Why did mathematicians conceptualized infinity? Do they use it in any mathematical systems?
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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!
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