Zeno of Elea (born circa 490 B.C.E.) was known for paradoxes involving infinity. The Greek philosopher Anaximander used the work apeiron to refer to the infinite. Around the 4th or 3rd century B.C.E., the Jain mathematical text Surya Prajnapti assigned numbers as either enumerable, innumerable, or infinite. The concept of infinity was understood long before Wallis gave it the symbol we use today. It's also possible the symbol is based on omega (Ω or ω), the last letter in the Greek alphabet. Wallis may have based the symbol on the Roman numeral for 1000, which the Romans used to indicate "countless" in addition to the number. The word "lemniscate" comes from the Latin word lemniscus, which means "ribbon," while the word "infinity" comes from the Latin word infinitas, which means "boundless." The symbol, sometimes called the lemniscate, was introduced by clergyman and mathematician John Wallis in 1655. In any case, (1/0)*0 can't be defined to make sense.The infinity symbol is also known as the lemniscate. Infinities propagate through calculations as one would expect: for example, 2 +, 4/ 0, atan () /2. Sometimes people also extend the real numbers to include -\infty, sometimes people don't: it depends on what you want to do with things. So we add the limit in and get a new number larger than all real numbers. Where do these choices come from? They do indeed come from taking limits of things like 1/x as x tends to zero: this is can be made arbitrarily large by taking x arbitrarily close to zero. The thing we commonly add is a symbol \infty and it is true that 1/0=\infty and 1/\infty=0 are definitions we make, but we do not define 0*(1/0) even in this extended set of numbers (just as we do not define 1/0 in R). The value of the first limit is 0 the value of the second limit is 1 the value of the third limit is. For your first and second questions, here are three limits of the form : In all three limits the first factor 'approaches' infinity while the second factor 'approaches' zero. We have to 'add' something to R to start to make sense of it. To qualify the answers above, I'm talking about limits. So, this is not actually a question about R. Let's switch to calling the number line the Real numbers which we will denote by R. So 1/0 or 1/infinity are not questions about the number line. So we're not actually talking about multiplying by the. We're usually talking about multiplying two functions, say f (x) and g (x), where f (x) approaches infinity and g (x) approaches zero. The first thing to remember is that the number line does not have infinity on it. When we say 'infinity times zero', we're usually referring to a specific type of limit problem. Just pick x to be bigger than 2/k, and you'll see that the limit is less than their guess, so they are wrong. For instance, suppose they think it's k, where k is really really small. What exactly do your friends think the value of 1/infinity is anyway? By using limits (so it's rigorous) you can show that if they think it's any number other than zero, they are wrong. It is true that in the limit as x->0, 1/x goes to infinity, but I'd be extremely weary of putting equals signs there without mentioning limits or you'll end up proving 1=2.
Clearly this is wrong and is a sign you've assumed you can do a certain operation you shouldn't. Using float (‘inf’) and float (‘-inf’): As infinity can be both positive and negative they can be represented as a float (‘inf’) and float (‘-inf’) respectively.
Below is the list of ways one can represent infinity in Python. I could just double each side to get 2 = 0*infinity, so that means 1=2. One can use float (‘inf’) as an integer to represent it as infinity. Now as x-> infinity, the 4/x and 2/x parts go to zero and you end up with y=3.īecause of this requirement to 'approach infinity' rather than just put in infinity, you can't say things like 1/0 = infinity means 1 = 0*infinity. Rather than just put in x=infinity, you consider what happens when x gets really big (or sometimes when x gets really small).įor instance, what is y = \frac for real-time views of the terrain to help you pick the best line along the trail.
When you're doing this kind of thing you need to instead use 'limits'. With 1000 HP and 0-60 in 3 seconds, a quiet revolution is coming. For a start, infinity isn't a member of the Real numbers, it's a member of the extended Reals. Why does the IEEE standard follow intuition 1. This feels counter-intuitive: Why is 0 × not 0 We can think of 0 × 0 as the limit of 0 × z as z tends to. However, IEEE 754 then specifies the result of 0 × as NaN. You cannot just plug in infinity and zero into certain equations like y = 1/x and then expect it to obey the 'normal rules'. Why is Infinity × 0 NaN IEEE 754 specifies the result of 1 / 0 as (Infinity).