Does anyone know why $\displaystyle Infinity$ is not classified as a real number?
Saying you can't reach infinity is the same as saying you can't rotate a line 90 degrees! do this to a line, making it vertical, and you have a slope of infinity, $\displaystyle \tan 90$, or $\displaystyle \frac {1} {0}$. infinity is a value that can be used in math.
Sir, by definition, a line is infinite, no matter what it's position. I recomend you have a look at this:
infinite definition | Dictionary.com
Maybe this will help you.
By the by, infinity, by DEFINITION, is immeasureable. Nubers are measurements of magnitude. Therefore, inifity is not a number. Now that's Logic 101.
Oh, what do we say about $\displaystyle \tan\frac{\pi}{2}$?
It's undefined.
You may enjoy reading about hyper real numbers as used in non-standard analysis.
Here is a website from which you can download for free a calculus textbook based on their use.
Chapter one contains the material you may be interested in.
The basic reason why ∞ cannot be included in the real number system is that it would violate the rules of arithmetic. For example, ∞ + 1 = ∞. Subtract ∞ from both sides and you get 1 = ∞ – ∞. But it's equally true that 2 + ∞ = ∞, and then if you subtract ∞ from both sides you find that 2 = ∞ – ∞. Put those two conclusions together and you get 1 = 2, which doesn't look good.
So either the whole of arithmetic has to be redesigned, or ∞ has to go. Mathematically, it's no contest. Infinity has to go.
I'm assuming you meant "numbers"
Numbers are not always "measures of magnitude." You can't measure negative numbers, yet they still count. And saying something is undefined is a cheap way of ignoring something you dont understand.
Negative numbers and zero also often defy the rules of math, yet they are still numbers
Nice way to quit, but I'm not allowed Coke--too caffeinated, my parents say I'll get hyper
Also, Wikipedia gives information, but won't let me present what I think.
By, the way, Plato, I'll check out that site when I get a chance. Thanks!
What rules of math are you referring to? If you mean things like "you can't take the square root of a negative number" or "you can't divide by 0", there are no "rules of math" that say you can.
The set of real numbers forms a "complete ordered field". The "rules of math" for the real numbers are precisely those of a complete, ordered, field:
1) The real numbers form a commutative group under addition.
_____a) (x+ y)+ z= x+ (y+ z)
_____b) x+ y= y+ x
_____c) there exist a number, 0, such that x+ 0= x for all x
_____d) for every x there exist a unique y such that x+ y= 0
2) xy= yx
3) there exist a number, 1, such that 1x= x for all x
4) For every x except 0, there exist y such that xy= 1
5) x(y+ z)= xy+ xz
6) There exist an order x< y such that
____a) if x< y then x+ z< y+ z
____b) if x< y and 0< z then xz< yz
____c) for any two number x and y, one and only one of
_______i) y= x
______ii) x< y
_____iii) y< x
is true.
7. All Cauchy sequences converge.
Which of those do negative numbers or 0 "defy"?
Yes, you're right. Sorry, I mistyped. I meant "rules of arithmetic." I meant they just don't follow normal rules in many cases. Also, Infinity follows all those rules you just stated, so it is a real number.
I thought you were leaving, von_nemo, but you're still coming back, as the thanks to hallsofivy show.
I went to wikipedia and found "Real numbers can be thought of as points on an infinitely long number line." An infinite number line would include infinity.
The number line is infinite. thus, infinity must be part of the number line. it is the only number which can reach far enough to fill the "last" spot.
.................................................. .................................................. ....... Oooookay... I'm gonna assume i'm right, but check about once a week.