The pattern he was describing at the end could be expressed with exponents. As a software developer I frequently deal with the powers of 2 as used in binary.
He started with 16 (or 2 to the fourth) and reduced the exponent by one each time until he reached 1 (or 2^0). We know that evaluates to 1 but he's describing it in terms of multiplying zero 2s together or an "empty product" of 2s.
The same pattern would apply using 3s, 4s, 5s, etc.
@xHAZARD78x Because of the properties of 1, it would prove wrong a lot of theorems. They would have to come up with new theorems, but they would be exactly the same except they would exclude 1. So it makes sense to alter the definition of 1 instead of keep it and have it change all the prime theorems. Because if it was really the same as all prime numbers, it wouldn't alter most of the definitions.
@MasterLyrics20 You cannot divide by 0. At first, my logic was that you can fit 0 pieces into 0 groups, thus 1. But, you cannot divide by 0. I'm not a mathematician, so don't ask me.
if every positive integer can be written as the product of primes then 1 must be prime. else how do you write 1 (a positive integer) as the product of primes?
seems your famous theorem is flawed. if 1 is prime then there's no uniqueness, and it if isn't then there's no way of writing the first positive integer as the product of primes...
I'm so confused, he said 2 is the product of just 1 prime number, itself, 2. But isn't the definition of product 2 or more number multiplied by one another.
@Noovil25 it means that each positive integer has one and only one product of prime numbers. so if one was a prime number there would be infinite products of prime numbers for each positive integer.
@figisligit thanks, but why does a positive integer have to be a unique product of primes in the first place? why can't it just be the way it is, with an infinite number of primes?
this kind of trickery can be important sometimes. For example in physics, our best current theories make predictions where the value of what is supposed to be a measurable, experimental quantity corresponds to a divergent sum. So physicists have learned to take (somewhat mathematically questionable) limits to get finite answers - and these answers agree very precisely with experiments! So sometimes you have to be willing to divide by zero. :)
@anticorncob6 a multiplication operation (a 'product') requires two numbers. Since there was only one number, a "x 1" is implied as the other number.
Note: There was an interesting sleight of hand going on in the video example. The powers notation was being used, but was referred to as multiplication. These aren't exactly the same thing. As an educational video, I would have expected better.
@aikimark1955 What I was trying to show you is that it is possible to multiply only one number, if I can convince you about zero numbers it's possible with one as well. You can make your own definition of "product", but mathematicians will not agree.
@anticorncob6 Since an empty product implies that "1" is the other value being multiplied, I would need more details from you or the author(s) of this video how an implied value (1) isn't really a value and doesn't actually use "1" as a prime number, which contradicts the assertion of numberphile.
@aikimark1955 There is NO rule in mathematics that states a product needs two numbers, it just looks funny so you assume it's not there. Your comment is entirely circular reasoning.
What about -1? If you counted -1 as prime, you could go into the negatives, but I don't know what you'd do with zero, it's not prime nor can it be multiplied by primes.
@anticorncob6 -1 doesn't count for basically the same reason as 1 doesn't. You could decide to include it in your list of what you call prime numbers, but then your concept of prime numbers wouldn't be quite as useful in doing mathematics. Things would would get more complicated, as you yourself point out.
So it makes sense not to include 1 or -1 (or 0 or 1.5897, etc) as primes. If we did, 'prime' wouldn't be such a useful concept! :)
But 2... where is the product if the definition of product is
The term "product" refers to the result of one or more multiplications. (implies 2 number at least)
2 would need to be the product of 2 numbers, 2 and 1, else 2 would also need to be refined to be somewhat unique, because product is defined as 2 or more numbers multiplied together, Will this theory mean you will now refine what product means?
@iamufreak Nah, the video states this a bit too non-scientifically for such questions. In fact when you think of any positive whole number and its 'prime representation'(I'm not English, I don't know the correct terms, unfortunately), it's actually not 6 = 2*3, it's rather 6=2^1*3^1*5^0*7^0(...), therefore 2=2^1*3^0... and the definition of a product is fully applicable here. (Just on a side note - without this, there would be a problem with all of the prime numbers, not just 2).
Hello numberphile. I have always asked myself this question. I have searched it online a couple of times and never found a really satisfying answer to this problem.
"If x^0 = 1 and 0^x = 0, then why does 0^0 = 1?"
As I said, I have tried looking this up and found the weirdest answer with all sorts of highly advanced formulas that the average person cant understand. I once even read this: "because mathematicians say so!"....
@GiaIsTheBest 0^x=0 only for positive x. 0 isn't positive so it isn't affected by this rule. You can think of 0^0 as the product of zero zeroes as x^y basically is the product of y x's. Anything of the form x^0 is the product of zero x's so basically it is the product of nothing, the empty product, which is 1 as explained in this video.
@GiaIsTheBest It's true that a^0 can also be thought of as a/a. So if what you are trying to say is that a^0 doesn't make sense when a=0 (or in other words that 0^0 doesn't make sense, because it's like 0/0) then you're right.
But keep in mind that 2^0 makes plenty of sense - that's like 2/2, which is 1.
