be careful when an imaginary number is raised to a fractional power

this is not the time to sing only you but  rather we have to sing only i ladies and   gentlemen have a look right here starting with  i and let's look at this as i to the first power   and everybody knows 4 over four is equal to one  but i know one is the same as four over four   so i'm gonna put this down as i raised to the  four over four power well let's continue by   looking at this as i to the fourth power and then  raised to the one over four power well i to the   fourth power is just equal to one and then this  right here let's just put down the fourth root   and finally we can see that the fourth root  of one is of course just equal to one right   well from here to here i have no problem with it  but if you look back what we are trying to say   is that i is equal to one oh my god what kind of  new discovery is this i have never knew about this   seriously i is equal to one no of course not  and you guys told me this already the mistake   is at step number two right here right well in  this video let me explain all the things that   i think you should know first from here to here  again we have no problem with the fourth rule of   one is equal to one but we just cannot say one is  equal to i so let me cross this out first and now   let's just look back is it okay for us to say i  is equal to the fourth root of one or maybe look   at this backwards the fourth root of one is equal  to i what do you guys think well the truth is i   will tell you this right here is somewhat okay so  let me just put down okay but in a quotation mark   in fact this is what i learned after i came to  the united states it seems that you can save   whatever you want to say as long as you put  a quotation mark around it now where people   seems to be you know they can understand you  automatically but anyway this is somewhat okay and   the reason is because they are in fact four fourth  roots so let me just write this down in words of one   and especially when we're dealing with complex  numbers when we're talking about the first root   usually we shall find out all the possibilities  and the secret is that in fact i is one of them and here let me just tell you guys  what are the four first truths   well we will have one negative one i and also  negative i and of course i is one of them   so now you see if you put this down what are you  trying to say first root of one is equal to i   you only put down one possibility hm how about the  other four right how about the other three i mean this is like what you know yes i tell i'm  telling you hey i'm going to treat you a   full course meal tonight but in the  end i'm just buying you the soup and   you have to pay for the entree dessert the  drink the second course things like that   how would you feel so i'm going to leave that  to you guys okay for that all right so now   let's really talk about this right here why are we  doing this this way could we have done it as i to   the one over fourth power first and then raise  to the fourth power that's the main issue that   we would like to talk about right here so let's  go ahead and put that down if we only put down i   to the fourth power and then raise to the one over  fourth power versus i to the one over four first   and then raised to the fourth power well which  one is correct or are they actually the same here let me tell you in fact this right  here will actually give you four answers   just like what we have been talking about right  here and here you actually will just get one   answer the truth is whenever we are  working with a fractional exponent   with the complex numbers we have to be super  super careful and now let me just explain to you   guys why these are and how they are right here now  let's take a look at what do we do when we have i   to the fourth power and then raise to the one over  fourth power well we are going to do this inside   out and we will find out the possibility check  this out when we have this of course i to the   fourth power is just equal to one and then we are  going to raise this to the one over fourth power   we are going to do everything in the polar form  and then let's see to do so i'm going to take one   right put on the complex languages right here this  is the real axis and this is the imaginary axis r   is equal to 1 and then the theta is just 2 um pi  so far so good so we are going to write the one   in terms of the polar form right so this right  here will be 1 times e to the i theta which is   2 um pi like so and then after that we will you  know raise this to the one over fourth power and   now you see we have this raised to the one over  fourth power we will actually get four answers   to do so what we can do is multiply the powers  we get e to the i and then we just have n over 2   pi and to get all the answers all we have  to do is plug in 0 1 2 and 3 into the n   and the truth is this right here is equal  to 1 and this right here is just equal to i   and this right here is equal to negative 1 and  this right here is equal to negative i as we can   see we do have four different answers okay so  that was good now let's check this out what   if we start off by looking at i to the one over  fourth first and then raised to the [Music] fourth   power like that well this time though because we  have the one over four right here for the i we   are going to take i and put that into the polar  phone first and then there will be four answers   here this is i and the polar form is going to be  the following i is right here and again let's just   make it prettier r e and then i am anyway r is  equal to one and then the angle theta is equal to   that's pi over 2 and don't forget just add the  2 pi so for i this is what we are going to get   r is 1 so we just have e and then i and let's  get the common denominator so we'll just get 4 n   plus 1 over 2 right and then we also have  the pi right here so this is what the i is   then we are going to raise this  to the one over fourth power and then we are going to raise this to the   fourth power all right so that's what we have now  check this out what we are going to do is we can   do this times that and then remember we have  four answers so just make sure you plug in n is   equal to zero one two and three and let's just  go ahead and do that right here i will just do   two things at one time and it's equal to zero one  two and three so what exactly are we going to get   for this part when n is equal to zero this is just  one half pi times one over four is just one over   eight and pi also the i so the first answer here  is e to the i pi over eight next we will have e   and then when you plug in one you get five  and then still have the eight on the bottom so   i five pi over eight and then e to the next one  you get nine so i nine pi over eight and lastly   e to the 13 right so that's i and  then 13 pi over 8. pretty good so   this part the inside we do have four answers  however we are going to take each one and then   raise to the power of four and now here's the  deal we're going to just multiply the exponents it still looks like we have