwise is incorrect. :)
    - Turneyboy 2y ago
      I'm not sure if you are kidding but just in case you are not this is very misleading and in fact misguided. Refering to polynomials as exponential just results in confusion essentially removing any meaning from the word. Any function can be written as something involving exponents, so that statement becomes meaningless.

Why?

Humans are notoriously terrible about estimating volumes when things are curved and volume functions are exponential. A great example of this done in 8th grade science classes across the US is to put 100ml of water in a 100ml graduated cylinder, 150ml in a 1L beaker, and ask the class which has more. Humans are awful at estimating how much volume the increased radius adds, and usually will say the 100ml. The problem only gets worse as we graduate from cylinders to spheres. We can all visually see which sphere is bigger, but cannot come close to estimating how much bigger one is than another.

AnimalMuppet 2y ago
Volume functions are not exponential. They are polynomial. (Fair point that people are lousy at estimating even polynomial functions, though...)
- dylan604 2y ago
  > They are polynomial I don't think this word means what you think it does. Or I don't. Exponents are just the number the value is raised. Squaring a value just uses an exponent of 2 where cubing uses an exponent of 3. Polynomials are x^2 + x + 1 type of equations. But admittedly, it has been 30+ years since I've thought about them at that level, so maybe I'm the one with fuzzy groking
  - AnimalMuppet 2y ago
    "Polynomial" meaning x^n. "Exponential" meaning e^x. Exponentials eventually grow much faster than polynomials, no matter what the exponent is. I mean, look, in v = x^3, the "3" is an exponent. But it's not an exponential function because the variable isn't in the exponent.
    - sfink 2y ago
      > Exponentials eventually grow much faster than polynomials, no matter what the exponent is. Since we're being pedantic, that last clause should be: "as long as the exponent is greater than 1."
      - margalabargala 2y ago
        We can go a step further with the pedantry, and say that the commenter above is using an unreasonably narrow definition of the work "exponential" and that there are others which allow x^2 to be described as "exponential". https://www.merriam-webster.com/dictionary/exponential https://www.merriam-webster.com/dictionary/exponential
        dylan604 2y ago
        Or one step further https://www.merriam-webster.com/dictionary/exponent https://www.merriam-webster.com/dictionary/exponent which is how I was taught. I only went to CalIII back in the early 90s, so who knows what's being taught now???
        sfink 2y ago
        We could, but I would describe it as "mathematically accurate". Which is not incompatible with "unreasonably narrow", given that the definition of "exponential" has recently gotten polluted enough that it is now often synonymous with "fast growing". But what's the point of arguing over definitions if we're going to start with a baseline of saying that there is no basis upon which to argue definitions other than recent conventional usage? > there are others which allow x^2 to be described as "exponential". Those same definitions allow x*1000 to be described as "exponential". (x*1000000 would be "more exponential"!) If you're describing something as exponential, then either you're just saying "fast growing", or you're trying to describe the type of growth. If you're describing the type of growth, then neither x*1000 nor x^2 is exponential. The fact that x^2 has an exponent in it is no more relevant than saying that x*1000=x*10^3 and x*10^3 has an exponent in it. (Again, I sadly accept that in today's world, "exponentially" is being used to mean "fast growing", or sometimes more specifically "faster than linear". If I'm trying to understand what someone means, then it doesn't matter whether I find that usage to be a good idea or not.)
  - pessimizer 2y ago
    Exponential is c^x=y Polynomial is x^c=y Logarithmic is c^y=x
- tommiegannert 2y ago
  Already eight years ago, I complained that people were using "exponential" where it doesn't make any sense. (See these two data points? Clearly exponential growth happend there. They're so far apart!) I believe the problem has increased exponentially since then. Now everyone is using exponentially in literally the same way as literally.
- margalabargala 2y ago
  Thanks for adding the mathematical definition! You might be interested to know that the first definition of "exponential" is "of or relating to an exponent". The second definition is, as you say, "involving a variable in an exponent". https://www.merriam-webster.com/dictionary/exponential https://www.merriam-webster.com/dictionary/exponential As this is an internet forum and not a rigorous mathematical setting, I assert that my use of "exponential" is correct in context and to claim otherwise is incorrect. :)
  - Turneyboy 2y ago
    I'm not sure if you are kidding but just in case you are not this is very misleading and in fact misguided. Refering to polynomials as exponential just results in confusion essentially removing any meaning from the word. Any function can be written as something involving exponents, so that statement becomes meaningless.