@GiaIsTheBest Well, dividing by zero is possible sometimes, just not when the numerator is finite. For example. sinx/x = 1. One difference here is that lim x^0 as x goes to zero is different from lim 0^x as x goes to zero. So you might want to say that there's no consistent way to take the limit, so we should say the limit DNE. On the other hand, lim(x^x)=1 as x goes to zero. So people usually say 0^0=1 for that reason.
@GiaIsTheBest Oh I was also going to say, this kind of trickery can be important sometimes. For example in physics, our best current theories make predictions where the value of what is supposed to be a measurable, experimental quantity corresponds to a divergent sum. So physicists have learned to take (somewhat mathematically questionable) limits to get finite answers - and these answers agree very precisely with experiments! So sometimes you have to be willing to divide by zero. :)
@GiaIsTheBest 0^0 = 1 is just an agreement, it's not derived from any arythmetics. However, there is a good reason for that. Let f(x)=x^x. Then f(1)=1^1=1. f(1/4)~=0.7, f(1/8)~=0.77, f(1/32)~=0.9 (...) and generally, the closer x gets to 0, the closer f(x) gets to 1. And, more to the point, you can get as close to 1 as you want. Meaning, that there is an x close to 0 that 1>f(x)>0.9999 for example. But it's never exactly 1. Therefore, if you visualize it, you draw a conclusion
@Bulasz "0^0 should be 1!". That's in simple terms, I can expand on this in a private message if anyone would like. Hope I helped : ) I can clarify anything, in a PM as well, if need be.
Finally somebody manages to explain to me why 1 is not a prime. I got marked down in a math test once because I included it in the list of primes which cost me a whole grade and my math teacher (who was a horrible person in the first place) could not explain to me why this is the case. Had a very negative effect on my future relationship with mathematics.
So, 1 is the product of primes by being the product of "no" primes. And we think that is more logical just telling folks adding and extra "1x" that they are just being silly? Reminds me of the old Hitchhiker's computer game. Your inventory always included "no tea." To get tea, you had to put down "no tea." (Hope I didn't spoil that for anyone)
The definition I know for prime numbers is that prime numbers have exactly two (unique) devisors (A and the number itself). Which is a defenition that 1 doesn't fit.
The definition you gave is one I learned at school.
@letimics There is no "actual definition", you can use whichever one actually represents all of the prime numbers and none more. Your does, except you have to add that p is positive.
I find it fascinating that in math you have to follow patterns to get the certain product. Without patterns, there would not be any mathematical values. Like 3 to the power of 0 is 1, because 3 to the power of 2 is 9 divide by three, 3, and divide by 3, 1 and then .333
Note there is another reason for one not being a prime:
In general, given a commutative ring, you can define prime elements similarly (what you do is say that a prime is an element for which if p divides a.b then either p divides a or p divides b). You usually also enforce that p is also itself not a unit (i.e. has no multiplicative inverse).
This set of numbers has nicer properties e.g. the property of being prime is equivalent to the property that it generates some non-trivial prime ideal.
I wish they'd just teach school kids to "multiply by the inverse of x" as opposed to "divide by x" if only to avoid all the annoying "divide by zero" crap.
@Hewpie Infinity isn't a number and is (at least, should be) no longer used be mathematicians. Division by zero isn't defined regardless of what is divided, be it 0 or 1.
@Hewpie Don't let @Sc0rpi0n0 confuse you; firstly, any real number divided by itself is going to equal 1, so that is not what defines it as a prime number, this is called the identity property. Secondly, 0/0 is not infinity, and though it makes cognitive sense, it can be disproved by the fact that anything divided by 0 would equal infinity. 0/0 is almost always considered undefined, with the exception of computer engineering, where, for whatever reason, 0^0 being equal to 1 makes formulae work.
@TheMuffinChef Thanks for that explanation. That clears up the issue about 0 not being a prime number.
What a mess zero makes! So 0 divided by 0 is not a number, and not infinity or 0. Anything raised to the power of 0 is always 1, unless it's negative, which makes it -1.
@Hewpie Not really; -0 doesn't exist because 0 is the only number with the property of never being negative nor positive. Also, even negative numbers divided by themselves are 1, not -1, because as in multiplication, the negatives cancel out with division.
So, 0/0 isn't infinity because dividing by any number is supposed to be unique, and anything divided by 0 would cognitively be infinity, so the whole thing is invalidated. It's not 0, and only in certain situations is it 1.
Another way to write these products-of-primes is as a polynomial. Just as you may have
a·x^2 + b·x + c = 0
that's also expressible as
a·x^2 + b·x^1 + c·x^0 = 0
So in this notation,
132 = 2^2 · 3^1 · 5^0 · 7^0 · 11^1 …
So the Fundamental Theorem of Arithmetic says that there's a unique list of exponents to put on the list of prime numbers to get the product which is that number. The number (one) is the whole number you get when all the exponents are (zero).
@kpYak With a number to the zeroth power x^0 (for example 2^0) you actually have x/x (like 2/2) or a division that results or have the quotient of 1, so 1 in this case is not a product and further shows that the product 2 is equal to 2*1 not 2=2, since 2/2*(2) = 1*(2) = 2. Without 1 as a prime it is impossible to have 2 as a product of primes.
