<p>2400 (sat)</p>
<p>I love it when someone gets the first digits right but can’t get the order of magnitude right, e.g., “I read in the paper today that the State is spending 17 million, or was it billion?, on x.”</p>
<p>I think it depends. A person can comprehend a million or billion dollars, but a million or billion baseballs would be out of the question. I mean, we wouldn’t be able to comprehend the actual physical amount of dollars, but we can comprehend the intrinsic value, like one million could buy you this, this, and this.</p>
<p>For me personally it’s only like six or seven. After that I just imagine them as groups of varying sizes. If you told me to think of ten baseballs I would think of a small group of baseballs. If I could somehow freeze that image I highly doubt it would be ten. It obviously gets worse and less accurate the higher the number. </p>
<p>I also think it has to do with the object. I’m better at imagining groups of 20-30 people rather than imagining 20-30 baseballs. That has more to do with my remembering what that amount of people looks like, rather than me being able to comprehend a number. I can also visualize 500ml of water easily, not as 500 individual 1ml water droplets but as a collective that I’m familiar with.</p>
<p>Counting objects is quite a narrow application of numbers, even in their discrete form. It may not be easy to make a mental picture of 100 objects, but most of us have the percentage system ingrained in our minds and we can easily recognize the significance of the difference between 94% and 95%, even if we cannot easily distinguish between 94 and 95 objects.</p>
<p>In fact, the elements of any finite set can be mapped to the natural numbers. You could assign consecutive numbers to the words in your vocabulary, and if you know x words, you could recognize the difference between x and x-1 just as easily as you could recognize the difference between 1 and 2.</p>
<p>I think we’re all interpreting “comprehend” differently. I took it to mean being able to actually visualize a collection of something. I didn’t take it to mean recognizing the difference between two collections. I personally can’t anymore accurately visualize 94% of 100 pens than I can visualize 94 pens, why? Because they’re the same number and I can’t accurately visualize that many pens.</p>
<p>Of course I can “comprehend” how many pens that is. It’s 94 pens, that’s more than 93 but less than 95. I know the significance of the difference between 1 million and 1 billion just as much as I do 10% and 15%. I may not be able to visualize a number in the millions, but I can definitely comprehend it in terms of application. If we get into numbers high enough and that I’ve never heard of, well then obviously I need to know what that number is before I can comprehend it.</p>
<p>So if comprehension is being to visualize objects or estimate how many objects there are without counting, then I’m sticking with my original answer of say 3 or 4. Yet if comprehension has to do more with being able to use those numbers and compare them to one another, then I feel the maximum number is whatever the highest number your familiar with is.</p>