# Pi Day, 3/14/15 at 9:26

This post will be posted automatically 3/14/15 at 9:26 EST đź™‚

For those who don’t know the significance of that time and date, here are the first digits of pi:

3.1415926

Also, here’s something else:

I <3Â Ď€

That’s internet for “I heart pi”, but it’s actually also the equation 1 < 3×3.14…, which is a true statement. Hooray for double meanings! Hooray for math! Hooray for Pi!

# Completed Game Idea: Up-or-Down

After creating a very complicated (but very cool) game called Djinn Combat (which is still in development) I wanted to create a simpler game that shared aÂ specific aspect with Djinn Combat that I really liked: that it could be played anywhere, at anytime, and with nothing but you and another player. So I created the game I call Up-or-Down.

The concept is simple enough to learn. The first playerÂ (P1) says a number 1-100, and the next playerÂ (P2) says a different number based on what the first person said. If theÂ P1 said an OddÂ number, then P2 has to say an OddÂ number that is AboveÂ the number that P1 said or an EvenÂ number that isÂ Below the number that P1 said. However, ifÂ P1 said an EvenÂ number, then P2 has to say an EvenÂ number that is Above the number that P1 said or an OddÂ number that isÂ Below the number that P1 said. ThreeÂ other restrictions are that the number can’t be less than 1 (it can be 1 though), it can’t be more than 100 (it can be 100 though), and it can’t be the number that was last said (after all, that isn’t above or below the number that was said).

For example, if P1 said 53 (an odd number),Â then P2 could say something like 67 (an odd number above 53) or something likeÂ 2 (an even number below 53). However, if P1 saidÂ 8 (an even number), P2 could say an even number that is above 8,Â  such as 100, or an odd number that is below 8, such as 7.

You can also have as many players as Â you want, although there are two different ways to play. In each version you have an order, such as P1 then P2 then P3 then P4 then P1 again. If P1 saidÂ 5, P2Â saidÂ 7, and P3 said 8, P3 would have lost, but P4 would continue the chain with P3’s last response, meaning that he could answer 10, or 80, or 1, etc. Note: even if the person who went out said aÂ really wrong number, like 1 million, the next player still has to say a number that is valid (between 1 and 100 and the correct even/odd-ness)

The different versions of multiplayer involve what happens after someone goes out. Version #1 isÂ a tournament version, in which if someone says the wrong number they lose and have to stop playing until the end of that game.Â Whoever the last one still in wins, and he/she gets to choose the first number forÂ the next game.

Version #2 is a training mode, in which no one goes out. If someone says the wrong number, then he/she gets to change his/her number to one that fits. This way people can more easily learn the game without having to wait until the next game every time they go out. However, you can still have a winner by keeping track of whoever the last person to make a mistake is.

Feel free to add any rules that you’d like, it’s pretty adaptable. Hopefully I’ll soon have a post on alternate ways to play to make it harder or different once you have it down. Either way, enjoy!

~ George

# How To do tan^-1 on a calculator

First off, I’m assuming that your calculator has threeÂ things: a “tan” button and a “tan-1” button. You do not have to know where theÂ “tan-1” buttonÂ is, but if it is a scientific calculator then it probably does have it. I’m also assuming that you know what theÂ “tan” and the “tan-1” buttons do, so this will not explain that.
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Your calculator should have a button labeled â€śtan-1â€ť, although usually you have to hit the â€ś2ndâ€ť key and then the â€śtanâ€ť key. If you are using the calculator that comes installed with Windows 7 (at least I’m pretty sure that’s where it comes from, if someone could verify that please)Â and already have it set to scientific calculator (hit the â€śviewâ€ť button in the top left corner then select â€śscientificâ€ť), then you hit the â€śInvâ€ť key (short for â€śInverseâ€ť, I think) and the â€śtanâ€ť button will turn into â€śtan-1â€ť.
Also, make sure that the calculator is set to degrees (or radians), because it is extremely frustrating to get the wrong answer and have no idea why. To set that there is often a button that says “DRG”, with which you can cycle through degrees, radians, and gradients, using arrow keys if your calculator has them. On the Windows 7 calculator there are three buttons in the top left, beneath where the output is shown. Click the one labeled with what you want.
Finally, if you already know how to get the “tan-1” button but donâ€™t know how to type it in, most fancy calculators (the \$10+ ones) require you to input the items in the order that you would write them (e.g. type â€śtan-1â€ť then â€ś500â€ť then â€ś/â€ť then â€ś1001â€ť then â€ś=â€ť). However, almost every other calculator that I use (including the Windows 7Â one and calculator apps) do it similarly, but certain functions (such as square root, log, and tan) are done after you finish typing the expression (e.g. type â€ś500â€ť then â€ś/â€ť then â€ś1001â€ť then â€śtan-1â€ť then â€ś=â€ť).
Of course, all of this also applies to the “sin”/”sin-1” and “cos”/”cos-1” buttons, but I was specifically asked about theÂ “tan-1” button, so that’s what I went with.
Â ~ George

# Age < 11

What if there was a boy who’s literal age was < 11? To clarify, it wouldn’t be that no one knows his age, it would be that his age simply wasn’t an exact number, just “under 11 years old”.

# Ponderings from Math Class

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 thatÂ 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 wandersÂ around makingÂ ponds, the word explains itself, you see.” đź™‚

(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’tÂ exactlyÂ 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 xâ‰ 1, both of which areÂ 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 đź™‚

(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, andÂ nobody would trust that their calculator wasn’t just rounding to the 20th decimal, causingÂ everybody would think they just had to restate the problem instead of giving an exact decimal.

More to Come!

~ George

# Spherical Properties

I was reading the Wikipedia article aboutÂ the eleven properties of a sphereÂ (I can’t remember why, but I do stuff like that a lot) and I noticed the first property, which I’ll quote:

The points on the sphere are all the same distance from a fixed point….

The article then says: Â “[This]Â is the usual definition of the sphere and determines it uniquely.”

The fixed point described is, of course, the center of the sphere, and the article says that only the sphere has the property described there.Â What I was wondering was, what if there is a fixed point forÂ every shape from which the distance between it and any point on the shapeÂ is the same? I’d imagine this would have to involve curving space and some sort of 4th spatial dimension (not including the ones predicted by string theory). This would have to include everything from planes torusesÂ (doughnut shaped, and yes, I GoogledÂ “what shape is a doughnut”. The Wikipedia article has some cool pictures in it :).

~ George