The 1 that is shown at the end is *not* a product or an empty product as it was called, the 1 is a quotient.
Calling numbers "prime" is just a name for a set of things that fit a definition. If the list is more useful in a certain way then we can make the word "prime" mean whatever we want it to mean.
I'm tempted to denounce this empty product non-sense, but if I've learned anything from particle physics, it's that you make rules based on what works, not what can be easily understood.
I was convinced by the theorem untill Brady asked if 1 was a whole number. The whole empty product thing doesn't fit into the theorem in my opinion. It kind of annoys me
We make the definitions up to suit ourselves. What it amounts to is that for number theory you repeatedly want to include a list of numbers that is all the primes and not one. So rather than come up with another name for that list, we just throw out one from the primes.
For me , the key part is that when braking down a number into prime factors, one might be included but it isn't actually necessary. And not being necessary definitely puts it into a different category to, say, seven.
So, a prime number is a positive integer which cannot be written as a product of any other prime numbers.
Then we start building the set from 1. It isn't prime because it can be written as 1 * 1 (which would otherwise be a product of two primes) so 1 is not in the set. We then take 2. We can write it 2 * 1, but we already know that 1 is not in the set, and multiplying by any other positive integer we get a number that's larger than 2 so 2 is prime. 3 is prime, but 4 = 2 * 2, so it's not prime...
In relation to the idea of the anti-product and the existence of the over all product break downs. He fails to mention the concept of 1 being the multiplicative identity. So when he says that 2 is the product of one prime, he does not mention what the other number it is multiplied by to make it a product. The number is the multiplicative identity, 1. The multiplicative identity is simple something that you multiply something else by to get the something else. What about: 3, 5, 7, 11, ...
@setelement The definition of product does not specify a minimum number of factors. 2 is the product of one prime, so there is no 'other number by which it is multiplied to make it a product'. Think of the product more as a powers. 2 is the product of one prime is the same as saying 2 to the power of one. 1 is the product of zero primes is the same as saying [any prime] to the power of zero.
BTW as the guy said prime numbers are like atoms that can be used to make other numbers. That is why they are called prime (first) numbers for goodness sake. It is not just a "category".
is that Brady from sixtysymbols?
MrJiggerify 1 hour ago
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wait this is the first number phile video i have watched but is that brady from sixtysymbols speaking?
MrJiggerify 1 hour ago
This has been flagged as spam show
wait this is the first numberphile video i have watched but is that brady from sixtysymbols speaking?
MrJiggerify 1 hour ago
wait this is the first numberphile video i have watched but is that brady from sixtysymbols speaking?
MrJiggerify 1 hour ago
Comment removed
MrInfernow 5 hours ago
@MrInfernow if you actually pay attention to what he says at around 4:47, you can have a product of just one prime number.
slasherrs9 4 hours ago
@slasherrs9 Ah ok, I actually didn't understand there, not that I wasn't paying attention.
MrInfernow 4 hours ago
zero is a whole number, but it cannot be represented as a unique list of primes...
plki901 14 hours ago in playlist Uploaded videos
@plki901 Actually zero is not a positive integer, since you don't have a negative equivalent (you don't get -0)
thehungrynerd 12 hours ago
@thehungrynerd Except in a 1's complement binary system which maps
0000000000000000 to zero
and
1111111111111111 to minus zero
CNN911Fakes 5 hours ago
"... only de bivided by.." haha
Surfurplex 19 hours ago
Yay!
stilsonkidd 1 day ago
The pattern he was describing at the end could be expressed with exponents. As a software developer I frequently deal with the powers of 2 as used in binary.
He started with 16 (or 2 to the fourth) and reduced the exponent by one each time until he reached 1 (or 2^0). We know that evaluates to 1 but he's describing it in terms of multiplying zero 2s together or an "empty product" of 2s.
The same pattern would apply using 3s, 4s, 5s, etc.
soul9528 1 day ago
i have to appologise to my math teacher
vmariani212 1 day ago
No, if there are no numbers, the product is 0. 1 IS a prime number, that's why it didn't work.
We just don't use it as one because it would make it complicated because of it's other special properties.
Austin101123 1 day ago
I learn more about the concept of mathematics in 5 minutes with this guy than I have in 12 years of math classes.
iangarnettify 1 day ago
At the end that's why ( 2^0 ) is equal to 1 :D
jjjhhhmmmxxx26 2 days ago
They didn't prove 1 isn't a prime. they proved their theorem wrong.
xHAZARD78x 2 days ago
@xHAZARD78x Because of the properties of 1, it would prove wrong a lot of theorems. They would have to come up with new theorems, but they would be exactly the same except they would exclude 1. So it makes sense to alter the definition of 1 instead of keep it and have it change all the prime theorems. Because if it was really the same as all prime numbers, it wouldn't alter most of the definitions.
iangarnettify 1 day ago
Simplify Point 9 Repeating, please. ;D
JayThePokemonMaster 2 days ago
Explain to me: what is the possibility of seeing the time 4:23 PM whenever I look at the clock when it's past 3:37 PM?
JayThePokemonMaster 2 days ago
what about 0?
