In case there's some question about whether i is a number:
- We know that x + 1 = 0 is a valid equation; however, if we only had regular "counting" numbers available, we couldn't solve it. We would have to invent a concept of negative numbers, which we are all comfortable with .
- Then we have the equation x2 + 1 = 0. It also had no solution without inventing the concept of imaginary numbers.
--> Imaginary numbers are negative numbers are both pretty abstract. We can't see or touch either, but they seem to be numbers in a purely mathematical sense.
My argument for both infinity and
i is their placement in the number system.
Number System:
All Numbers- Imaginary Numbers- Numbers using the quantity
i[/i(Complex Numbers). i defined as being the square root of -1.
Real Numbers- A number that can be expressed on a number line.
Irrational Numbers- A number that cannot be written as a result of the calculation: a/b.
Rational Numbers- Can be written as a result of the calculation: a/b. This includes repeating decimals, such as 5/7 or 1/3.
Integer- A number that is in the set of Natural numbers, 0, and the negatives of the natural numbers.
Whole Number- A number is in the set of Natural numbers, and 0.
Natural Number- A number in the set [1, 2, 3, 4, ...]
This is by no means the entire set of numbers, just what I think of when I thinking about what a particular value is. Since i fits, it's a number, and since ∞ does not, it is not a number.
Ok, so the next question is, is 0 a number? It seems that 0 is just a placeholder. Doesn't "0" represent the lack of a number, not an actual number?
I agree that zero is a lack of a number, but for practical uses, we have to regard it as a whole number.
If 0 is a number, is it even or odd? What is the definition of even and odd that shows that 0 is even? Can it be shown that 0 also fits into the odd category?
An even number is a number that can be expressed as (2*x). Seeing that 2*0=0, 0 is even.
Is the 0 in 2304 the same as the number 0? How are they related?
The 0 in 2304, means that in base 10, the second place, which is 10^1, has no value. 0 used in that text means that in whatever place it is, regardless of base, it means that that particular place has no value.
What is x in 2x = 6? Is it a number? It doesn't look like a number, but to anybody with more than a few years of math experience, it's immediately obvious that it's 3. If x can be considered a number, is it possible that our definition of numbers isn't as clear as we think?
x in 2x=6 is a variable. It is a symbol used to represent a quantity. In this case however, the hidden value "x", is 3. Which is a number.
What about x in 0x = 6? Is x still a number? Any number, real, rational, irrational, imaginary, etc. can be place in front of the x. But when the number 0 is, x no longer has a value. It seems like 0 isn't behaving as a number here.
In this case, x cannot be defined, as we cannot divide by zero.
Or, is there a case for considering 1/0 to be a number? Is the result infinite or undefined? Why? Is there a case for looking at it either way?
I like to view it in baby terms. Lets say we have "1" of something. If we divide it upon 0 people, has it been divided at all, or has it been divided to 0 people, but just in unknown quantities? I had to look this up, and this is what wikipedia said:
Division: 0 / x = 0, for nonzero x. But x / 0 is undefined, because 0 has no multiplicative inverse, a consequence of the previous rule. For positive x, as y in x / y approaches zero from positive values, its quotient increases toward positive infinity, but as y approaches zero from negative values, the quotient increases toward negative infinity. The different quotients confirms that division by zero is undefined.
I think this is the most important question of all:
How are 0 and infinite related? Think about how to get the result of infinite in a finite equation. The only way to get it is to use a 0. This means that 0 and infinite are related, and that infinite can be derived from 0. Because, as it was already discussed here, infinite isn't a number, does that show that 0 isn't either?
The only possible way I could think to produce ∞ from an equation would be x*0=0. But as that is undefined, I do not see any other way.
---------------Whew! That was long post
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Reminds me of last year's math homework...
! 0! = 1.
. 71 Degrees Fahrenheit. I believe it hasn't snowed in FL since the 1980's, and that was up by the panhandle...
I was just about to say the same thing.
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