1 ^ ∞, It's Not What You Think

1 to the 1 is 1 1 squar is 1 1 cubed is 1 1 to the pi is 1 1 to the zero is 1 in fact it's true for negative numbers as well further than that one to any real number is one so one to the Infinity power should be one seems like this should be true it certainly feels true but is it when we ask questions like this we also have to ask what do you mean what EX exactly is one to the infinity we know how exponents work for finite numbers we take the base and multiply it by itself exponent times but what does an infinite exponent mean one way you can try to handle Infinity is by using the calculus technique of limits we can approach Infinity via functions letting the variables in those functions grow larger and larger out to positive Infinity the problem with this is we can approach our question of one to the Infinity in a multitude of ways consider these two functions 1 + 1X X and 1 + 1x^2 to the X take these functions and let X get bigger and bigger that is take the limit of these functions as X grows to Infinity 1 /x and 1x^2 would get very small towards zero and the exponents of X would grow very large out to Infinity a essentially we have 1 + 0 to the infinity or 1 to the infinity these Expressions appear to approach one to the Infinity power yet Computing these limits using calculus techniques reveal their true answers which are both different this is why when we discuss limits one to the infinity is considered an indeterminate form we can't determine its exact value without further inspection another approach to dealing with this issue is by using logarithms logarithms are the inverse of exponents and we can often use them to deal with exponents in particular this property of logarithms is often used to turn exponential Expressions into linear Expressions which are easier to deal with so maybe we can try this with one to the infinity now completely glossing over the fact that this doesn't work if the exponent is infinity we run into a different problem even if we could drop this Infinity power down the logarithm of 1 is0 and we have Infinity * 0 this is its own basket of worms and I've dealt with it in another video the good news is using limits or logarithms weren't really addressing my original question I don't care about approaching or getting close to one to the infinity I want to know if theoretically I could type into my calculator one to the power of infinity and hit enter what would the result B this brings up our next issue doing that doesn't make sense exponentiation is only traditionally defined for numbers and infinity is not a number it's a different kind of concept and although it might be kind of disappointing for good reason one to the Infinity power is traditionally undefined but viewers of this channel know that I hate to stop there and although one to the infinity is traditionally undefined we could devise a new mathematical system where we do Define it ironically an intuitive approach here is actually using limits we sort of naturally think that one to the infinity is one multiplied itself in a NeverEnding way so we could Define this symbol one with an exponent of infinity to mean the limit as X approaches Infinity of 1 to the X power now we have no trouble Computing this since 1 to the x is 1 for all real values of X and now taking the limit of any constant is that constant so if you'd like to make this definition you could say 1 to the infinity equals 1 just like you probably want it to be just keep in mind it's not traditionally defined this way and if you try to use this definition out of context talking with any other mathematically trained person including myself they might disagree with you

Channel: BriTheMathGuy

Views: 943,656

Keywords: math, maths

Id: drsd7Mc-gvI

Length: 4min 28sec (268 seconds)

Published: Mon Jan 30 2023