MasterLyrics20 2 days ago
@MasterLyrics20 You cannot divide by 0. At first, my logic was that you can fit 0 pieces into 0 groups, thus 1. But, you cannot divide by 0. I'm not a mathematician, so don't ask me.
experiment117 2 days ago
what the hell. you better explain that empty product thing"!
juacoredx 2 days ago
or or OR .50 x2 :P 0.50 x 2 = 1 right?
xXgAvInArNoLdXx 2 days ago
@xXgAvInArNoLdXx .5(.5) not .5(2) :)
A4N3O2N1 2 days ago
Could you also make a video on whether zero is a natural number? I'm curious as to what the consensus (if there is one) is.
themissingn 2 days ago in playlist Uploaded videos
how did I get here from ponies?
MrBronyface 2 days ago
One does not simply become a prime number.
seadranx 2 days ago
hey numberphile, can u make a heap of math magic vids coz that's my favorite kind of magic
magicturtle24 4 days ago
if every positive integer can be written as the product of primes then 1 must be prime. else how do you write 1 (a positive integer) as the product of primes?
seems your famous theorem is flawed. if 1 is prime then there's no uniqueness, and it if isn't then there's no way of writing the first positive integer as the product of primes...
WhiteHenny 4 days ago
@WhiteHenny "the product of no primes"? you're 'avin a laff
WhiteHenny 4 days ago
his face at 3:44 :D -> :| -> :D
chriskevini 5 days ago 2
so is that related to why anything^0 is 1?
pyrea17 5 days ago
Physics student here
At 3:06, I facepalmed. Mathematicians... o.0
TakesTwoToTango 5 days ago
i'm in love with numbers
SYamooraSY 5 days ago
love the enthusiasm of that guy xD. I didn't find the very last explanation very satisfying though..
MippieMan 5 days ago 8
Is he on some kind of stimulant?
blaticavenue 1 week ago
asshol ,like if this didnt help!!!
hypnoticderank 1 week ago
I'm so confused, he said 2 is the product of just 1 prime number, itself, 2. But isn't the definition of product 2 or more number multiplied by one another.
DuhuhudeHarmon 1 week ago
And the number 1 was never heard from again...
shadowblue13 1 week ago
ooh, i've always wondered this. Thanks for making it so clear now
mrbluesky6 1 week ago in playlist Uploaded videos
um are in colage or are you a relly young prof. ps im12
Teddy5775 1 week ago
Finally the notation of x^0 = 1 makes sense.
xanshriekal 1 week ago 3
Like the Beatles reference @ 1:10
spairpartsdealer 1 week ago
@spairpartsdealer That's from Three Dog Night, not the Beatles.
SomeoneCommenting 1 week ago 8
love the videos i am going to show my math teacher this vid
tcara11 1 week ago
Interesting O_o
CJFRANKS7 1 week ago
1 is prime
TheDarthvader14 1 week ago
@TheDarthvader14 nope, 1^0 =1
BlackManSlim562 1 week ago
His face is uncomfortably close to the camera.. i can almost smell his english breathe D:
ito8790 1 week ago in playlist Uploaded videos 4
@ito8790 "breath"
jayynecobb 1 week ago
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What happened to him at 03:44 ????????
abhim7 1 week ago
Comment removed
abhim7 1 week ago
Oh i thought his name was called "singingbanana"...rofl
SkinCollecter2 1 week ago in playlist More videos from numberphile
Dr Grime and all his primes!!
stephanied2604 2 weeks ago in playlist More videos from numberphile
Forgot to mention the two factors of primes.
RandomNinjaOfEvil 2 weeks ago
Did anyone else notice the spoonerism at 0:31?
RandomNinjaOfEvil 2 weeks ago
Is Zero considered an amount? as in 'I have zero amount of apples'
If so then 1 should be prime
...Wait then wouldn't that mean zero is prime...?
This is why I didn't like Maths at school.
clownstangler 2 weeks ago in playlist Uploaded videos
loneliest number
shugochara78 2 weeks ago
Two men think the're Jesus.
One of them must be wrong.
xchvcwryb 2 weeks ago
but what do you mean by "unique"? why does including 1 makes it not unique anymore?
Noovil25 2 weeks ago
@Noovil25 it means that each positive integer has one and only one product of prime numbers. so if one was a prime number there would be infinite products of prime numbers for each positive integer.
figisligit 2 weeks ago
@figisligit thanks, but why does a positive integer have to be a unique product of primes in the first place? why can't it just be the way it is, with an infinite number of primes?
Noovil25 2 weeks ago
How I wish I have more time to watch all Brady's videos, skipping a day at school should allow me to catch up?
Noovil25 2 weeks ago
A way to think of prime numbers is to say, "what numbers can lower a product to it's simplest form?" 1 would not help at all so it's not prime.
darkfire8008 2 weeks ago
I should stop watching all these before I decide to become a math major O.o
houseofcards613 2 weeks ago 2
Comment removed
xShortBusHerox 2 weeks ago
i keep on telling my teacher this... but she just isn't buying it.
RonnieSpeaksTv 2 weeks ago
The virginity rating is strong for this one...