four different answers  right but you know what let me tell you guys that   the truth is this right here is equal to i this  right here 5 pi over 2 which is also that angle   it's also i likewise 9 pi over 2  it's also right there is i and lastly   is 3i no just kidding i don't know why i  wanted to do that but anyway i'm sorry i   so this is not the time to think only  you but rather we have to sing only i   i know it sounds so conceited but uh no in  math it's okay the answer is just only i   we don't have my answer so you see this is  actually a legitimate way to take care of that   uh fractional situation and you might be wondering  seriously why is that when we do the exponents in   a different order how can we end up with different  results well you can bring on the complex numbers   and let me tell you though in fact the problem the  real problem with this is that we didn't reduce   we didn't reduce the fraction so instead  of looking at i to the four over four   like this for that you get two different  answers but if you look at let's say i to   this three over four which the fraction is in  the reduced form already well you end up with   the same answer regardless which way you do  it so now let me show you guys that real quick what if we do i and then do the third power  first and then raise to the one over fourth power   in this case you see three over four you cannot  reduce the fraction for that well anyway inside   wise this is going to be negative i and then we  are going to look at this raise to the one over   fourth power and you know the usual business let's  take the i to the let's take the negative i now   to the complex plank which is right here this time  though the angle is uh three pi over two and then   you just add a two m pi right okay and then again  let's put on re and also the i m because i made a   horrible mistake before so i really feel like i  should put down r e and also the i m if you guys   know what i'm talking about thank you anyway so  negative i is going to be well r e i theta so r   is equal to 1 and then we have e and then i and  let's just get the common denominator so we have   4n and then plus 3 over 2 and then the pi so this  is what we have for negative i and we are going to   raise this to the 1 over 4 power well again we  can just plug in 0 1 2 and 3 and also multiply   the exponents so let's go ahead and do that real  quick and perhaps i will multiply the exponent   first it's slightly serious thing anyway this  time we get 4 m plus three over eight and then pi   and then n is equal to zero one two three plugging  plugging plugging so we do have these four answers   that's it now have a look right here half  half half a look right here suppose today   we did that by looking at i to the one over  four first and then raised to the third power   well i to the one over four is equal to what  we talked about earlier so i'm just going to   put down the result now we are going to raise  everybody right individually to the third power   i'll do the same yes they are this checks  with that right so check check and then   we have the 15 so this right here so checks  so two checks two checks now here's the deal   27 pi over eight is over 2 pi already  so what you have to do is just subtract   2 pi and then give you a common denominator so  this is actually the same as 11 pi over eight   that's pretty much the period of sine cosine so  this right here is three checks one two three   this is actually the same as that one two three  right so again this is the same as e to the   i 11 pi over 8. and lastly  if you look at 39 pi over 8   you have to subtract multiple of 2 pi and just  make sure that the angle is within 0 to 2 pi   well in that case you are going to subtract  4 pi and that will give you 7 pi over 8.   so this guy 1 2 3 4 is actually the same as 1 2  3 4. this is the same as e to the i seven pi over   eight so ladies and gentlemen as you can see when  you have this right here in fact it doesn't matter   how you do it so when exactly can we do that  well let me summarize all this for you guys   well let's say z is a complex number and  then we have m over n so this right here   okay is equal to either z to the  1 over n and then to the nth power   and you can also define this to be z to the  nth power and then to the one over this power   so they are okay under the assumption that if the  greatest common divisor of m and n is equal to one if the m and m they don't have  any common factors other than one   then you can do either way just what we are doing  right here but suppose this is reducible come on   just go ahead and reduce the fraction if you  ever have to do that i to the six over eight well   in this case seriously just go ahead and do   i to the three over four and then be happy come on  all right if g c t of m n is not equal to 1 then   just reduce the fraction and over n all right or  you do it this way the right way to deal with the   fractional exponent is that you do z to the 1  over n first and then raised to the m's power   so this shows you why earlier we should do i to  the 1 over 4 first and then to the fourth power   right so hopefully this helps if you find this  video interesting then you can continue the   learning at brilliant it is a math and science  website app with a focus on problem solving   and they have over 60 interactive courses in math  science and computer science and they are always   adding new ones i like their courses because  they are interactive and have storytelling   which make the learning process a lot more fun  they always challenge me to pay attention to   technical details just like what we did today  and they teach us how to think outside the   box it doesn't matter what your level is because  they have something for everyone go ahead and get   started with their basic courses such as algebra  geometry or go ahead and get to the advanced   topic that calculus will quantum computing  better yet use the link in the description   blender works black and red pen to get 20 percent  of discount for the annual premium subscription   thank you for checking them out and thanks  to brilliant for sponsoring this video   so i think that was the reason why you  guys told me this right here is the   mistake right but you know what in my opinion  this right here i think it actually has a   bigger issue and of course i'm putting this  in a quotation mark because why seriously why   why in the world would you like to write  i to the first power as i to the 4 over 4   it doesn't do us any good and then it becomes  so complicated after that oh my god seriously yeah so in my opinion that's where the bigger  issue is but let me tell you though if you really   want to deal with i to the 4 over 4 just like  what we talked about earlier be sure you do the   1 over 4 power first and then do  the fourth power on the other side   or better you just write it as i to the  first power man who did that yes it was me

Channel: blackpenredpen

Views: 358,959

Keywords: i^(4/4), imaginary numbers, complex exponential, power of i

Id: awrgXX0Qnjs

Length: 17min 35sec (1055 seconds)

Published: Fri May 07 2021