Why did mathematicians conceptualized infinity? Do they use it in any mathematical systems?
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I’m glad you asked because it gives me a chance to share [this video](https://youtu.be/mrVg9GM5h7Q) which was a huge help to me. I have difficulty understanding this stuff in pure numbers, but she shows the physical, practical problems that eventually led to understanding infinities.
It is used to calculate PI and some other numbers. An old method of calculating PI that is no longer used is to imagine a polygon inside a circle with all of the corners touching the circle. Calculating the circumference of a square gives a bad approximation of PI. An octagon gives a more accurate approximation. A 16 sided polygon is even more accurate. As the number if sides tends towards infinity, the approximation becomes more accurate. Isaac Newton found a better method using an infinite series that is easier to calculate.
Infinity wasn’t conceptualized, it just shows up. Divide 1 by 3. You get 0.3333…. How many 3s are there? Whoops! We found an infinity.
Infinity as a concept is also very common in calculus. Calculus deals with continuous change, meaning it analyzes how something behaves at every single point in time. How many points does a curve have? Infinitely many! One part of calculus deals with measuring the area under a curve, and the way mathematicians do this is by slicing that curve into infinitely many, infinitely small rectangles and then summing up the areas of those rectangles to get the total area! Very cool.
There’s also derivatives, which is just a fancy name for finding a tangent to a point on a curve, it is done by drawing a line between two infinitely close points! Also very cool.
I mention these things because they are also very useful in the real world. For example: artificial intelligence! Derivatives put the “learning” in machine learning. Many modern AI models use something called backpropagation, which is just a fancy name for taking a bunch of derivatives and adjusting model parameters based on what you got. This is just one example, but applications of calculus can be found almost everywhere in the modern world, it is one of the most “useful” areas of mathematics!
I’d recommend Netflix’s “A trip to infinity”, it’s a fun little docu / interview movie about how many versions of infinite there is.
18th century mathematician Leonhard Euler said “Infinity is just a way to reason about limits.” Limits are roughly a calculus-thing so until you start working with that it’s just a placeholder symbol. It’s an idea and not a number.
Numerical analysts use an explicit infinity symbol in interval analysis. The idea is that each interval is represented by an upper bound and lower bound. Various arithmetic and set operations are defined for intervals.
Plus and minus infinity (and undefined) are defined with consistent properties.
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