NoahVoorhees 2 weeks ago
New level of loneliness : "1"
Forever Alone !
bharathpv9 2 weeks ago
This has been flagged as spam show
this kind of trickery can be important sometimes. For example in physics, our best current theories make predictions where the value of what is supposed to be a measurable, experimental quantity corresponds to a divergent sum. So physicists have learned to take (somewhat mathematically questionable) limits to get finite answers - and these answers agree very precisely with experiments! So sometimes you have to be willing to divide by zero. :)
rophekatv 2 weeks ago
You showed a non-product for your '2' value. There was no multiplication.
aikimark1955 2 weeks ago
@aikimark1955 Yes there was. You can multiply only one number.
anticorncob6 2 weeks ago
@anticorncob6 a multiplication operation (a 'product') requires two numbers. Since there was only one number, a "x 1" is implied as the other number.
Note: There was an interesting sleight of hand going on in the video example. The powers notation was being used, but was referred to as multiplication. These aren't exactly the same thing. As an educational video, I would have expected better.
aikimark1955 2 weeks ago
@aikimark1955 "(a 'product') requires two numbers."
No it doesn't.
Do some research, I say look up "empty product".
anticorncob6 2 weeks ago
@anticorncob6 From Wikipedia:
"It is equal to the multiplicative identity 1". If you are trying to avoid "1" as a prime, then an empty product is not the way to do it.
aikimark1955 2 weeks ago
@aikimark1955 What I was trying to show you is that it is possible to multiply only one number, if I can convince you about zero numbers it's possible with one as well. You can make your own definition of "product", but mathematicians will not agree.
anticorncob6 1 week ago
@anticorncob6 Since an empty product implies that "1" is the other value being multiplied, I would need more details from you or the author(s) of this video how an implied value (1) isn't really a value and doesn't actually use "1" as a prime number, which contradicts the assertion of numberphile.
aikimark1955 1 week ago
@aikimark1955 There is NO rule in mathematics that states a product needs two numbers, it just looks funny so you assume it's not there. Your comment is entirely circular reasoning.
anticorncob6 1 week ago
Derp...?
jojonr619 2 weeks ago
optimis prime excluded himself after he saw this video
luke53285 2 weeks ago
e^(iπ)=-1
campion1211 2 weeks ago in playlist Uploaded videos
What about -1? If you counted -1 as prime, you could go into the negatives, but I don't know what you'd do with zero, it's not prime nor can it be multiplied by primes.
Oh wait, then 15 would equal -1 * -1 * 3 * 5
:(
anticorncob6 2 weeks ago
@anticorncob6 -1 doesn't count for basically the same reason as 1 doesn't. You could decide to include it in your list of what you call prime numbers, but then your concept of prime numbers wouldn't be quite as useful in doing mathematics. Things would would get more complicated, as you yourself point out.
So it makes sense not to include 1 or -1 (or 0 or 1.5897, etc) as primes. If we did, 'prime' wouldn't be such a useful concept! :)
DoctorFastest 2 weeks ago
But 2... where is the product if the definition of product is
The term "product" refers to the result of one or more multiplications. (implies 2 number at least)
2 would need to be the product of 2 numbers, 2 and 1, else 2 would also need to be refined to be somewhat unique, because product is defined as 2 or more numbers multiplied together, Will this theory mean you will now refine what product means?
iamufreak 2 weeks ago
@iamufreak Nah, the video states this a bit too non-scientifically for such questions. In fact when you think of any positive whole number and its 'prime representation'(I'm not English, I don't know the correct terms, unfortunately), it's actually not 6 = 2*3, it's rather 6=2^1*3^1*5^0*7^0(...), therefore 2=2^1*3^0... and the definition of a product is fully applicable here. (Just on a side note - without this, there would be a problem with all of the prime numbers, not just 2).
Cheers!
Bulasz 2 weeks ago
Hello numberphile. I have always asked myself this question. I have searched it online a couple of times and never found a really satisfying answer to this problem.
"If x^0 = 1 and 0^x = 0, then why does 0^0 = 1?"
As I said, I have tried looking this up and found the weirdest answer with all sorts of highly advanced formulas that the average person cant understand. I once even read this: "because mathematicians say so!"....
Could you guys please clearify this for us?
Thank you for your time!
GiaIsTheBest 2 weeks ago
@GiaIsTheBest 0^x=0 only for positive x. 0 isn't positive so it isn't affected by this rule. You can think of 0^0 as the product of zero zeroes as x^y basically is the product of y x's. Anything of the form x^0 is the product of zero x's so basically it is the product of nothing, the empty product, which is 1 as explained in this video.
TaiFerret 2 weeks ago
@TaiFerret Well I have heard this before but I still do not get this:
I have heard that 0^x = 0 only counts for positive x's because if you fill in a negative x, you will start to divide which is impossible regarding 0.
But as far as I know, x^0 is also dividing (2^0 = 2/2). And as we all know, dividing by 0 is impossible.
Could you please explain this?
GiaIsTheBest 2 weeks ago
@GiaIsTheBest It's true that a^0 can also be thought of as a/a. So if what you are trying to say is that a^0 doesn't make sense when a=0 (or in other words that 0^0 doesn't make sense, because it's like 0/0) then you're right.
