Does 1/0 equal infinity?
1/0 doesn’t, technically, equal anything; it’s undefined. If you divide 1 by a number that’s very close to zero, you get a number that’s very large, at least in absolute value—some of the numbers that are very close to zero are negative, so, for example, 1/(-0.000001) = -100000.
So, the way it’s expressed in Calculus terms is that the limit of 1/x as x approaches 0 from the right is positive infinity, and the limit of 1/x as x approaches 0 from the left is negative infinity.
I’m a bit confused about what you meant when you said in calculus terms.
It means that are you attempt to get arbitrarily close to 1/0 by, say, calculating 1/2, then 1/3, then 1/4 … then 1/323034234, then 1/323034235, then 1/323034236… then 1/3453049530458304598058032023850234750239845703249875, etc., you get closer and closer to infinity. For the other side, calculating -1/2, then -1/3, then -1/4 … then -1/323034234, then -1/323034235, then -1/323034236… then -1/3453049530458304598058032023850234750239845703249875, etc., you get closer and closer to negative infinity.
no what I mean is in calculus is 1/0 treated as equal to infinity?
No. In calculus 1/0 is undefined.
However, in calculus we’re often concerned with values arbitrarily or infinitesimally close to 0 without actually being 0.
So what Thudlow Boink is saying is that for f(x) = 1/x, for values of x very close but not equal to 0, f(x) will approach infinity.
That doesn’t change the fact that 1/0 is still undefined.
Here’s one explanation: http://tutorial.math.lamar.edu/Classes/CalcI/InfiniteLimits.aspx
Most of the time, “infinity” is not considered to be a number. You are of course free to set up a system of mathematics in which infinity is a number, but that will lead to other issues.
For example, consider the equation “X + 1 = X” One might think that this equation is true if X = infinity, but consider what happens when you subtract X from both sides of the equation. If you proceed according to the usual rules of algebra, you get 1 = 0. So you can see that if you think of infinity as a number, it has the potential to foul up the usual approach to algebra.
Check post #4. For the function 1/x, as you head toward x=0 from the negative side, the result tends toward negative infinity; as you head toward x=0 from the positive side, the result tends toward positive infinity.
So is 1/0 equal to negative infinity, or positive infinity? Neither. Both. Everything in between. You can’t pin a value on it; all you can say is that it’s undefined, because zero goes into one as many (or as few) times as you’d like.
1/0 is NOT equal to infinity in any sense. You can’t confuse 1/0 with the limit of 1/x as x approaches zero. They are different things.
This is the easiest way to get a handle on it. Division is the opposite of multiplication. 6 / 3 = 2 and 2 * 3 = 6, but if we try to assign any numeric value to the solution of 6 / 0 = x, then x * 0 needs to equal 6, but there’s nothing you can multiply by zero to get any answer other than zero.
Post #4 is actually completely wrong. Accidentally so, I’m sure, but wrong.
The examples get closer and closer to zero instead of infinity. If instead you go 1/2 1/0.356 1/0.000034 1/0.0000000000000000000000000001, and so on, we’ll be approaching infinity in no time flat.
You can only even talk about whether something is “equal to infinity” in a context where you’ve defined what it means for something to be “equal to infinity.” In modern mathematics, it’s perfectly standard and well-defined to talk about a limit being equal to infinity (some details are given in my earlier link), but not for a numerical expression to equal infinity.
:smack: You’re right. I totally missed that.
It is if you’re working on the Riemann Sphere, upon which “infinity” is a perfectly valid number and a lot of things people intuit about it are true. But that is not a standard interpretation for most first-year-calc uses. In particular, the distinction between +inf and -inf is lost
What is confusing is that mathematicians use the term “infinity” in at least two major senses:
(1) As a cardinal number, i.e., as the number of elements in a set. The numbers 1, 2, 3, etc., are cardinal numbers, because you can have sets with 1, 2, 3, etc., elements. Those are examples of finite cardinal numbers. Now, if you take the set of finite cardinal numbers, i.e., the set {0, 1, 2, 3, …}, you find that the cardinal number of that set is not finite. Loosely, you could call the number of elements in that set “infinity” – but then we find that there is an infinite number of larger infinite numbers, so to be precise this one is called “aleph zero” or “aleph null”. However, aleph null is the first number that mathematicians usually think about when you say “infinity” – and it is not equal to 1/0.
(2) As a term in defining limits in analysis or calculus. Here infinity is not a number at all, but part of either:
(a) a statement about what happens to a function f(x) when you make the variable x indefinitely large, or
(b) a statement that a function f(x) becomes indefinitely large when the variable x gets near to a particular value.
Each of these kinds of statements come into play when the function f(x) is 1/x:
(a) The limit of 1/x as x approaches infinity is 0.
(b) (For positive values of x) the limit of 1/x as x approaches 0 is infinity.
However, in neither case does either x or 1/x actually equal infinity: it just “approaches infinity”, which means that you can take an indefinitely large value for the variable or function.
FWIW, when we say “undefined”, we don’t mean there’s a mathematical entity known as “undefined” that has its own properties. It’s not jargon.
The word literally means what it says. Division by zero is NOT defined, nor can it be defined in a way that avoids contradictions or violates other properties of fields. Basically, there can be no definition of division by zero (with some caveats I will address below).
So, 1/0 is a lot like posing the question “What is the color of a duck’s quack?” The individual words make sense, and the question is easily understood. But it doesn’t make any sense. Maybe you can come up with some kind of clever pseudo-answer, but there’s no real answer to the question that makes any sense in the real world because the question itself doesn’t make any sense.
That said: we can generate an “extended” real number system where we do define division by zero so that 1/0 = mynumber and -1/0 = -mynumber, where mynumber > any traditional real number and -mynumber < any traditional real number, but this extended real number system is not a field nor is it a ring.
I won’t go into what ‘fields’ and ‘rings’ are, but they do have nice properties that make arithmetic and algebra work in the ways we expect.
The wiki article on the extended reals is a pretty good starting place.
Note that I used “mynumber” rather than infinity to attempt to avoid some confusion over the connotations normally associated with infinity.
Mostly, we avoid using the extended reals, because the limitations can be severe. I will note that the use of the extended reals is common for limits, sequences, and series.
So, in calculus, you might actually see the use of it.
For example, if you had two sequences {a_n} and {b_n} such that {a_n} converged to 1 and {b_n} converged to 0, it is valid to speculate about the convergence of the sequence {a_n / b_n}. The individual terms approach 1/0 and the sequence may very well diverge. In these cases, we say the limit is “infinity”, by which we mean the sequence is divergent or “infinity” limited to the context of the extended reals.
I notice that there’s another extension of the reals that uses a single, unsigned infinity, rather than two signed infinities.
If you weren’t concerned with calculus and limits, would there be reasons to prefer one of those systems over the other?
I came across infinity in the real-projective-line, 1-divided-by-0 sense in the study of computable reals. See, for instance, “Exact real computer arithmetic with continued fractions” by Vuillemin.
It would depend on what you are concerned with. Having a single point at infinity makes a lot of sense in the complex plane, but you don’t want to give up signed infinities for a lot of what you do on the real line.