One of the things that has me so tied up for time is my math class, precal/trig in one semester (5 credits!). However, during class I often think of random questions or observations, and I think that instead of making a post for every single one, it’d be better if I just make this single post and keep updating it. If I make a post based on something in here, even better! Quite a few of these will probably only make much sense in my head, but since half the point of this blog is to make people think about things they wouldn’t usually think about, I think it’ll work :). Anyhow, on to the ponderings…
(1) First off, every time I hear the word “pondering” I think of a joking definition made by a past teacher of it. Someone asked what pondering meant, and he replied something like, “pondering is what it’s called when someone wanders around making ponds, the word explains itself, you see.” 🙂
(2) What if 4 > 4 was somehow a valid statement? (((I actually have a way to make it so, but it involves something that I’m not anywhere near posting)))
(3) 1 = 1/1 = 2/2 = 3/3… Somewhat similar to the last question, what if it wasn’t exactly equal, or in some way it wasn’t true. They actually aren’t exactly the same, after all; you type them different, so if you were to measure the bit count of them they wouldn’t be equal
(4) How can you have $i ? Is it equal to $1? $-1?
(5) A joking quote from the teacher: “I’m pointing big!”. What would that actually mean? “Point” means a specific place, but point “big” could mean a general direction, or what?
(6) Interesting Observation: It is impossible to plot a perfect graph, because there is always a smaller number to calculate
(7) Interesting Observation: In precal, you are told that the only numbers you have to worry about being undefined are dividing by zero (infinity), even roots of a negative number (i), taking log of a negative number (also i), and taking log of zero (infinity again). However, you also can’t do this: 0^0, which is undefined (note: not infinity or i, “indeterminate”; even Wolfram Alpha can’t figure it out: http://www.wolframalpha.com/input/?i=0%5E0). So if you had a problem in which you needed to find the domain of 0^x, the answer would be {x|x<>0}. Also, a^0 = 1 = a/a, so 0^0 = 0/0, which is also indeterminate.
(8) Funny comment my teacher made: “Now back to more pleasant things, like math!” (after a conversation about exponential growth and Ebola)
(9) log base a of x = y is equivalent to a^x=y, so long as x>0 and x≠1, both of which are impossible (at this level of math?), for example, log 0 = y would mean that 10^SomeRealNumber = 0, but what if it worked? Same with log (-100) = x, which means 10^x=-100, although I think that one is possible with imaginary numbers
(10) Interesting Observation (that I’m positive lots of people know): You can have any base for a number system (ours is base-10) and still use decimals, it just depends on when you shift over a decimal. For example, our base 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (one 10), 11 (1 and one 10), 12 (2 and one 10)…; base 8: 0, 1, 2, 3, 4, 5, 6, 7, 8 (one 8), 11 (1 and one 8), 12 (2 and one 8), 13, (3 and one 8)…; also, base 12: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c (one c), 11 (1 and one c), 12 (2 and one c)….
It can get pretty hard to imagine, but you could do it, especially if you were raised on it
(11) 0 < 8 > 0 is a perfectly valid statement, but is there any context in which it would be useful? At the very least it’s symmetrical 🙂
(12) It’d be interesting to create a problem that has an exact answer and is made up of whole numbers, yet the exact answer is 20 digits long (and not just a irrational number, like e, or a fraction, like 1/3). Also, if you wanted to be mean, you could say that the answer has to be exact, and nobody would trust that their calculator wasn’t just rounding to the 20th decimal, causing everybody would think they just had to restate the problem instead of giving an exact decimal.
More to Come!
~ George