But keep in mind that 2^0 makes plenty of sense - that's like 2/2, which is 1.
DoctorFastest 2 weeks ago
@DoctorFastest I am not saying that 2/2 doesn't make sense. It totally does. But 0/0 (or 0^0) does not.. because dividing by 0 is impossible.
GiaIsTheBest 2 weeks ago
@GiaIsTheBest Well, dividing by zero is possible sometimes, just not when the numerator is finite. For example. sinx/x = 1. One difference here is that lim x^0 as x goes to zero is different from lim 0^x as x goes to zero. So you might want to say that there's no consistent way to take the limit, so we should say the limit DNE. On the other hand, lim(x^x)=1 as x goes to zero. So people usually say 0^0=1 for that reason.
This
DoctorFastest 2 weeks ago
@GiaIsTheBest Oh I was also going to say, this kind of trickery can be important sometimes. For example in physics, our best current theories make predictions where the value of what is supposed to be a measurable, experimental quantity corresponds to a divergent sum. So physicists have learned to take (somewhat mathematically questionable) limits to get finite answers - and these answers agree very precisely with experiments! So sometimes you have to be willing to divide by zero. :)
DoctorFastest 2 weeks ago
@GiaIsTheBest 0^0 = 1 is just an agreement, it's not derived from any arythmetics. However, there is a good reason for that. Let f(x)=x^x. Then f(1)=1^1=1. f(1/4)~=0.7, f(1/8)~=0.77, f(1/32)~=0.9 (...) and generally, the closer x gets to 0, the closer f(x) gets to 1. And, more to the point, you can get as close to 1 as you want. Meaning, that there is an x close to 0 that 1>f(x)>0.9999 for example. But it's never exactly 1. Therefore, if you visualize it, you draw a conclusion
Bulasz 2 weeks ago
@Bulasz "0^0 should be 1!". That's in simple terms, I can expand on this in a private message if anyone would like. Hope I helped : ) I can clarify anything, in a PM as well, if need be.
Cheers!
Bulasz 2 weeks ago
0!
Ldraconian 2 weeks ago
Finally somebody manages to explain to me why 1 is not a prime. I got marked down in a math test once because I included it in the list of primes which cost me a whole grade and my math teacher (who was a horrible person in the first place) could not explain to me why this is the case. Had a very negative effect on my future relationship with mathematics.
GodJasperX 2 weeks ago
So, 1 is the product of primes by being the product of "no" primes. And we think that is more logical just telling folks adding and extra "1x" that they are just being silly? Reminds me of the old Hitchhiker's computer game. Your inventory always included "no tea." To get tea, you had to put down "no tea." (Hope I didn't spoil that for anyone)
HaslamCorp 2 weeks ago
TERM UP THE DAMN VOLUME. i have to blast my speakers to hear you
MakiahLangan 2 weeks ago
i feel like im watching a "calmer" version of math's mansion.
yeoldpepsi 2 weeks ago
The definition I know for prime numbers is that prime numbers have exactly two (unique) devisors (A and the number itself). Which is a defenition that 1 doesn't fit.
The definition you gave is one I learned at school.
Muscleduck 2 weeks ago
The definition of prime that I learned excluded 1 automatically, let me know if you think it is correct/sound?
A prime number p has exactly 2 factors, or divisors: {1,p}.
For p=1, its set of factors is {1}. It only has one factor, not two like all primes must.
ishouldtellyou714 2 weeks ago
@ishouldtellyou714 It's pretty good, but it should be:
A prime number p has exactly 2 unique positive factors.
'Unique' just replaces the second part of your sentence, but 'positive' is crucial, since 2 has 4 divisors in total - 1, -1, 2 and -2.
Cheers : )
Bulasz 2 weeks ago
@Bulasz
gah I always forget to include 'unique' in my statements!
but I didn't even think about possible negative factors, thanks for helping me to be more specific. =)
ishouldtellyou714 2 weeks ago
Hello Numberphile!
I wanted to ask, whether you plan to go to more abstract part of mathematics, like Category Theory or Sets?
nirthfurzahad 2 weeks ago
isn't the actual definition of a prime number: "a number p for which if p divides a*b then p divides a or p divides b"?
letimics 2 weeks ago
@letimics e.g. 4 divides 16 and 4 divides 32 => 16*32=512 => 4 divides 512 too
Copernicium277 2 weeks ago in playlist Uploaded videos
@letimics There is no "actual definition", you can use whichever one actually represents all of the prime numbers and none more. Your does, except you have to add that p is positive.
Bulasz 2 weeks ago
I find it fascinating that in math you have to follow patterns to get the certain product. Without patterns, there would not be any mathematical values. Like 3 to the power of 0 is 1, because 3 to the power of 2 is 9 divide by three, 3, and divide by 3, 1 and then .333
samylvern 2 weeks ago
4:42 that's a funky ass 4
mashedbatata 2 weeks ago
This has been flagged as spam show
Note there is another reason for one not being a prime:
In general, given a commutative ring, you can define prime elements similarly (what you do is say that a prime is an element for which if p divides a.b then either p divides a or p divides b). You usually also enforce that p is also itself not a unit (i.e. has no multiplicative inverse).
