+Ahmed Medhat I don't have a specific video series on group theory (I will do one in the future) but I do have a series called Geometry and Groups, which explores the links between geometry and group theory.
I know that Complex Numbers are closed under all possible operations but what about dividing by zero? That is undefined ofcourse but then what about mathematical closure? Does 1/0 render mathematics not closed or is the "undefined" answer sufficient enough? Because before imaginary numbers, the sqr(-1) was also undefined, until it was defined to be the number i. Is that a hint that mathematics are not closed even in the level of complex numbers because of this 1/0 problem? Or it being undefined solves our problem forever?
on my text book it says when performing the operation on any 2 numbers in the set, will result in another number in the set. So does that mean in this situation you could also do c+a=b?
+Matthew James See Topology by Munkres, 2nd ed, Chapter 1, Section 4, Ex 5. Part (d) asks you to prove closure of the integers. A good starting point is to prove that the integers are inductive: that is, for all integers k, 1 is also an integer and so is k + 1.
i understand.. but im a bit lost with the division part. i mean, what if i say: 2/1=1. isnt that closed??? 1 is an integer, right??? so how does this work!!!
I believe Sal made a mistake when he assumed that Natural Numbers (Integers) are closed under subtraction. It's not the case that if you take two integers and subtract them from one another, the result is another integer. Take 2 and for 4. 4 - 2 = 2, but 2-4= -2 and -2 isn't an integer. Therefore, the conclusion is that Natural Numbers are only closed on ADDITION and MULTIPLICATION.
The set of natural numbers that you have mentioned is a different set, it is denoted N, and it takes only the positive integers. However, the set of integers takes all the positive and negative integers and the zero, and it is denoted Z. To clarify, the set of natural numbers is a part of the set of integers, but they are not the same thing. I hope this was helpful.
All natural numbers are integers but not all natural numbers are integers. For example -3 is not a natural number it is an integer. Integer is closed under addition, multiplication and subtraction only and Natural numbers are closed under addition and multiplication only.
This guy is amazing, he has always helped me with my math issues. God bless him.
Some of the best explanations allll over the world 🌎!!! Thanks for khan academy!.....
thank god my teacher was like “figure it out yourself” so here i am
Thanks, that helps a lot in understanding how to tackle proofs
What is a set????????
khan the savior god bless.
how do you right so good with your mouse on desktop
The Legendary Eagle drawing tablet
It could be a touch screen computer
Can you do group theory? I'm taking further maths A level and it will be really useful, thanks for the awesome videos!
+Ahmed Medhat I don't have a specific video series on group theory (I will do one in the future) but I do have a series called Geometry and Groups, which explores the links between geometry and group theory.
I'll check them out.
yaa please.!!
I know that Complex Numbers are closed under all possible operations but what about dividing by zero? That is undefined ofcourse but then what about mathematical closure? Does 1/0 render mathematics not closed or is the "undefined" answer sufficient enough?
Because before imaginary numbers, the sqr(-1) was also undefined, until it was defined to be the number i. Is that a hint that mathematics are not closed even in the level of complex numbers because of this 1/0 problem? Or it being undefined solves our problem forever?
Pls discuss with example in real life of real analysis every thing. I want to say pls discuss in broad view, ignited mind
on my text book it says when performing the operation on any 2 numbers in the set, will result in another number in the set. So does that mean in this situation you could also do c+a=b?
Yes
THANK YOU SO MUCH!!!
Thank you
where can I see the proof of closure under addition? Great video btw!
+Matthew James See Topology by Munkres, 2nd ed, Chapter 1, Section 4, Ex 5.
Part (d) asks you to prove closure of the integers. A good starting point is to prove that the integers are inductive: that is, for all integers k, 1 is also an integer and so is k + 1.
Thank u
i understand.. but im a bit lost with the division part. i mean, what if i say: 2/1=1. isnt that closed??? 1 is an integer, right??? so how does this work!!!
Why isn't this on khan academy's website?
Connor Grady idk
im in 7th grade and we have to know all about closure FML
OMG SAMEE IM DOING FOR NUMBER THEORYY
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I don't know this kind of English
1:25 leave a like on this comment if you understood👌😂
Whole numbers are not closed under substraction
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Mixed up english or what
I believe Sal made a mistake when he assumed that Natural Numbers (Integers) are closed under subtraction. It's not the case that if you take two integers and subtract them from one another, the result is another integer. Take 2 and for 4. 4 - 2 = 2, but 2-4= -2 and -2 isn't an integer. Therefore, the conclusion is that Natural Numbers are only closed on ADDITION and MULTIPLICATION.
The set of natural numbers that you have mentioned is a different set, it is denoted N, and it takes only the positive integers. However, the set of integers takes all the positive and negative integers and the zero, and it is denoted Z.
To clarify, the set of natural numbers is a part of the set of integers, but they are not the same thing.
I hope this was helpful.
Natural numbers are not integers. NN are subset of integers
Lol -2 is an integer. Integer is either positive or negative numbers.
All natural numbers are integers but not all natural numbers are integers. For example -3 is not a natural number it is an integer. Integer is closed under addition, multiplication and subtraction only and Natural numbers are closed under addition and multiplication only.
i think you got whole numbers mixed up with integers.... -2 isnt a whole number, but it is an integer!