why do you use | for division? In an earlier video you use | to shorthand "such that" with {x | P(x)}. In the context of this video it is clear that | is representing division, however if I was just looking at the formula, I might deduce a|b as a "such that" b rather than a divide by b. Why not use / a/b?
Sorry I'm late, but two different things here. 1) the | symbol only (at least that I've seen) means "such that" if it's in a set. It is used in set--builder notation and has some property or function that is true after the |. So you'd never read a|b as "a such that b is true" because it's not a set. You can also read it as {x in domain | P(x)} = "All values of x where P(x) is true" (he said that in the video I think you're referencing). So if it's not a set or in set-builder notation, it doesn't mean "such that" or "where". And I believe they don't use a/b because that's an actual fraction, not a notation. This might not seem different, but b|a is a statement (either true or false), not an expression. a/b is an expression that is equal to another number. 2) The less exciting answer - it's just how math is. You can ask why we use + or - instead of sigma or delta, and I'm sure there's a long history of when it first started and everything like that, but there's not really a real answer to why + is used specifically. It may seem obvious to use + to us now, since we're so used to it, but think of every symbol we could have used... why +? Math also likes the 'share'. Example: the inverse of cos (cos^-1) doesn't actually mean 1/cos in the way that 5^-1 means 1/5, even though they both use the exact same notation. Instead, cos^-1 is more meant to mean 'the opposite of' than it's reciprocal. Hopefully this helps
a/b means "a divided by b". if you would be programming this, "divide by" would be a function that returns numbers, while "divisable by" is typically a function that returns booleans (True or False). so the type of first function would be Number -> Number -> Number, while the second function would have Integer -> Integer -> Boolean type. so people use different operators for this.
Great video! Regarding the definition of divisibility: the d =/= 0 requirement was added to set any expression that is divided by 0 as undefined, right? Since, if d were allowed to be 0, then though no integer k exists such that, for instance, 5 = 0 * k, it would now be the case that 0 does not divide 5, instead of that 5 / 0 is not defined. Similarly, if n = d = 0, then 0 would divide 0 but the result would not be a number but rather the entire set of integers since for any integer k it is the case that 0 * k = 0.
I can't pay fees rn but I'll be your potential patreon after I get a job. For now I watch ads and click on em. 😬
It took me a while to get the last proof. When I wrote it down myself it made sense 🤣 Thanks for all your effort again 😀
why do you use | for division? In an earlier video you use | to shorthand "such that" with {x | P(x)}. In the context of this video it is clear that | is representing division, however if I was just looking at the formula, I might deduce a|b as a "such that" b rather than a divide by b. Why not use / a/b?
Sorry I'm late, but two different things here.
1) the | symbol only (at least that I've seen) means "such that" if it's in a set. It is used in set--builder notation and has some property or function that is true after the |. So you'd never read a|b as "a such that b is true" because it's not a set. You can also read it as {x in domain | P(x)} = "All values of x where P(x) is true" (he said that in the video I think you're referencing). So if it's not a set or in set-builder notation, it doesn't mean "such that" or "where". And I believe they don't use a/b because that's an actual fraction, not a notation. This might not seem different, but b|a is a statement (either true or false), not an expression. a/b is an expression that is equal to another number.
2) The less exciting answer - it's just how math is. You can ask why we use + or - instead of sigma or delta, and I'm sure there's a long history of when it first started and everything like that, but there's not really a real answer to why + is used specifically. It may seem obvious to use + to us now, since we're so used to it, but think of every symbol we could have used... why +? Math also likes the 'share'. Example: the inverse of cos (cos^-1) doesn't actually mean 1/cos in the way that 5^-1 means 1/5, even though they both use the exact same notation. Instead, cos^-1 is more meant to mean 'the opposite of' than it's reciprocal.
Hopefully this helps
Let me introduce you to p.. //listing all the possible meanings//
a/b means "a divided by b". if you would be programming this, "divide by" would be a function that returns numbers, while "divisable by" is typically a function that returns booleans (True or False). so the type of first function would be Number -> Number -> Number, while the second function would have Integer -> Integer -> Boolean type. so people use different operators for this.
Are you writing everything backwards? I’m calling the police.
He's writing normal and mirroring the video
@@SpiderManAndMore he has a ring on his left hand so i dont think the video is mirrored
@@shannonheinig8912 This sneak, has rings on both is left and right hand!
This is the funniest comment ever and serious people ruined it
I really love the way you speak,,,makes listening to you very easy😇
You're a cool teacher 👍
Great video. Really cleared all my doubts. I was having trouble understanding Elementary Number Theory by David Burton.
I’m literally going to cry, this was so helpful. I’ve been struggling to understand this in class and I have a test tomorrow
Great video! Regarding the definition of divisibility: the d =/= 0 requirement was added to set any expression that is divided by 0 as undefined, right? Since, if d were allowed to be 0, then though no integer k exists such that, for instance, 5 = 0 * k, it would now be the case that 0 does not divide 5, instead of that 5 / 0 is not defined. Similarly, if n = d = 0, then 0 would divide 0 but the result would not be a number but rather the entire set of integers since for any integer k it is the case that 0 * k = 0.
maths is easier all of a sudden
wtf