This set of numbers has nicer properties e.g. the property of being prime is equivalent to the property that it generates some non-trivial prime ideal.
jamma246 2 weeks ago
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jamma246 2 weeks ago
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jamma246 2 weeks ago
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schoolmonkey13 2 weeks ago
Arghh, so much talk about dividing by zero.
I wish they'd just teach school kids to "multiply by the inverse of x" as opposed to "divide by x" if only to avoid all the annoying "divide by zero" crap.
jamma246 2 weeks ago
how can 2 by nothing be a product??? aaaahhhh
PinkCammy 2 weeks ago
"One is the loneliest number that you'll ever do."
Do you know ANBB? Because they've got a song with exactly the same words.
--
As always, great video!)
ScarfaceDeb 2 weeks ago
So is zero a prime? If not why not?
Hewpie 3 weeks ago
@Hewpie 0 divided by 0 = infinity. A prime number divided by itself must = 1.
sc0rpi0n0 3 weeks ago
@sc0rpi0n0 0/0 is not equal to infinity.
puttolaz 3 weeks ago
@puttolaz Dividing by 0 gives you an undefined answer.
Gytax0 2 weeks ago
@sc0rpi0n0 Thanks for your prompt reply. Wait a minute - isn't 0 divided by 0 equal to 1?
In that case 0 is a prime... Aargh! my head's going numb... Gaah! blbb blb bloosh! *dribble*
Hewpie 3 weeks ago
@Hewpie Infinity isn't a number and is (at least, should be) no longer used be mathematicians. Division by zero isn't defined regardless of what is divided, be it 0 or 1.
Anastius 2 weeks ago
@Hewpie Don't let @Sc0rpi0n0 confuse you; firstly, any real number divided by itself is going to equal 1, so that is not what defines it as a prime number, this is called the identity property. Secondly, 0/0 is not infinity, and though it makes cognitive sense, it can be disproved by the fact that anything divided by 0 would equal infinity. 0/0 is almost always considered undefined, with the exception of computer engineering, where, for whatever reason, 0^0 being equal to 1 makes formulae work.
TheMuffinChef 2 weeks ago
@TheMuffinChef Thanks for that explanation. That clears up the issue about 0 not being a prime number.
What a mess zero makes! So 0 divided by 0 is not a number, and not infinity or 0. Anything raised to the power of 0 is always 1, unless it's negative, which makes it -1.
Does -0 do anything?
Hewpie 2 weeks ago
@Hewpie Not really; -0 doesn't exist because 0 is the only number with the property of never being negative nor positive. Also, even negative numbers divided by themselves are 1, not -1, because as in multiplication, the negatives cancel out with division.
So, 0/0 isn't infinity because dividing by any number is supposed to be unique, and anything divided by 0 would cognitively be infinity, so the whole thing is invalidated. It's not 0, and only in certain situations is it 1.
TheMuffinChef 2 weeks ago
@Hewpie Yes.the place value of 0 helps us shortened our number.ex Roman Numeral-MMMCCCIII = 3303 so 0 actually does something and many more stuff.
TheUltimateXslayer 2 weeks ago
@TheUltimateXslayer In Egyptian hieratic writing, the number 3303 consists of only 3 characters.
TaiFerret 2 weeks ago
@Hewpie No, negative numbers to the 0 power are 1 also. Think about -1^2, -1^1, -1^0, 1/-1, ...
LokiClock 2 weeks ago
that guy looks like an idiot on 00:32
munir489 3 weeks ago
@munir489 Though he made you look like an idiot.
Gytax0 2 weeks ago
....Im gonna go watch "One" by Three Dog Night...THEYLL tell me the truth!
vallangaard 3 weeks ago
I am sorry. but he has a small head and a big mouth. it makes me smile.
andyman01869 3 weeks ago
"Product of Primes" sounds plural. Even "Product" sounds like there ought to be more than one number in the event.
HaslamCorp 3 weeks ago
One is not alone. It's out there with Pluto.
HaslamCorp 3 weeks ago 37
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ViveleRussian 3 weeks ago
one plus one equals two
Zee96969696 3 weeks ago
YES YES YES
THE VIEWERS ASK AND YE SHALL RECEIVE.
heyandy889 3 weeks ago
And to go just slightly off-topic, if you take the formula
N = 2^a·3^b·5^c·7^d·11^e· …
and let the exponents a,b,c,d,e,… go negative as well, then you get the positive Rational numbers.
R = 2^a·3^b·5^c·7^d·11^e· …
kpYak 3 weeks ago 8
Another way to write these products-of-primes is as a polynomial. Just as you may have
a·x^2 + b·x + c = 0
that's also expressible as
a·x^2 + b·x^1 + c·x^0 = 0
So in this notation,
132 = 2^2 · 3^1 · 5^0 · 7^0 · 11^1 …
So the Fundamental Theorem of Arithmetic says that there's a unique list of exponents to put on the list of prime numbers to get the product which is that number. The number (one) is the whole number you get when all the exponents are (zero).
1 = 2^0 · 3^0 · 5^0 · 7^0 · 11^0 …
kpYak 3 weeks ago
@kpYak With a number to the zeroth power x^0 (for example 2^0) you actually have x/x (like 2/2) or a division that results or have the quotient of 1, so 1 in this case is not a product and further shows that the product 2 is equal to 2*1 not 2=2, since 2/2*(2) = 1*(2) = 2. Without 1 as a prime it is impossible to have 2 as a product of primes.
The 1 that is shown at the end is *not* a product or an empty product as it was called, the 1 is a quotient.
SuperFinGuy 3 weeks ago
@SuperFinGuy Prime numbers do not need to be a product of primes.
TheMuffinChef 2 weeks ago
@TheMuffinChef yes but whole numbers do, in order to satisfy the theorem.
SuperFinGuy 2 weeks ago
Calling numbers "prime" is just a name for a set of things that fit a definition. If the list is more useful in a certain way then we can make the word "prime" mean whatever we want it to mean.
kpYak 3 weeks ago
Keep in mind that the number 1 is the multiplicative identity. Your methods seem to be ignoring that.
technicallyabsurd 3 weeks ago
What if -1 is a prime number also, just not in the realm of the natural ones?
SREproducciones 3 weeks ago
@SREproducciones Then numbers have an infinite amount of prime factorizations as
(-1)^n is 1 if n is even, sorry, I use to love the concept but it doesn't work.
anticorncob6 2 weeks ago
I disagree, this girl I met was very lonely, and she was number 26...it said so in her dress.
xSeiZmikx 3 weeks ago in playlist More videos from numberphile
I'm tempted to denounce this empty product non-sense, but if I've learned anything from particle physics, it's that you make rules based on what works, not what can be easily understood.
gregiep 3 weeks ago
1 is a victim of mathematicians, same as Pluto the victim of astronomers.
animimm 3 weeks ago
@animimm Who do you think should be determining what numbers qualify as prime and what space rocks get to be planets?
joeloud1 3 weeks ago
I was convinced by the theorem untill Brady asked if 1 was a whole number. The whole empty product thing doesn't fit into the theorem in my opinion. It kind of annoys me
razveck 3 weeks ago
We make the definitions up to suit ourselves. What it amounts to is that for number theory you repeatedly want to include a list of numbers that is all the primes and not one. So rather than come up with another name for that list, we just throw out one from the primes.
For me , the key part is that when braking down a number into prime factors, one might be included but it isn't actually necessary. And not being necessary definitely puts it into a different category to, say, seven.
chrisofnottingham 3 weeks ago
So, a prime number is a positive integer which cannot be written as a product of any other prime numbers.
Then we start building the set from 1. It isn't prime because it can be written as 1 * 1 (which would otherwise be a product of two primes) so 1 is not in the set. We then take 2. We can write it 2 * 1, but we already know that 1 is not in the set, and multiplying by any other positive integer we get a number that's larger than 2 so 2 is prime. 3 is prime, but 4 = 2 * 2, so it's not prime...
ReasonSharp 3 weeks ago
@ReasonSharp Or, a prime is a positive integer with exactly two divisors.
Anastius 3 weeks ago
@Anastius Indeed, both definitions define the same set of numbers, that is, primes.
ReasonSharp 3 weeks ago
In relation to the idea of the anti-product and the existence of the over all product break downs. He fails to mention the concept of 1 being the multiplicative identity. So when he says that 2 is the product of one prime, he does not mention what the other number it is multiplied by to make it a product. The number is the multiplicative identity, 1. The multiplicative identity is simple something that you multiply something else by to get the something else. What about: 3, 5, 7, 11, ...
setelement 3 weeks ago
@setelement The definition of product does not specify a minimum number of factors. 2 is the product of one prime, so there is no 'other number by which it is multiplied to make it a product'. Think of the product more as a powers. 2 is the product of one prime is the same as saying 2 to the power of one. 1 is the product of zero primes is the same as saying [any prime] to the power of zero.
z0tx 3 weeks ago
What about Zero?
Nebulae8 3 weeks ago
Three Dog Night was correct: "One is the loneliest number.
Tallguyification 3 weeks ago
"You can tell it's an important theorem because it has a name."
Now THAT is a reason! :D
TrelosLikos 3 weeks ago
00:53 histoicly
albinbjarhall 3 weeks ago
So we've been sending out the wrong numbers in SETI all along? No wonder our neighbours have been ignoring us...
Misterb0z 3 weeks ago
My favourite things about these videos are Brady's questions. Always asks what I'm thinking.
xenon1987 3 weeks ago
Do a video about imaginary numbers.
livedandletdie 3 weeks ago
do a video about the amazing properties of the number 64
electelectro 3 weeks ago
I'd be amazed actually if one could multiply two primes that aren't 1 and get the whole number 1.
SuperFinGuy 3 weeks ago
@SuperFinGuy i think amazed would be an understatement.
electelectro 3 weeks ago
BTW as the guy said prime numbers are like atoms that can be used to make other numbers. That is why they are called prime (first) numbers for goodness sake. It is not just a "category".
SuperFinGuy 3 weeks ago
"1 is the lonliest number you'll ever do"
The5T1N9R4Y 3 weeks ago